1. The Child and Family Services Act, subject to subrule (6).
2. The Children's Law Reform Act.
3. The Divorce Act (Canada).
4. The Family Law Act.
5. The Family Responsibility and Support Arrears Enforcement Act, 1996. O. Reg. 120/07, s. 1.
(2) The purpose of this rule is to promote the active management, in accordance with subrule 2 (5), of cases to which this rule applies by conferring specified family law jurisdiction on a Family Case Manager. O. Reg. 120/07, s. 1.
"Family Case Manager" means a person appointed under section 86.1 of the Courts of Justice Act by the Lieutenant Governor in Council as a case management master who is assigned to manage cases for the purposes of this rule. O. Reg. 120/07, s. 1.
(b) the exercise of those powers and the performance of those duties and functions are subject to the restrictions specified in subrules (5) and (6). O. Reg. 120/07, s. 1.
(vii) to grant summary judgment. O. Reg. 120/07, s. 1.
(b) the jurisdiction of the Family Case Manager with respect to cases under Part III of that Act is not as broad as it is with respect to cases under the other Acts to which this rule applies but is subject to such further limitations as are specified in this rule. O. Reg. 120/07, s. 1.
1. With respect to cases under an Act to which this rule applies other than the Child and Family Services Act, any power described in rule 14 other than a power described in subrule 14 (21).
2. With respect to cases under Part III of the Child and Family Services Act, any power described in rule 14 other than a power described in subrule 14 (21), (22) or (23). O. Reg. 120/07, s. 1.
1. An order under rules 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 18, 19 and 20.
2. An order for costs under rule 24 relating to a step in the case that the Family Case Manager dealt with.
3. An order under rule 25 relating to an order made by the Family Case Manager.
4. An order to change a temporary order made by the Family Case Manager.
5. An order under section 10 (Leave for blood tests and DNA tests) of the Children's Law Reform Act.
6. A temporary order for or relating to custody of or access to a child under section 21, 23, 25, 28, 29, 30, 32, 34, 39 or 40 of the Children's Law Reform Act.
7. A temporary order for custody of or access to a child under section 16 of the Divorce Act (Canada).
8. An order appointing a mediator under section 31 of the Children's Law Reform Act or section 3 (Mediation) of the Family Law Act.
9. A temporary order for or relating to support under section 33, clause 34 (1) (a), (e), (f), (g) or (h), subsection 34 (5) or section 37, 42 or 47 of the Family Law Act.
10. A temporary order for support under section 15.1 (Child support order) or 15.2 (Spousal support order) of the Divorce Act (Canada).
11. A temporary order under section 40 of the Family Law Act.
12. A temporary order dealing with property other than real property.
13. A support deduction order under section 10 (Support deduction orders to be made) of the Family Responsibility and Support Arrears Enforcement Act, 1996.
14. An order limiting or suspending a support deduction order.
15. An order under section 8 (Director to cease enforcement, termination of support obligation) of the Family Responsibility and Support Arrears Enforcement Act, 1996 that terminates a support obligation or orders repayment from a person who received support.
16. An order that is necessary and incidental to the power to make a temporary order that is within the jurisdiction of the Family Case Manager. O. Reg. 120/07, s. 1.
1. An order under subrule 3 (5) (order to lengthen or shorten time), if the motion is made on consent.
2. An order under rule 5 (where a case starts and is to be heard), if the motion is unopposed or made on consent.
3. An order under rule 6 (service of documents).
4. An order under subrule 7 (5) (adding a party), if the motion is unopposed or made on consent.
5. An order under section 39 (parties and notice) or under subsection 48 (3) (transfer of a proceeding) of the Child and Family Services Act, if the motion is unopposed or made on consent.
6. A finding that there is no person who should be presumed to be, or recognized in law as, the father of a child under section 8 of the Children's Law Reform Act.
7. An order granting an adjournment, if the motion is made on consent. O. Reg. 120/07, s. 1.
(10) An order for adjournment made under paragraph 7 of subrule (9) in a child protection case shall provide for the continuation of any temporary order made under subsection 51 (2) (custody during adjournment) of the Child and Family Services Act that applies in respect of the case being adjourned. O. Reg. 120/07, s. 1.
(11) Revoked: O. Reg. 151/08, s. 9.
(12) Subject to subrule (13), the Family Case Manager may conduct a case conference, settlement conference or trial management conference instead of a judge under rule 17. O. Reg. 120/07, s. 1.
(13) In a case under Part III of the Child and Family Services Act, the Family Case Manager shall not conduct a settlement conference without the consent of the parties and of the child's representative. O. Reg. 120/07, s. 1.
ii. the Family Case Manager shall not make a final order unless the parties consent to the order.
ii. the only temporary or final orders that the Family Case Manager may make under clause 17 (8) (b) are those described in subrule (9) of this rule.
3. A party to the conference may not request that the conference be conducted by a judge under subrule 17 (9).
4. Despite clause 17 (10) (a), a case may be scheduled for trial if the Family Case Manager conducted a settlement conference. O. Reg. 120/07, s. 1.
(b) the powers relating to garnishment orders set out in subrules 29 (5) and (19). O. Reg. 120/07, s. 1.
(16) Despite anything to the contrary in this rule, the Family Case Manager may at any time order that a matter assigned to him or her be adjourned and sent to a judge. O. Reg. 120/07, s. 1.
(17) Subrules 38 (5) to (45) apply with necessary modifications to an appeal from a temporary order of the Family Case Manager. O. Reg. 120/07, s. 1.
(18) Subrules 38 (1), (2) and (3) apply with necessary modifications to an appeal from a final order of the Family Case Manager. O. Reg. 120/07, s. 1.
(19) This rule is revoked on June 30, 2012. O. Reg. 120/07, s. 1; O. Reg. 51/10, s. 1.


Join host Melissa (@MelissaSChapman) and the #MobileLiving team at 3 pm ET (12 noon PT) on Friday, November 2 as we chat about Mobile Gaming!
Smartphones and tablets have inspired millions of people to start playing video games. Join us as we explore our fascination with mobile gaming, look at some of the latest trends and new releases, and maybe even brag about a few of our high scores!
RSVP and attend the chat for a chance to win Samsung Level On PRO Wireless Headphones or a UE Boom 2 Bluetooth Speaker!
Join us on TweetDeck or HootSuite (#MobileLiving) on Friday, November 2 between 3:00 – 4:00 pm ET.


According to JLL, CEE's real estate investment transactional volume for 2018 was in excess of €13.23 billion and recorded an 11 percent increase over 2017 (€11.92 billion). We have registered continued appetite from investors for the full range of assets across the entire region. Poland smashed its previous record volume set in 2006 (5.05 billion euro) with a spectacular €7.2 billion traded, and secured a massive regional share of 54 percent. Poland was followed by the Czech Republic (2.51 billion EUR - 19 percent share), Hungary (1.85 billion EUR - 14 percent), Romania (900 million EUR - 7 percent) and Slovakia (820 million EUR - 6 percent).
"With 13.23 billion euro in 2018, the CEE commercial real estate investment volume recorded an 11 percent increase over 2017, and for the third year running set a new record transaction volume for the region. We have seen the most significant volume increase in Poland in 2018 with €7.2 billion traded, a substantial uplift in volumes over previous years. For 2019, we expect continued, strong interest for product in CEE markets although perhaps not quite matching the spectacular levels seen last year. One of the most interesting aspects has been the growth of domestic capital in the Czech Republic and Hungary with 50 percent and 60 percent of volumes respectively coming from domestic capital sources. We expect this to continue in 2019. Our current forecast for the full year suggests that CEE regional volumes will total around 11.0 billion euro for 2019", says Mike Atwell, Head of Capital Markets Czech Republic & CEE, JLL.
"For 2019, we expect continued, strong interest for product in the CEE markets although perhaps not to quite to levels seen in 2018. Our current forecast for the full year suggests that CEE regional volumes will reach ca. €11.0 billion. In addition, we assume that in some countries the share of local investors will increase compared to those from abroad."
The Czech Republic still reports strong macro-economic performance and continues to be considered as the most stable country with the lowest investment risk rating within the CEE region.
Office, industrial as well as retail occupational markets have registered strong performance, leading to record low vacancies across the sectors, supporting the strong fundamentals of the real estate investment market. On the basis of positive macro-economic results, the Czech Republic continues to be highly popular amongst both international and domestic capital. The volume of investments remained very strong, especially in the second half of the year, reaching an annual total of € 2.51 billion. Unlike most other CEE countries, Czech domestic investors dominated the market with a 50 percent share of volumes, followed by a 40 percent share of capital coming from Germany.
Investor activity and appetite for investment product continues to be strong, however, it is limited by a lack of supply of prime assets and high price expectations of sellers. This has resulted in lower 2018 investment volumes on the Czech real estate market by approximately 29 percent compared to 2017. However, the Czech Republic confirms its role as one of the most attractive destinations in the CEE region in all sectors.
In H2 2018, we recorded further yield compression with our view on prime yields as follows: prime offices compressed to 4.50 percent, prime shopping centres remained at 4.85 percent and the industrial and logistics prime yield stands at 5.50 percent. Prime retail parks are at 6.00 percent while prime high-street assets would trade at 3.50 percent.
According to JLL, the total office investment volume recorded for H2 2018 reached €502 million, accounting for approximately 36 percent of the total investment volume. Thanks to high occupancy and strong demand, Prague keeps its key destination status. The most significant office transactions in H2 2018 were the acquisition of Trimaran and Element by Allianz Real Estate, followed by Forum Karlin being sold to Amundi Czech Republic and Florence Office Center being sold to Korean investor, Shinhan.
The most significant retail transaction of H2 2018 was the acquisition of Forum Nova Karolina by Reico. The remaining retail transactions were regional shopping centres, supermarkets and retail parks, typically acquired by Czech domestic investors.
With the largest transaction of the year being the acquisition of a CTP portfolio by Deka Immobilien worth €458 million, the industrial and logistics sector volume totalled €488 million in H2 2018.
The Polish investment market, which saw about €5.0 billion traded in 2017, continued to grow for the 4th consecutive year with an impressive 2018 year-end result of €7.2 billion traded. This volume was unprecedented and so far outperformed the previous best result for country investment volumes (2006) by almost €2.2 billion. 2018 was a unique year in Poland for many reasons. It saw the noticeable yield compression in most asset classes. There were almost 100 transactions concluded throughout the year including the biggest retail portfolio sale in the history of the Polish market (€1.0 billion), the largest number of portfolio transactions traded in a single year in the industrial sector (7) and the highest investment volumes reported in the main sectors: office (€2.75 billion), retail (€2.47 billion) and industrial (€1.84 billion).
"Since 2012, investments in commercial real estate in the CEE region have been growing at a swift pace, with an average of 40% per annum, and in 2018 we have historically recorded the highest investment volume - over €13 billion. This trend and the ever-expanding mix of investors show that we are a strategically important region. The Polish market itself attracted 20 percent of all Asian investments in 2018. We assume that the Czech Republic is still expecting this Asian wave to come", said Hana Kollmanová, Director of JLL Capital Markets.
Although the first half of 2018 started weak for Hungary, the second half of the year proved to be especially strong, reaching a level of €1.3 billion, the highest H2 volume since 2007. The strong volume was the result of four large transactions, which generated nearly 60 percent of the total volume. As a result, the most active sectors were offices (45 percent) and retail (41 percent), followed by hotels (6%), industrial (4 percent) and assets for development purposes (4 percent).
According to JLL, in 2018, the property investment volume for Romania is estimated at ca. €900 million, a value slightly below the one registered in 2017 (€963 million). However, several transactions in different stages of negotiations were postponed and they are most likely be concluded during the first half of 2019. The overall number of transactions decreased, although, the average deal size increased, standing at approximately €31 million. Bucharest accounted for over 78 percent of the total investment volume. Market volumes were dominated by office transactions (50 percent), while retail accounted for 35 percent.
Following robust investment activity in 2016 and 2017, the Slovak investment market has experienced yet another strong year resulting in a 2018 transactional volume of €820 million, much more than expected by JLL (650 million).
The sector split of income producing assets comprised €340 million in retail, €265 million in office, €170 million in industrial and €9 million in hotels. Compared to the 4-year averages, the retail sector performed 85 percent above the average level; the office sector had the highest investment volume ever achieved at 180 percent above average and the industrial sector recorded a 6 percent decline caused mainly by a lack of product and some deals remaining in the disposal process. Positive macroeconomic and market conditions, high occupancy rates, GDP growth and economic and political stability are the reasons why Slovakia is attracting more and more investors. The outlook for 2019 is therefore very optimistic.


You might dream of traveling, playing golf, or spending time with the grandchildren when you retire. You're probably not dreaming about paying on your debt. Ideally, none of us would be worried about debt in our later years.
The reality is that half of us will have debt in our retirement years.
Ensure that you're aware of all the benefits you should be receiving. There are benefits available to help with utilities, food, health care, and other items. Check with the National Council on Aging to explore all the options available to you.
Avoid being bashful. You worked all of your life and deserve these services. Every little bit helps and leaves more money for debt payments.
Consider cutting back. Like anyone else with more bills than income, it might be time to downsize your lifestyle. A smaller, less expensive house can free up a lot of money each month. Ensure your entertainment dollars aren't wasted. Make every dollar count.
Consider your retirement expenses before retiring. Consider the lifestyle you'd like to live and calculate your expenses. Most retirees are surprised by how much retirement costs. They fail to account for the cost of enjoying the time they previously spent working.
You probably don't want to sit at home all day. Going to the movies, going out for lunch, travel, golf, and shopping all cost money.
Project your expenses and determine how much money you'll have left over each month for your debt.
Work for a few more years. There are always options if you need to continue working. A few extra years can make a huge difference.Your current investments have more time to grow. You can also avoid dipping into your savings.
Refinance your home. If you have a lot of equity in your home, consider pulling it out with a refinance. You can invest that equity and use it for paying your debt or living expenses. Be wary of adjustable interest rates. Predictability may be more important for your situation.
Pay your mortgage more quickly. If you're still working, make extra payments on your home to eliminate the mortgage as soon as possible. Imagine how nice it would be to retire without a mortgage payment.
Consider a reverse mortgage. There are many rules and regulations regarding this type of mortgage. However, you can live in your home and don't have to make payments as long as you live in the home. It's important to educate yourself before considering a reverse mortgage. Consult with an expert.
Consult with professionals. If you have excessive debt and minimal income, a financial expert can be of great assistance.
Consider bankruptcy. Whether this is a viable option will depend on your state of residence and your financial situation. For example, some states will permit you to keep up to $1 million of equity in your home. Others only permit a few thousand dollars.
If you have excessive consumer debt and minimal assets, bankruptcy might be an option. Consult with a knowledgeable bankruptcy attorney before taking action.
No one likes to think about dealing with debt in their later years. However, statistics show that roughly 50% of retirees have considerable debt. Get started today and prepare for the future. If you're already retired, consider getting an expert opinion regarding your debt. There is often a solution you never considered.
Previous articleWho Else Wants to Travel Without Aches and Pain?


Many people have heard of the IRS Fresh Start program, but they don't know what it really is or how it helps them. The truth is the Fresh Start initiative created many helpful programs. These programs allow taxpayers to resolve back taxes in ways that were not available before.
When you owed back taxes prior to this initiative, there were limited ways you could resolve your debt. Now, there are multiple ways to resolve your tax issues—because of this program. So what is the IRS Fresh Start program?
What Did the IRS Fresh Start Initiative Make Possible?
Threshold for Tax Liens – The first change the program brought about is that it raised the amount of taxes a person has to owe before a lien could be placed. A federal tax lien can't be placed now unless the taxpayer owes $10,000 or more in back taxes—with a few exceptions.
Penalty Relief – The program brought about penalty relief, which is when the IRS decides to forgive your penalties if you have a "good" reason for being behind on your taxes. This can be a tremendous relief for taxpayers who owe a lot in penalties.
Installment Agreements – The IRS Fresh Start program also made it easier for taxpayers to access payment plan options. Now, installment agreements are offered to individuals and small businesses who owe less than $50,000 in back tax debt.
Now you know what the IRS Fresh Start program is all about. This initiative has brought about many benefits to taxpayers. Learn more about the different programs that may be able to help you. Get your free tax debt analysis by calling the W Tax Group at 1-877-500-4930, or fill out the form below to reach an IRS lawyer.


text:
News Irish News
Bathing spots stripped of top marks
Ed Carty
TWENTY-ONE bathing spots have been stripped of top marks for water quality because of last year's record wet summer, environment watchdogs have revealed.
https://www.independent.ie/irish-news/bathing-spots-stripped-of-top-marks-29238379.html
https://www.independent.ie/incoming/91d36/29238378.ece/AUTOCROP/h342/swimming.jpg
Four of the country's 136 popular swimming areas were given a poor grade after failing to meet basic standards - Fountainstown in Cork, Rush in north Dublin, Ballyheigue in Kerry and Clifden in Galway.
The Environmental Protection Agency (EPA) said run-off from farmland sprayed with slurry and manure or saturated ground where cows, sheep and horses graze was mostly to blame.
Peter Webster, EPA senior scientific officer, said Ireland's results were similar to the UK where rains wreaked havoc with quality at 10% of beaches.
"The fall is disappointing but I'm surprised it was not more. At times the rain was the equivalent of biblical proportions," he said.
The number of beaches and bathing places meeting stricter good standards dropped to 66.9% - 91 of 136 - compared to 83%, or 112, in 2011.
Out of 107 days in the May to September bathing season it rained for 80, the EPA said, with beaches in Cork and Kerry suffering the worst from Atlantic weather systems.
Mr Webster said it is less of a problem on Mediterranean beaches because the sun kills off pollution caused by faeces.
"Sunshine is the best disinfectant, and we barely saw it last year," he said.
Only Dun Laoghaire-Rathdown County Council, Sligo, and Leitrim County Councils had good quality at all their bathing spots - eight out of the total.
Under the EPA rules there is a 5% risk of picking up an illness or infection if you swim in water failing to meet the mandatory standards.
"But you are probably at greater risk of catching a cold on a bus on the way," Mr Webster said.
The EPA scientist said he does not expect 2013 to be as bad as last year,
Mr Webster also defended councils for closing beaches at risk of causing infections, including in Clare where bosses were heavily criticised for damaging tourism at Lahinch and Spanish Point last June.
"If you get pollutants coming into those horseshoe bays and you've a westerly breeze, there's nowhere for it to go. And you don't need an awful lot of poo in the water to breach the limits," he said.
"Imagine your average dog poo was the equivalent three years worth of bacteria, the actual number of bacteria we are looking for to meet the minimum values is one and a half minutes. Mandatory standards is 15 minutes.
"One dog poo faeces is sufficient to contaminate an Olympic sized swimming pool."
Clifden in Galway which was closed to swimmers because of long-running problems with discharges from the nearby sewage works is not likely to be up to standard until 2015.
Fountainstown, at the mouth of Cork harbour, suffered from episodic but persistent pollution while Ballyheigue failed after one very uncharacteristic sample exceeding E. coli limits. In Rush, one bad sample for E. coli was blamed on a wastewater pumping station.
Aoife Walsh Emergency services are currently responding to reports that a car has entered the River Lee in Cork.
Met Éireann issues low temperature and ice warning for every county - except Donegal
Denise Calnan and Gabija Gataveckaite Forecasters have issued a low temperature and ice warning for the majority of the country as temperatures are to plummet to minus four degrees tonight.
Aoife Walsh and Wayne O'Connor Prince William and Kate Middleton are expected to visit Ireland in March, with preparations being but in place for a trip to Dublin.
Tributes for student killed after men gate-crashed house party Irish News
Emergency services at River Lee following reports of car entering water Irish News
Election 2020: The 10 towns that will decide who wins Election 2020
Cash-strapped FAI to lose out on vital income as Aviva unavailable for friendlies Irish News
Who wears the pants? A belated leap for Aer Lingus
The last time Aer Lingus launched new uniforms, in 1998, Google had just begun and...

redpajama_set_name: RedPajamaCommonCrawl


Amish handcrafted primitive pine Colonial Buffet made by Vintage Creations By Sam. This item is handcrafted, unique variations should be expected. Pictured in Antique 2-Tone Finish, Buttercream with Heritage Stain.


text:
\section{Introduction}
\label{sec:Introduction}
Cylindrical floating bodies are ubiquitous in ocean engineering. In part, this is due to the fact that the most accessible cases in the linear theory of wave-structure interaction conform to the geometries for which the field equation -- Laplace's equation -- is separable. In three spatial dimensions, the most straightforward such geometry is the circular cylinder.
Due to the accessibility, and wealth of prior studies (a brief review of which is given later in the Introduction), the heaving truncated cylinder is an exceedingly popular model for a wave energy converter (WEC), able to absorb energy from waves independent of direction.  Due to this very ubiquity, we feel the design parameters -- in this case limited to cylinder size and damping -- deserve a separate, careful investigation. Furthermore, we aim to extend prior work by considering all three modes of motion available to an axisymmetric body.

The intention of this study is to present a rather comprehensive account of the hydrodynamics of a WEC based on relative motion of two cylindrical bodies, which are allowed to move in heave, as well as sway and roll. This entails an investigation of different WEC sizes and damping coefficients and their performance in wind-generated sea surface waves given by a Pierson-Moskowitz (PM) spectrum. The spectral description of the sea surface allows one to derive values for the \emph{significant displacements} of the WEC, and thus add a measure of survivability to the design considerations. Finally, an illustrative grading system can be devised to categorize the various performance metrics of the self-reacting WECs.

Following the work of Falc\~ao \cite{Falcao2010} we classify WECs into three different types based on different working principles: oscillating water columns, overtopping devices, and oscillating body devices. For the most part, offshore devices fall into the last category. Furthermore, since energy production by oscillating bodies is governed by the wave-induced motion, the dimensions of such WECs are constrained significantly by the incident wavelengths, and will likely be roughly similar under given conditions. Subsequently, oscillating body devices may be further subdivided into those reacting against a fixed frame of reference (either the sea bed or a bottom fixed structure), and those reacting against one or more floating bodies, called \emph{wave-activated bodies} or \emph{self-reacting devices.} These last devices may be installed in deep water, where the large distance between the sea-bed and the surface might otherwise be prohibitive. The mooring system for such devices has the sole role of counteracting drift and current forces, allowing the weight of moorings and anchors to be relatively small (see \cite{Cerveira2013} and references therein).

We consider vertically floating cylinders, for which a rich literature exists. The radiation problem in heave only was addressed by Ursell in 1949 \cite{Ursell1949}, and the scattering problem by Dean and Ursell \cite{Ursell1959}.  Miles and Gilbert \cite{Miles1968} later employed a variational approximation to provide the far field potential for scattering by a circular dock, along with the lateral forces on the dock. However, their results were subsequently found to contain several inaccuracies, in particular in their calculations of the radiation forces. This prompted Garrett\cite{Garrett1971} to take up the problem afresh, and establish the scattering forces for a circular dock. Subsequently, Black et al.\cite{Black1971} revisited the application of variational methods to the radiation and scattering problem by several cylindrical geometries, employing Haskind's theorem to give the wave forces.
This latter, variational approach did not yield the added mass and damping coefficients. Hence in 1981, Yeung\cite{Yeung1981} studied the radiation problem of a vertical cylinder floating on the water surface and undergoing the combined motions of heave, sway and roll, and obtained these hydrodynamic coefficients. More recently, Bhatta\cite{Bhatta2007} also gave the added mass and damping coefficient of a vertical cylinder undergoing heave motion, in terms of the two dimensionless ratios characterizing the problem (depth to radius and draft to radius).
While prior work had focused on the finite depth case, in 2013, Finnegan et al \cite{Finnegan2013} treated by means of an analytical approximation due to Leppington the forces on a truncated vertical cylinder in water of infinite depth.

In the context of wave energy, the consideration of floating cylinders as models of WECs goes back at least to Berggren \& Johansson \cite{Berggren1992}, who  approximated a device described by Hagerman by two floating, axisymmetric cylinders oscillating in heave, albeit without any considerations of captured power. More recently, Garnaud and Mei \cite{Garnaud2009} revisited the single buoy with the intention of studying it in densely packed arrays, giving the captured power for buoys hanging from a large frame. This floating point absorber was also employed, e.g.\ by Child and Venugopal \cite{Child2010} in their discussion of optimization of WEC arrays, or by Borgarino et al \cite{Borgarino2012} as a generic model to investigate wave interaction effects. Similarly, Teillant et al \cite{Teillant2012} employ an axisymmetric, heaving 2-body device for their study of WEC economics, without detailed hydrodynamic considerations. Meanwhile, a slightly different construction of the floating body was considered by Engstr{\"{o}}m et al \cite{Engstrom2009}, who added a sphere under the floating cylinder.  This two-body configuration of floating cylinder and submerged sphere was then assumed connected to the sea bed by a generator, and its performance analyzed. Zheng et al \cite{Zheng2005}, in a generalization of Berggren \& Johansson to three modes of motion, considered the hydrodynamics of two unconnected, coaxial floating cylinders, again without considering power capture. The power capture for a heave-only two-cylinder WEC was recently obtained for attacking monochromatic incident waves by Wu et al \cite{Wu2014}, albeit with a rather terse discussion of their results.

We combine features of several previous authors, and consider the novel case of two floating cylinders with an idealized power take--off (PTO), represented by a linear damper of constant characteristics, whose optimization is part of the design procedure.\footnote{While studies on PTO control show a promising potential for enhancing performance, particularly for devices with a narrow-banded natural response, practical and robust applications must still be developed (see Hong et al \cite{Hong2014}).}. We undertake our parametric study with an eye towards applications, and thus consider irregular waves in the form of a PM spectrum (see e.g.\ recent work on optimizing a floating box-barge under irregular waves by \ B\'{o}dai \& Srinil \cite{Bodai2015}) While scatter diagrams may be available for some sites where an assessment of the wave resource has been carried out, where this is not the case estimates based on wind speed will need to be made. To this end, we present our data nondimensionalized on the basis of wind speed, which uniquely determines the PM spectrum. Values of significant wave height and peak period may be readily derived therefrom, and the data recast in these terms if desired.

The paper is organized as follows: in Section \ref{sec:physical preliminaries} we present the physical set-up of the problem. This consists in presenting the twin cylinder WEC and characterizing its geometry, and subsequently presenting the PM spectra for design and survivability considerations. In Section \ref{sec:governing equations} we present, very briefly, the basic mathematical formulation of the governing equations and sketch the solution procedure. Subsequently, we employ the hydrodynamic coefficients and forces found from solving the equations of Section \ref{sec:governing equations} to characterizing WEC design under monochromatic waves in Section \ref{sec:design of the wec}, and under irregular waves given by a Pierson-Moskowitz spectrum in Section \ref{section: irregular design}. A discussion of these results with a view to applications is given in Section \ref{section:discussion}, which is subdivided into Section \ref{subsec:power capture} on power capture, Section \ref{subsec: survivability} on survivability, and Section \ref{subsec:grading} which presents a synthesis of the preceding sections. Finally, Section \ref{sec:conclusions} presents some concluding remarks and perspectives.

\section{Physical preliminaries}
\label{sec:physical preliminaries}
\subsection{Geometry}

The geometry and basic parameters of the twin-cylinder WEC are depicted in Fig \ref{fig:geometry}. The $Oxy$ plane is the still water surface and the $z$-axis points upwards. ($r, \theta$) are polar coordinates in the horizontal plane, such that $x = r\cos \theta$ and $y = r \sin \theta.$ The upper cylinder floats on the water surface with a draft $H_1$. To provide for flotational stability, it is important to note that the mass of this cylinder is not uniformly distributed, but is divided into two parts with drafts $l_1$ and $l_2$ and densities $\rho_1$ and $\rho_2$, respectively. The lower cylinder is entirely submerged with a draft $H_3$, and like the upper cylinder is assumed divided into two parts with densities $\rho_3$ and $\rho_4$ and drafts $l_3$ and $l_4$, respectively. The distance between the two cylinders is $H_2$. Both of them have the same radius $R$, and the water depth $h$ is taken to be very large compared to the attacking wave length, with the intention of approximating deep-water conditions.

As shown in Fig.\ref{fig:geometry}, the two cylinders are connected by a continuously distributed dashpot, which connects the upper edge of the lower cylinder with the lower edge of the upper cylinder at $r=R$. The integrated dashpot coefficient is $C$, which results in a dashpot coefficient per length  $\frac{C}{2\pi R}$. The dashpot is considered to represent a PTO, which generates energy from both the relative heave and roll motion of the cylinders.\footnote{Due to the small effect of sway motions, it is not necessary to consider power take-off in the sway mode for this device geometry.}

\begin{figure}[h!]
\begin{center}
\begin{lpic}[l(0mm),r(0mm),t(2mm),b(1mm)]{TWINCYLINDERS_NEW(0.2)}
\lbl[r]{0,110;$H_3$}
\lbl[r]{0,230;$H_1$}
\lbl[r]{70,165;$H_2$}
\lbl[r]{70,212;$l_2$}
\lbl[r]{70,240;$l_1$}
\lbl[r]{70,115;$l_3$}
\lbl[r]{70,90;$l_4$}
\lbl[r]{210,220;$\rho_1$}
\lbl[r]{210,190;$\rho_2$}
\lbl[r]{210,100;$\rho_3$}
\lbl[r]{210,75;$\rho_4$}
\lbl[r]{355,145;$h$}
\lbl[r]{205,315;$z$}
\lbl[r]{315,270;$x$}
\lbl[r]{270,285;$y$}
\lbl[r]{300,160;$C$}
\end{lpic}
\end{center}
\caption{Schematic depiction of the WEC geometry.}
\label{fig:geometry}
\end{figure}

Since the two cylinders are axisymmetric, only these three modes are studied. The heave and sway motions will give rise to relative motions in $z$ and $x$ directions, respectively. For waves propagating in the $x-$direction, the two cylinders roll around the $y$-axis in the mean free surface ($z=0$), yielding a relative angle about this axis.

This formulation of the problem leaves us with thirteen parameters ($\{H_i \mid i\in \{1,2,3\}\}, \{l_j,\rho_j \mid j \in \{1,2,3,4\} \}, R,\text{ and } C$) characterizing the WEC. Before proceeding, we will make several restrictions to ensure that the problem remains manageable; nevertheless, we shall see that a wealth of interesting phenomena and properties of the WEC are still accounted for.

For simplicity, we will take the drafts and distance between the cylinders identical to their radius, and denote the single size parameter by $q$, i.e.
\begin{equation} \label{eq: cylinder sizes}
H_1=H_2=H_3=R \equiv q .
\end{equation}
For the density distribution of the cylinders, we shall assume
\begin{equation}
\rho_1=\rho_3=\frac{3}{4}\rho, \;\; \rho_2=\rho_4=\frac{3}{2}\rho,\;\; l_1=l_3=2l_2=2l_4=\frac{2}{3}q,\label{casestudy}
\end{equation}
where $\rho$ is the density of the water.
Thus, the design problem  is reduced to two parameters, a size $q$ and dashpot coefficient $C$, whose interplay with incoming waves of certain frequencies is the issue at hand. We shall see that suspending the lower cylinder at a depth $2q$ below the still water surface has the desired effect of rendering its motion rather small, and thus creating a relatively stable point for the upper cylinder to react against.

There are several reasonable criteria which may govern the design of a WEC. Evidently, the WEC should capture as much of the incoming wave energy as possible. At the same time, as economic viability is the prime driver behind wave energy technology, the costs -- which may be assumed to scale with device size -- should be kept low; in practical terms, this means that device size should be kept small. Competing with this are concerns over the survivability of the converter, which dictate that displacements of the WEC not be too large under severe conditions, favoring larger devices. We shall return to these issues in detail in later sections.

\subsection{The Pierson Moskowitz spectrum}
\label{subsect:the pm spectrum}
One of the most common descriptions of a sea-state for engineering purposes is the unidirectional Pierson Moskowitz (PM) spectrum, here given as a function of wavenumber $k:$
\begin{equation}
S(k)=\frac{0.00405}{k^3}\exp{\left\{-0.55411\frac{g^2}{U^4 k^2}\right\}},\label{pm2}
\end{equation}
where $U$ is the mean wind speed at a height of $10$ m above the mean surface level, and $g$ is the gravitational acceleration. This empirically derived formula gives the energy distribution for wind waves in deep water, and differs from the JONSWAP spectrum only by the addition of a spectral--peak enhancement factor.

This spectrum (\ref{pm2}) readily yields a number of important values associated with the sea-state:
\begin{align}
& H^{(1/3)}={0.24181 U^2}/{g},  \label{eq:PM to Hs}\\ \label{eq:PM to peak wavenum}
& k_p={0.66570 g}/{U^2},
\end{align}
where $ H^{(1/3)}$ is the significant wave height and $k_p$ is the wave number of the spectral peak for a given wind speed $U$. This makes it easy to present subsequent results in an alternative form when desired.
A monochromatic wave with wavenumber $k_p$ and the same wave energy density as the PM spectrum will have an amplitude
\begin{equation} \label{eq:monochrom ampl}
a_0(k_p) = {0.08549 U^2}/{g}.
\end{equation}

For subsequent illustration it will be necessary to have some concrete, physical examples, which means specifying a sea-state via a wind speed value $U.$ Our design conditions (denoted by a subscript $d$) will correspond to a wind speed $U_d = 10$ m/s, while we will consider two ``severe states" (denoted by subscripts $s1$ and $s2$) with regard to the survivability, corresponding to wind speeds $U_{s1} = 15$ m/s and $U_{s2} = 20$ m/s. These are summarized in Table \ref{SwSs}.

\renewcommand{\arraystretch}{1.2}
\begin{table}[!htb]
\caption{The wind speed $U,$ significant wave height $H^{(1/3)},$ peak wavenumber $k_p,$ and peak wavelength $\lambda_p = 2 \pi/k_p$ associated with PM spectra used for design and survivability considerations.}
\centering
\begin{tabular}{@{} l c c c @{}}
  \toprule
  &$S_d$&$S_{s1}$&$S_{s2}$\\
  \cline{2-4}

  $U \text{ (m/s)}$	&10&15&20\\
$H^{(1/3)}\text{ (m)}$&2.47& 5.55&9.87 \\
$k_p\text{ (1/m) }$	&0.065 & 0.029 &0.016	  \\
$\lambda_p\text{ (m)}$	&96.30 & 216.67 &385.19	  \\
\bottomrule
  \end{tabular}
\label{SwSs}
\end{table}

\section{Governing equations}
\label{sec:governing equations}

Our approach to solving the wave-structure problem for the twin-cylinder WEC relies on domain decomposition and eigenfunction expansion methods (in the context of floating cylinders, see Black \& Mei \cite{Black1970}, who give a comprehensive description of the method, or more generally, Linton \& McIver \cite{Linton2001}, or Zheng et al \cite{Zheng2005} for a recent application to floating cylinders).  As the full formulation is rather lengthy, we only indicate the most important equations, and refer the interested reader to work cited above.

The fluid is assumed to be incompressible and inviscid, and the flow irrotational. Introducing a velocity potential $\Phi(r,\theta,z,t),$ and assuming periodic motion of frequency $\omega,$ the potential is separated into the spatial and temporal parts,
\begin{equation}
\Phi(r,\theta,z,t)=\phi(r,\theta,z)e^{i\omega t},\label{sep}
\end{equation}
where $\phi(r,\theta,z)$ satisfies the Laplace equation:
\begin{equation}\label{ge:total}
\phi_{rr}+\frac{1}{r}\phi_{r}+\frac{1}{r^2}\phi_{\theta\theta}+\phi_{zz}=0,
\end{equation}
subject to the linearized boundary conditions on the free surface  $z = 0$ and on the bed $z = -h$:
\begin{align}\label{bc:total}
& \phi_z-\sigma\phi = 0, \text{ on } z=0,\;\;r>R, \\
& \phi_z = 0, \text{ on } z=-h, \label{bc:total:2}
\end{align}
where $\sigma=\omega^2/g.$

At the interface between structure and fluid, the normal velocity of the structure must equal that of the adjacent fluid particles, written in terms of the potential \eqref{sep}:
\begin{equation} \label{eq:body boundary condition - general}
\frac{\partial \Phi}{\partial n} = V_n,
\end{equation}
where $V_n$ is the component of the structure's velocity in the direction of the outward pointing normal vector $n,$ which may be applied at the equilibrium surface under the assumptions of linearity. Owing to this very linearity, we continue with a decomposition of the problem into two parts: one due to the waves ($\phi_S$) scattered from the structure (which is assumed fixed) by the incident wave field, and one due to the waves ($\phi_R$) radiated by the motion of the structure, such that $\phi = \phi_S + \phi_R.$ $\phi_S$ is decomposed further into the potential due to the incident wave $\phi_I$ and that due to the waves diffracted from the fixed structure $\phi_D,$ where
\begin{equation}
\frac{\partial \phi_D}{\partial n} = -\frac{\partial \phi_I}{\partial n} \text{ on the body surface } S.
\end{equation}
The remaining radiated part of the potential $\phi_R$ must then satisfy \eqref{eq:body boundary condition - general}, where the normal velocities are to be determined from the equations of motion. We shall consider an incident monochromatic wave with amplitude $a_0,$ so that $\phi_I$ is known \emph{a priori.}

Introducing the as-yet unknown displacements of the upper ($j=1$) and lower ($j=2$) cylinder for the three modes of motion
\begin{eqnarray}
\zeta_{zj}&=&\zeta_{zj0}e^{i\omega t}\;\;\text{for heave},\\
\zeta_{xj}&=&\zeta_{xj0}e^{i\omega t}\;\;\text{for sway},\\
\theta_{j}&=&\theta_{j0}e^{i\omega t}\;\;\text{for roll},
\end{eqnarray}
where $\zeta_{zj0}$, $\zeta_{xj0}$ and $\theta_{j0}$ are the complex amplitudes of the corresponding displacements, we can write the boundary condition \eqref{eq:body boundary condition - general} for the spatial part of the total potential $\phi$ in the following form
\begin{align}
& \phi_z = i\omega\zeta_{z10} - i\omega \theta_{10} r\cos\theta, \quad &\text{ on } z=-H_1, \, r<R,\label{bc:total:3} \\
& \phi_z = i\omega\zeta_{z20} -i\omega \theta_{20}r\cos\theta, \quad &\text{ on } z=-(H_1+H_2), \, z=-(H_1+H_2+H_3), \, r<R \label{bc:total:4} \\
& \phi_r = i\omega \zeta_{x10}\cos\theta -i\omega \theta_{10}(z_0-z)\cos\theta, \quad &\text{ on } -H_1<z<0, \, r=R,\label{bc:total:5} \\
& \phi_r = i\omega \zeta_{x20}\cos\theta -i\omega \theta_{20}(z_0-z)\cos\theta, \quad &\text{ on } -(H_1+H_2+H_3)<z<-(H_1+H_2), \, r=R,\label{bc:total:6}
\end{align}
where \eqref{bc:total:3} is posed on the bottom of the upper cylinder, \eqref{bc:total:4} on the top and bottom of the lower cylinder, \eqref{bc:total:5} on the sides of the upper cylinder, and \eqref{bc:total:6} on the sides of the lower cylinder.
These conditions are supplemented by Sommerfeld's radiation condition, requiring the diffracted and radiated waves to be outgoing as $r \rightarrow \infty.$
\begin{figure}[h!]
\begin{center}
\begin{lpic}[l(0mm),r(0mm),t(2mm),b(1mm)]{Subdomains_NEW(0.2)}
\lbl[c]{200,215;$j=1$}
\lbl[c]{200,100;$j=2$}
\lbl[r]{355,145;$h$}
\lbl[r]{205,315;$z$}
\lbl[r]{315,270;$x$}
\lbl[r]{270,285;$y$}
\lbl[Wc]{205,160;$\textbf{II}$}
\lbl[c]{205,45;$\textbf{III}$}
\lbl[c]{100,160;$\textbf{I}$}
\lbl[c]{310,160;$\textbf{I}$}
\end{lpic}
\end{center}
\caption{Domain decomposition for the twin-cylinder problem.}
\label{fig:subdomains}
\end{figure}
Due to the configuration of two axisymmetric floating cylinders, we must consider three fluid regions, one between the coaxial cylinders (region II), one between the lower cylinder and the bed (region III), and one outside the vertical extension of the cylinders (region I), as depicted in Figure \ref{fig:subdomains}. Subsequently the scattering problem is divided into three problems, one in each subdomain, and the radiation problem for each of the three modes of each of the two cylinders is divided into three problems. The reader may appreciate the effort involved in keeping track of, solving, and subsequently matching solutions, of 21 problems for the potentials involved. These potentials are then applied in calculating the forces on the two cylinders, in the form of pressures from the surrounding fluid.

The full expressions for the exciting, hydrodynamic, and hydrostatic forces are lengthy and will not be given. We note only that we have found excellent agreement between our results and published work  \cite{Garrett1971,Yeung1981,Berggren1992,Zheng2005,Bhatta2007,Garnaud2010,Bhatta2011}.

The forces due to the fluid, as well as the forces due to the dampers
are employed with Newton's second law to yield the body motions. The first two \eqref{eq:mo1}-\eqref{eq:mo2} equate the vertical (heave) forces with the masses and accelerations of the upper and lower cylinder, respectively. The next \eqref{eq:mo3}-\eqref{eq:mo4} are those for the horizontal (sway) forces. The final pair \eqref{eq:mo5}-\eqref{eq:mo6} equate the torques about the $y-$axis to the angular acceleration times moment of inertia of the upper and lower cylinder, respectively.
\begin{align}
  &F_{z1}+F_{z1\rightarrow z1}+F_{z2\rightarrow z1} +F_{hs,z1}+F_{d,z1} =-\omega^2\zeta_{z10}M_1, \label{eq:mo1}\\
  &F_{z2}+F_{z1\rightarrow z2}+F_{z2\rightarrow z2} +F_{hs,z2}+F_{d,z2} =-\omega^2\zeta_{z20}M_2, \label{eq:mo2}\\
  &F_{x1}+F_{x1\rightarrow x1}+F_{x2\rightarrow x1}+F_{y1\rightarrow x1}+F_{y2\rightarrow x1} =-\omega^2\zeta_{x10}M_1, \label{eq:mo3}\\
  &F_{x2}+F_{x1\rightarrow x2}+F_{x2\rightarrow x2}+F_{y1\rightarrow x2}+F_{y2\rightarrow x2} =-\omega^2\zeta_{x20}M_2, \label{eq:mo4}\\
  &F_{y1}+F_{x1\rightarrow y1}+F_{x2\rightarrow y1}+F_{y1\rightarrow y1}+F_{y2\rightarrow y1} +F_{hs,y1}+F_{d,y1} =-\omega^2\theta_{10}I_1, \label{eq:mo5}\\
  &F_{y2}+F_{x1\rightarrow y2}+F_{x2\rightarrow y2}+F_{y1\rightarrow y2}+F_{y2\rightarrow y2} +F_{hs,y2}+F_{d,y2} =-\omega^2\theta_{20}I_2, \label{eq:mo6}
\end{align}
The terms appearing in the above equations have the following interpretation:
\vspace{1em}
\begin{table}[h!]
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{} l m{9cm} @{}}
$F_{xi}, \, F_{yi}, \, F_{zi}, \, i \in \{1,2\}$  &  {Exciting forces/torques on cylinder $i$ in the $x,y,$ and $z$ directions}  \\
$F_{\alpha i \rightarrow \beta j}, i,j \in \{1,2\}, \, \alpha,\beta \in \{x,y,z\}$  &  Hydrodynamic force/torque of the $\alpha$ motion of cylinder $i$  in the $\beta$ direction of cylinder $j.$ \\
$F_{hs,yi}, \, F_{hs,zi}, \, i \in \{1,2\}$  &  Hydrostatic forces in the $y$ and $z$ direction on cylinder $i.$ \\
$F_{d,yi}, \, F_{d,zi}, \, i \in \{1,2\}$  &  Damping forces in the $y$ and $z$ direction on cylinder $i.$  \\
$\zeta_{xi0}, \, \zeta_{zi0}, \, \theta_{i0}, \, i \in \{1,2\}$  & Displacement amplitudes of cylinder $i$ in sway ($\zeta_x$), heave ($\zeta_z$), and roll ($\theta$). \\
$M_i, \, i \in \{1,2\}$  & Mass of cylinder $i.$ \\
$I_i, \, i \in \{1,2\}$  & Moment of inertia of cylinder $i$ about the $y$-axis.
\end{tabular}
\end{table}
\vspace{1em}

\noindent The masses and moments of inertia have the explicit form (see \eqref{eq: cylinder sizes}, \eqref{casestudy})
\begin{align*}
& M_1=M_2 = \rho\pi q^3, \\
& I_1=\frac{73}{108} \rho\pi q^5 , \\
& I_2=\frac{757}{108} \rho\pi q^5 ,
\end{align*}
and the damping forces are
\begin{align*}
&F_{d,z1}= - i\omega C(\zeta_{z10}-\zeta_{z20})e^{i\omega t}, \\
&F_{d,z2}=  i\omega C(\zeta_{z10}-\zeta_{z20})e^{i\omega t}, \\
&F_{d,y1}=- \frac{1}{2}i\omega C R^2(\theta_{10}-\theta_{20})e^{i\omega t}, \\
&F_{d,y2}= \frac{1}{2}i\omega C R^2(\theta_{10}-\theta_{20})e^{i\omega t}.
\end{align*}
After the displacements of the cylinders are obtained, the captured power can then be calculated as follows:
\begin{equation}
P_a=\frac{1}{2} C \omega^2 (\zeta_{z10}-\zeta_{z20})(\zeta_{z10}^{*}-\zeta_{z20}^{*})
+\frac{1}{4} C \omega^2 R^2 (\theta_{10}-\theta_{20})(\theta_{10}^{*}-\theta_{20}^{*}),\label{Pa}
\end{equation}
where $\zeta_{zj0}^{*}$ and $\theta_{j0}^{*}$ are the complex conjugates of $\zeta_{zj0}$ and $\theta_{j0}$, respectively.

\section{Design of the WEC for monochromatic waves}
\label{sec:design of the wec}

We now undertake to examine the design of the WEC, based on the three parameters characterizing the environmental conditions $\rho, \, g,$ and $U,$ the two WEC parameters $q$ and $C,$ and the seven WEC performance parameters calculated from the wave-structure interaction problem $P_a, \, \zeta_{x10}, \, \zeta_{x20}, \, \zeta_{z10}, \, \zeta_{z20}, \, \theta_{10}$ and $\theta_{20}.$  An application of Buckingham's $\pi$ theorem \cite{Crowe2001} yields that there will be nine dimensionless quantities that characterize this problem: $\frac{q}{U^2/g}$, $\frac{C}{\rho U^5/g^2}$, $\frac{P_a}{\rho U^7/g^2}$, $\frac{\zeta_{zj0}}{U^2/g}$, $\frac{\zeta_{xj0}}{U^2/g}$, and $\theta_{j0}.$ In the sequel, we will make use of a $\sim$ to denote nondimensional variables, i.e., the nine dimensionless quantitites above will be $\tilde{q}, \, \tilde{C}, \tilde{P_a}, \tilde{\zeta}_{zj0}, \tilde{\zeta}_{xj0},$ and $\tilde{\theta}_{j0}.$

\subsection{The WEC in heave motion under a monochromatic wave}
\label{subsec:heave only}
For simplicity of presentation and ease of understanding we initially consider only the heave mode, motivated by the fact that, while sway and roll are generally coupled, they are both independent of heave. The response of the WEC under incoming monochromatic waves is first considered, where our physical test-case corresponds to a monochromatic wave of wavelength $96.3$ m equal to that at the peak of the design spectrum $S_d,$ and an amplitude $a_d = 0.87 \text{ m,}$ such that the total energy density of the wave is equal to that of $S_d,$ see \eqref{eq:monochrom ampl} and Table \ref{SwSs}.

\subsubsection{Step 1: determination of the WEC's size}
We first choose the dashpot coefficient $C$ to be zero, which means that the two cylinders are freely floating.
In this case, once the incident monochromatic wave is given, the only WEC parameter to be determined is $q$. Dimensional analysis can then be applied to the problem of determining the quantity of interest $q$ for the motions of the upper cylinder $\zeta_{z10}$ and the lower cylinder $\zeta_{z20}$ separately. Once again, Buckingham's $\pi$ theorem yields that, for the variables $\rho,g, U,q,$ and $\zeta_{zj0},$ there exist exactly two nondimensional quantities, which must be related by a relation
\begin{equation} \label{eq:heave-only disp max}
\tilde{\zeta}_{zj0}=\Psi_{1j}(\tilde{q}).
\end{equation}
The maximum displacement of the cylinder $j$ as a function of size $\tilde{q}$ thus corresponds to the extrema of $\Psi_1.$
Equation \eqref{eq:heave-only disp max} is plotted in Figure \ref{fig:free-heave-z} for the upper and lower cylinders.
\begin{figure}[H]
    \centerline
    {\includegraphics[scale=0.35,angle=270]{freely-heave-z-nondim-U10.pdf}}
    \caption{Displacement amplitudes for each of the two freely floating cylinders ($C=0$) under the design monochromatic wave. $\tilde{\zeta}_{z10}$: upper cylinder, thick line; $\tilde{\zeta}_{z20}$: lower cylinder, thin line.}
  \label{fig:free-heave-z}
\end{figure}
As we are ultimately interested in relative displacements of the cylinders, the global maximum of $\Psi_1(\tilde{\zeta}_{z10})$ and the local maximum of $\Psi_1(\tilde{\zeta}_{z20} )$ which occur at $\tilde{q}=0.97$ yield the chosen design size.
\subsubsection{Step 2: determination of the dashpot coefficient $C$}
The maximum displacement in Fig.\ \ref{fig:free-heave-z} is related to the resonance between the cylinders and the incident monochromatic wave. The introduction of a damper, while changing the magnitude of the displacement, can be shown to have no effect on the location of the resonant maximum, which remains $\tilde{q}=0.97$ (see Fig.\ \ref{fig:free-heave-z}) even for various values of $C$. Thus, the size of the WEC determined from the freely floating case is used to specify the dashpot coefficient $C$.

Given the unique relationship between $q$ and $\zeta_{zj0},$ independent of $C,$ described above, the dimensional analysis of the power absorption involves the quantities $\rho, g, U, C, q$ and $P_a,$ which can be written in three dimensionless ratios:
\begin{equation}
\tilde{P}_a=\Psi_2(\tilde{C};\tilde{q}),
\end{equation}
where $\Psi_2$ is plotted in Fig.\ \ref{fig:pald-heave} for the WEC size as determined in the last section ($\tilde{q}=0.97$).

\begin{figure}[H]
    \centerline
    {\includegraphics[scale=0.35,angle=270]{pald-heave-nondim-U10.pdf}}
    \caption{The relationship between the captured power $\tilde{P}_a$ and the dashpot coefficient $\tilde{C}$ for the heave motion induced by the design monochromatic wave, where $\tilde{q}=0.97$.}
  \label{fig:pald-heave}
\end{figure}

We elect to determine the dashpot coefficient $C$ from the maximum of captured power $P_a$ in Figure \ref{fig:pald-heave}, calculated from the heave terms only in (\ref{Pa}). This results in $ \tilde{C}=0.32$ and $ \tilde{P}_a=0.0034$.

Thus, the WEC design for a monochromatic wave has been determined. Taking the design wave introduced in the beginning of Section \ref{sec:design of the wec} as a physical example, the WEC has the dimensions  $q=9.9 m$ and $C=3.3\times10^5 Ns/m$, and can capture $P_a=3.5\times10^5 \text{ Watt}$ from a monochromatic wave 96.3 m long and 0.87 m in amplitude.

\subsection{General motions of the WEC in monochromatic waves}

Having treated the simpler case of heave-only motion, we now consider the general case in which the WEC is additionally allowed to undergo sway and roll motions. Akin to the previous section which only dealt with the heave motion, the design procedure of the WEC in the general case is also divided into two steps, as illustrated in detail below.

\subsubsection{Step 1: Determination of the WEC's size $q$}
We start with the freely floating case, where the dashpot coefficient $C=0$. Using the equations of motion (\ref{eq:mo1})-(\ref{eq:mo6}), we can obtain the displacements of the two cylinders in the $x$ and $z$ directions, and the angle around the $y$ axis.

Once the monochromatic wave is given, or equivalently, once the mean wind speed for the corresponding PM spectrum is given, the physical process of determining the size of the WEC can be written in the following dimensionless form:
\begin{eqnarray}
\tilde{\zeta}_{zj0}&=&\Psi_{1j}(\tilde{q}),\\
\tilde{\zeta}_{xj0}&=&\Psi_{2j}(\tilde{q}),\\
\theta_{j0}&=&\Psi_{3j}(\tilde{q}),\\
\end{eqnarray}
where $\zeta_{zj0}$ and $\zeta_{xj0}$ denote the amplitudes of the vertical and horizontal displacements respectively, $\theta_{j0}$ is the amplitude of the angle around the $y$ axis, and  $j=1,2$ corresponds to the upper and lower cylinder, respectively.
We now seek the maxima of the functions $\Psi_{1j}, \Psi_{2j}$ and $\Psi_{3j},$ presented in Fig.\ref{freefloat-3m}.

Due to the increase in number of modes, the picture of the displacements is more complex than in the preceding section. It may be observed that the heave mode is decoupled from the sway and roll modes, yielding again the global maximum at $\tilde{q}=0.97.$ The sway and roll modes are coupled, and are observed to present a global maximum for relative displacement at $\tilde{q}=0.61,$ resulting in an ambiguous situation for determining the size of the WEC.

\begin{figure}[H]
  \centering
  \subfigure[upper cylinder]{
    \label{fig:freefloat-3m:a}
    \includegraphics[width=2.5in]{upper-freelyfloating-3m-nondim-U10.pdf}}

  \subfigure[lower cylinder]{
    \label{fig:freefloat-3m:b}
    \includegraphics[width=2.5in]{lower-freelyfloating-3m-nondim-U10.pdf}}
  \caption{Displacement amplitudes of the two freely-floating cylinders ($C=0$) in heave, sway and roll under the design monochromatic wave. $\tilde{\zeta}_{zj0}$: amplitude of the vertical displacement; $\tilde{\zeta}_{xj0}$: amplitude of the horizontal displacement; $\theta_{j0}$: amplitude of the angle around the $y$ axis. $j=1,2$ correspond the the upper and lower cylinders, respectively. }
  \label{freefloat-3m}
\end{figure}

\subsubsection{Step 2: Determination of the dashpot coefficient $C$}
As in the preceding section, we now suppose that the size of the WEC is given. The captured power $P_a$ then depends on the dashpot coefficient $C$. The determination of optimal power absorption as a function of dashpot coefficient is described in dimensionless form by
\begin{equation}
\tilde{P_a}=\Psi_4(\tilde{C};\tilde{q}),
\end{equation}
where, as we have seen, there is some flexibility in choice of $q.$ The function $\Psi_4$ is plotted in Fig. \ref{fig:pald-3m} for both $\frac{q}{U^2/g}=0.61$ and $\frac{q}{U^2/g}=0.97$. For the device operating optimally in heave ($qg/U^2 = 0.97,$ thick line) there is a unique maximum at $\tilde{C} = 0.34$ with $\tilde{P_a}=0.0035$ (denoted Case E), very close to the heave-only case discussed in Section \ref{subsec:heave only}. For the roll--sway optimized device ($qg/U^2 = 0.61,$ thin line) there are two local maxima $\tilde{C}=0.035$ and $\tilde{C}=1.34,$ with corresponding $\tilde{P_a} = 0.0012$ and $0.0013,$ (denoted Case A1 and A2) respectively.

\begin{figure}
    \centerline
    {\includegraphics[scale=0.35,angle=270]{pald-caseAE-3m-nondim-U10.pdf}}
    \caption{The captured power $P_a$ of the WEC in the combined motions versus the dashpot coefficient $C$. Thin line: $\frac{q}{U^2/g}=0.61$; thick line: $\frac{q}{U^2/g}=0.97$.}
  \label{fig:pald-3m}
\end{figure}

The situation for monochromatic incident waves is summed up in Table \ref{3m:mono:AH}, which shows the nondimensional size, optimal damping, captured power, and displacement amplitudes for the cases discussed above. As we have observed, introducing roll and sway motions leads to a two-fold branching in the design procedure. Firstly, in free motion, one value of $\tilde{q}$ is found to yield the largest roll and sway displacements, while another value yields the largest heave displacements. While the heave-optimized case has a unique maximum $\tilde{P}_a$ as a function of damping, the roll/sway-optimized case admits two local maxima of $\tilde{P}_a,$ one with relatively low, the other with relatively high damping $\tilde{C},$ compared to the heave case (see Figure \ref{fig:pald-3m}).

This opens up the possibility that the overall optimal design may not coincide with a design optimized for roll/sway or heave alone, but occupying some middle ground.
The performance of such intermediate devices (Cases B, C, and D), as well as devices somewhat larger than Case E (Cases F, G and H) is explored for the monochromatic design wave in Table \ref{3m:mono:AH}. In each of Cases A1 through H, a damping ${C}$ has been chosen to maximize the captured power.

\begin{table}[!htb]
\renewcommand{\arraystretch}{1.3}
\caption{The size, damping, displacement and captured power of 3-mode WECs in monochromatic waves.}
\centering
\begin{tabular}{@{} llllllllll @{}}
  \toprule
  &A1&A2&B&C&D&E&F&G&H\\
 \midrule
 $\tilde{q}$ & 0.61 & 0.61 &0.70&0.79&0.88&0.97&1.06&1.15&1.24\\

$\tilde{C}$& 0.035&1.34 &0.90&0.67&0.51&0.34&0.56&1.09&1.91\\

$\tilde{P}_a$        &0.0012&0.0013&0.0015&0.0021&0.0028&0.0035&0.0028&0.0020&0.0015\\

$\tilde{\zeta}_{z10}$	 &0.11 &0.089&0.093&0.11&0.14&0.19&0.13&0.063&0.039\\
$\tilde{\zeta}_{z20}$	 &0.036&0.053&0.033&0.026&0.023&0.021&0.0097&0.0042&0.0031\\
$\tilde{\zeta}_{x10}$	 &0.12 &0.063&0.060&0.055&0.050&0.048&0.047&0.040&0.038\\
$\tilde{\zeta}_{x20}$	 &0.018&0.016&0.015&0.013&0.012&0.011&0.0093&0.0069&0.0057\\
${\theta}_{10}(rad)$	 &0.70 &0.041&0.052&0.069&0.068&0.056&0.042&0.028&0.022\\
${\theta}_{20}(rad)$	 &0.028&0.031&0.021&0.014&0.0094&0.0066&0.0051&0.0035&0.0027\\
\bottomrule
  \end{tabular}
\label{3m:mono:AH}
\end{table}

Here we see that a shift in device size from the smaller, primarily rolling/swaying devices, towards larger, primarily heaving devices has a positive impact on captured power, up to device E. Thereafter, an increase in device size leads to a reduction in captured power, as the larger devices operate preferentially at smaller wavenumbers.

This situation is depicted in Figure \ref{fig:pa-k-AH-3m}, which shows $\tilde{P}_a^* \equiv P_a^*/(\rho U^3),$ the dimensionless captured power per unit wave amplitude squared, where $P_a^* \equiv P_a/a_0^2.$  To illustrate the associated displacements, Figure \ref{fig:dis1-k-AH-3m} shows the displacement in heave for the upper cylinder $\zeta_{z10}$  divided by $a_0$. Note that for case A1, the maximum value of $\zeta_{z10}(k)/a(k)$ is 4.4 (not shown).

\begin{figure}[H]
  \centering
  \subfigure[Case A1, A2, B and C]{
    \label{fig:pa-k-AH-3m:a}
    \includegraphics[width=2in,angle=270]{k-paa02-AC-non.pdf}}

  \subfigure[Case D and E]{
    \label{fig:pa-k-AH-3m:b}
    \includegraphics[width=2in,angle=270]{k-paa02-DE-non.pdf}}
  \subfigure[Case F, G and H]{
    \label{fig:pa-k-AH-3m:c}
    \includegraphics[width=2in,angle=270]{k-paa02-FGH-non.pdf}}
  \caption{The dimensionless captured power per unit wave amplitude square $P_a^{*}(k)/(\rho U^3)$ under different monochromatic waves as a function of wavenumber.}
  \label{fig:pa-k-AH-3m}
\end{figure}

\begin{figure}[H]
  \centering
  \subfigure[]{
    \label{fig:dis1-k-AH-3m:a}
    \includegraphics[width=2.1in,angle=270]{Az1-AC.pdf}}

  \subfigure[]{
    \label{fig:dis1-k-AH-3m:b}
    \includegraphics[width=2.1in,angle=270]{Az1-DH.pdf}}
  \caption{The dimensionless displacement $\zeta_{z10}/a_0$ of the upper cylinder in the combined motions under different monochromatic waves.(a)vertical displacement of cases A1, A2, B and C; (b)vertical displacement of cases D, E, F, G and H.}
  \label{fig:dis1-k-AH-3m}
\end{figure}

\section{Design of the WEC for a PM spectrum}
\label{section: irregular design}
Up to this point, we have considered WEC design for monochromatic waves. In brief: a given wind speed $U$ determines the two necessary parameters, wavenumber $k_p$ and amplitude $a_0$ in terms of the PM spectrum. With a monochromatic wave fully described by $(k_p,a_0),$ we may initially assume freely floating cylinders, and choose their size $\tilde{q}$ for maximum displacement in roll and sway (as these modes are coupled), for maximum displacement in heave, or at some intermediate value. In each case, a damping $\tilde{C}$ is chosen to maximize the captured power for this incident design wave, leading to the cases A1 through H above. As demonstrated in Figures \ref{fig:pa-k-AH-3m} and \ref{fig:dis1-k-AH-3m}, the motions and performance of a device designed for a wave $(k_p,a_0(k_p)$ may change considerably for other waves.

For practical reasons, our primary interest must be focused on irregular waves, where we may elect to tune the device to operate optimally at the peak of the spectrum, but must consider its performance for a broad band of incident waves. We begin with some preliminaries regarding the behavior of the WEC in irregular seas.

For a monochromatic wave, the absorbed power $P_a$ (see \eqref{Pa}) and displacements $\zeta_{\alpha j 0}$ are written as
\begin{align*}
& P_a (q,C,k,a_0) \equiv a_0^2 \hat{P}_a(q,C,k), \\
& \zeta_{\alpha j 0}(q,C,k,a_0) \equiv a_0 \hat{A}_{\alpha j}(q,C,k),\\
& \theta_{j 0}(q,C,k,a_0) \equiv  a_0 \hat{A}_{y j}(q,C,k).
\end{align*}
where $j =1,2$ denotes the upper and lower cylinders, $\hat{P}_a$ is the absorbed power per unit wave amplitude square, and $\hat{A}_{\alpha j}$ with $\alpha \in \{x,y,z\}$ denote the relative amplitudes of sway, roll and heave motions, respectively.
For a given spectrum $S(k)$ the total absorbed power by a device of type $(q,C)$ is
\begin{equation}
P^{\text{total}}_a = \int_0^\infty 2 \hat{P}_a(k) S(k) dk.
\end{equation}

Just as the spectrum describes the distribution of wave energy among different frequencies, and allows for statistical inferences such as a definition of the significant wave-height, so analogously we may consider a \emph{displacement spectrum}
\begin{equation}
E_{\alpha j}(k) \equiv S(k) (\hat{A}_{\alpha j})^2,
\end{equation}
and define the \emph{significant displacement} by
\begin{equation}
H_{\alpha j}^{1/3} = 4 \cdot \left(\int_0^\infty E_{\alpha j}(k) dk\right)^{1/2}.
\end{equation}
Here $H_{\alpha}^{(1/3)}$ ($\alpha=x,y,z$) is the the distance from the displacement's trough to crest and
\begin{eqnarray}
\zeta_{zj0}^{(1/3)}=\frac{1}{2}H_{zj}^{(1/3)}, \label{a1} \\
\zeta_{xj0}^{(1/3)}=\frac{1}{2}H_{xj}^{(1/3)},\label{a2}  \\
\theta_{j0}^{(1/3)}=\frac{1}{2}H_{yj}^{(1/3)},\label{a3}
\end{eqnarray}
are the so-called ``significant amplitudes of the displacement" in $z$ and $x$ directions, and the angle around $y$ axis, respectively.

\begin{table}[!ht]
\renewcommand{\arraystretch}{1.3}
\caption{The nondimensional significant amplitudes of displacement in heave $\tilde{\zeta}^{(1/3)}_{zj0},$ sway $\tilde{\zeta}^{(1/3)}_{xj0},$ and roll $\theta_{j0}^{(1/3)}$ (rad), along with dimensionless captured power $\tilde{P}_a^{\text{total}}$ for the WECs A1 through H attacked by a design spectrum $S_d,$ a severe spectrum $S_{s1}$ ($U_{s1} = 1.5 U_d$), and a second severe spectrum $S_{s2}$ ($U_{s2} = 2 U_d$).}
\centering
\begin{tabular}{@{} llllllllll @{}}
\toprule
  &A1&A2&B&C&D&E&F&G&H\\
  \midrule
 $\tilde{q}$ &0.61 & 0.61 &0.70&0.79&0.88&0.97&1.06&1.15&1.24\\

$\tilde{C}$& 0.035&1.34 &0.90&0.67&0.51&0.34&0.56&1.09&1.91\\
\midrule
  $S_{d}$\\

$\tilde{P}_a^{\text{total}}$ & $7.01\times10^{-5}$&0.00065&0.00081&0.0010&0.0011&0.0010&0.0011&0.00089&0.00086\\

$\tilde{\zeta}_{z10}^{(1/3)}$	 &0.22&0.097&0.096&0.11&0.12&0.14&0.12&0.094&0.071\\

$\tilde{\zeta}_{z20}^{(1/3)}$	 &0.044&0.065&0.040&0.030&0.027&0.025&0.021&0.017&0.013\\

$\tilde{\zeta}_{x10}^{(1/3)}$	 &0.10&0.077&0.073&0.068&0.064&0.059&0.056&0.054&0.051\\

$\tilde{\zeta}_{x20}^{(1/3)}$	 &0.022&0.019&0.018&0.016&0.015&0.014&0.012&0.011&0.0093\\

${\theta}_{10}^{(1/3)}$	 &0.60&0.050&0.059&0.073&0.084&0.096&0.068&0.045&0.032\\

${\theta}_{20}^{(1/3)}$	 &0.012&0.015&0.0093&0.0058&0.0039&0.0027&0.0020&0.0016&0.0012\\
\midrule
  $S_{s1}$ \\

$\tilde{P}_a^{\text{total}}$ & 0.00016&0.0025&0.0030&0.0037&0.0043&0.0046&0.0061&0.0060&0.0074\\

$\tilde{\zeta}_{z10}^{(1/3)}$	 &0.37&0.31&0.29&0.29&0.32&0.35&0.34&0.31&0.29\\

$\tilde{\zeta}_{z20}^{(1/3)}$	 &0.15&0.25&0.18&0.14&0.13&0.12&0.11&0.11&0.10\\

$\tilde{\zeta}_{x10}^{(1/3)}$	 &0.25&0.21&0.22&0.21&0.21&0.21&0.20&0.19&0.19\\

$\tilde{\zeta}_{x20}^{(1/3)}$	 &0.090&0.076&0.081&0.081&0.078&0.075&0.070&0.065&0.060\\

${\theta}_{10}^{(1/3)}$	 &0.98&0.16&0.14&0.17&0.21&0.29&0.23&0.16&0.12\\

${\theta}_{20}^{(1/3)}$	 &0.13&0.17&0.11&0.075&0.058&0.046&0.039&0.034&0.029\\
\midrule
  $S_{s2}$ \\

$\tilde{P}_a^{\text{total}} $ & 0.00019&0.0044&0.0049&0.0056&0.0064&0.0068&0.0097&0.011&0.014\\

$\tilde{\zeta}_{z10}^{(1/3)}$	 &0.55&0.54&0.51&0.52&0.53&0.56&0.56&0.54&0.53\\

$\tilde{\zeta}_{z20}^{(1/3)}$	 &0.34&0.47&0.39&0.33&0.30&0.29&0.28&0.27&0.27\\

$\tilde{\zeta}_{x10}^{(1/3)}$	 &0.45&0.42&0.42&0.42&0.42&0.42&0.42&0.41&0.40\\

$\tilde{\zeta}_{x20}^{(1/3)}$	 &0.20&0.19&0.19&0.20&0.20&0.19&0.19&0.18&0.17\\

${\theta}_{10}^{(1/3)}$	 &1.07&0.317&0.21&0.22&0.27&0.37&0.31&0.24&0.19\\

${\theta}_{20}^{(1/3)}$	 &0.31&0.40&0.25&0.18&0.14&0.12&0.10&0.088&0.077\\
\bottomrule
  \end{tabular}
\label{3m:severe2:AH}
\end{table}

Applying the concepts developed above to the problem of power absorption from an incident, broad-banded sea, we evaluate the above expressions for the spectra introduced in Section \ref{sec:physical preliminaries}. The results are given in nondimensional form in Table \ref{3m:severe2:AH}, which shows the captured power and displacement amplitudes for the spectra $S_d, \, S_{s1}$ and $S_{s2},$ nondimensionalized by $U=U_d.$ This may be compared to the analogous Table \ref{3m:mono:AH} for the monochromatic case. In the following section, we turn to a discussion of these results.

\section{Discussion}
\label{section:discussion}
\noindent As we have mentioned above, several competing criteria exist in determining WEC size. Those we shall consider in depth are limited to power capture, which naturally should be maximized, and survivability -- measured in terms of device motions.

We note that the Cases A1 through H presented above are ordered by increasing size $q$ which may be assumed correlated to the cost \emph{per device,} all other things being equal. Due to the burgeoning state of wave energy technology, it seems premature to speculate any further about cost, given that it depends not only on device size, but also design specifics such as materials and component costs, as well as costs related to regular maintenance or major overhaul, both factors which will in turn be affected by size.

In the following sections, we will delve into a detailed analysis of the WEC behaviour with a view to power capture and survivability. Subsequently, a synthesis of these two viewpoints is attempted, bearing in mind the primary aim of providing quantitative information relating to the design of oscillating body converters in a range of different, broad-banded sea states.

\subsection{Power capture}
\label{subsec:power capture}
The most straightforward metric to evaluate concerns the power captured by a WEC. For a design PM spectrum $S_d$ corresponding to a wind speed $U_d=10$ m/s, and severe spectra $S_{s1}$ and $S_{s2}$ corresponding to $U_{s1} = 15$ and $U_{s2} = 20$ m/s, respectively, the dimensional size, damping and absorbed power of WECS A1 through H are presented in Table \ref{table:dimensional power capture}.

\begin{table}[h]
\centering
\renewcommand{\arraystretch}{1.3}
\caption{Dimensional absorbed power $P_a$ (Watt) for cases A1 through H, for an incoming monochromatic wave ($P_a^m$) and the design PM spectrum with $U = 10$ m/s ($P_a^d$), both with the same energy density of $3.7 \text{ KJ}/\text{m}^2.$ Also given are the absorbed power for the severe spectra $S_{s1}$ ($P_a^{s1}$) and $S_{s2}$ ($P_a^{s2}$).}
\label{table:dimensional power capture}
\begin{tabular}{@{} llllllllll @{} }
\toprule
 & A1&	A2&	B&	C&	D&	E&	F&	G&	H \\
 \midrule
q [m] &6.2 &	6.2	&7.1&	8.1&	9.0&	9.9&	10.8&	11.4&	12.7 \\
C $(\cdot 10^5$) [Ns/m] &	$0.364$	& $14.0$	& $9.37$	& $6.98$& 	$5.31$& 	$3.54$& 	$5.83$& 	$11.3$	& $19.9$
 \\
$P_a^m$ $(\cdot 10^5$) [W] &	$1.25$	& $1.35$ &	$1.56$ &	$2.19$	& $2.92$ &	$3.64$ &	$2.92$	& $2.08$ &	$1.56$ \\
$P_a^{d}$ $(\cdot 10^5$) [W] & $0.0730$	& $0.672$ &	$0.849$ 	& $1.04$ & $1.14	$ & $1.06$ &	$1.17$	& $0.931$ & $0.893$  \\
\midrule
$P_a^{s1}$ $(\cdot 10^5$) [W] & $0.165$	& $2.65$ &	$3.13$ &	$3.80$ &	$4.43$ &	$4.77$ &	$6.36$ &	$6.27$ &	$7.73$ \\
$P_a^{s2}$ $(\cdot 10^5$) [W]& $0.197$ &	$4.62$	& $5.09$	& $5.88$ &	$6.67$ &	$7.10$ &	$10.1$ &	$11.0$ &	$15.0$ \\
\bottomrule
\end{tabular}
\end{table}

We recall the monochromatic wave used for device design, with a wavelength of 96.3 m and an amplitude of 0.87 m, with an energy density of 3.7 KJ/m$^2$ equal to that of the design PM spectrum parametrised by a wind-speed $U = 10$ m/s. The picture which emerges from comparing the absorbed powers in the monochromatic and spectral cases is quite striking. While the narrow-banded response of device A1 (see Figure \ref{fig:dis1-k-AH-3m:a}) yields a performance comparable to slightly larger devices for monochromatic waves, power absorption declines drastically for an incident PM spectrum.

Likewise, though the heave-optimized device E is clearly superior to devices of similar size (D and F) for monochromatic waves, this situation sees a dramatic reversal in the case of incident irregular waves. This shows the potential utility of de-tuning in WEC design, as devices on either side of the heave-optimum outperform it for irregular seas. In particular, it demonstrates the pitfalls of a design based on monochromatic waves.

Dimensional values of captured power are also provided for the two severe spectra, $S_{s1}$ corresponding to a wind speed $U_{s1} = 15$ m/s, or an energy density of 18.7 KJ/m$^2$, as well as $S_{s2},$ corresponding to a wind speed $U_{s2} = 20$ m/s and an energy density of 59.6 KJ/m$^2$. As expected, the larger devices benefit most from this increased wave resource, while a sea composed of increasingly long waves ($\lambda_p$ for $S_{s1}$ is 217 m, and for $S_{s2}$ is 385 m, see Table \ref{SwSs}) begins to saturate the power capture capabilities of the smallest devices. In the following sections on survivability and grading of WECs, we shall explore the feasibility of operating WECs in such large sea states.

\subsection{Survivability}
\label{subsec: survivability}
We come now to the less well-defined of the two concepts with a bearing on the performance of a twin-cylinder WEC: survivability. The disparity between the motions and resulting loads experienced by a WEC in normal operation, and those during severe conditions may be immense. Following Brown et al \cite{brown2010towards} we distinguish between the reliability of a WEC, related to failure during normal operation, and survivability. This latter concept applies to sea states outside of the intended operating conditions -- when the average conditions of the ocean environment exceed the safe operational limits of the device.

While it is clear that WECs must be robust in design, as during a ten-year operational period a converter may expect to see some tens of millions of waves, during particularly severe events, power production will need to be halted in order to avoid damage to the device or loss of station-keeping.

\begin{table}[h]
\renewcommand{\arraystretch}{1.3}
\centering
\caption{Relative heave displacements versus draft $\zeta^r_{z} = (\zeta_{z10}^{(1/3)}-\zeta_{z20}^{(1/3)})/q$, and relative roll displacement $\theta^r = \theta^{(1/3)}_{10}/(\pi/2)$ }
\label{table:survivability}
\begin{tabular}{@{} llllllllll @{} }
\toprule
 & A1&	A2&	B&	C&	D&	E&	F&	G&	H \\
 \midrule
 $S_d$ \\
$\zeta_z^{r}$ & \cellcolor{orange!25}0.29	&\cellcolor{green!25}0.05	&\cellcolor{green!25}0.08&\cellcolor{green!25}	0.10&	\cellcolor{green!25}0.11&\cellcolor{green!25}	0.12&\cellcolor{green!25}	0.09&\cellcolor{green!25}	0.07&\cellcolor{green!25}	0.05 \\
$\theta_{1}$&\cellcolor{red!25}0.38&	\cellcolor{green!25}0.03&	\cellcolor{green!25}0.04	&\cellcolor{green!25}0.05&	\cellcolor{green!25}0.05&	\cellcolor{green!25}0.06&\cellcolor{green!25}	0.04&	\cellcolor{green!25}0.03&	\cellcolor{green!25}0.02 \\
$S_{s1}$ \\
$\zeta_z^r$ & \cellcolor{red!25}0.36&	\cellcolor{green!25}0.10	&\cellcolor{yellow!25}0.16	&\cellcolor{yellow!25}0.19	&\cellcolor{yellow!25}0.22&\cellcolor{yellow!25}	0.24	&\cellcolor{yellow!25}0.22&	\cellcolor{yellow!25} 0.18&\cellcolor{green!25}	0.15 \\
$\theta_{1}$ & \cellcolor{red!25}0.62&	\cellcolor{green!25}0.10&	\cellcolor{green!25}0.09&	\cellcolor{green!25}0.11	&\cellcolor{green!25}0.13&	\cellcolor{green!25}0.18&	\cellcolor{green!25}0.15	&\cellcolor{green!25}0.10&	\cellcolor{green!25}0.08 \\
$S_{s2}$ \\
$\zeta_z^r$& \cellcolor{red!25}0.34	&\cellcolor{green!25}0.11&\cellcolor{yellow!25}	0.17	&\cellcolor{yellow!25}0.24	&\cellcolor{orange!25}0.26&\cellcolor{orange!25}	0.28	&\cellcolor{orange!25}0.26	&\cellcolor{yellow!25}0.24&\cellcolor{yellow!25}	0.21 \\
$\theta_{1}$& \cellcolor{red!25}0.68	&\cellcolor{yellow!25}0.20&\cellcolor{green!25}	0.13	&\cellcolor{green!25}0.14	&\cellcolor{green!25}0.17&	\cellcolor{yellow!25}0.24&	\cellcolor{yellow!25}0.20&\cellcolor{green!25}	0.15&	\cellcolor{green!25}0.12 \\
\bottomrule
\bottomrule
\end{tabular}
\end{table}

We have developed an example framework for survivability for the twin-cylinder WEC in the three spectral sea states considered, which is presented in Table \ref{table:survivability}. The maximum allowed relative vertical travel $\zeta^{(1/3)}_{z10}-\zeta^{(1/3)}_{z20}$ is limited to $q/3,$ while the maximum allowable roll is $30^\circ.$ Those cases which exceed these values are marked red (PDF only). A vertical travel of more than $q/4$ or a roll of more than $22.5^\circ$ is marked orange, while a vertical travel of more than $0.15q$ or a roll more than $13.5^\circ$ is marked yellow (PDF only). Device motions smaller than these are marked green (PDF only). Recall that these nondimensional quantities depend only on the relations $U_{s1} = 1.5\cdot U_d$ and $U_{s2} = 2\cdot U_d$ as specified in Section \ref{sec:physical preliminaries}, and the concomitant changes in significant wave-height and peak wavenumber.

For illustrative purposes, if the design spectrum $S_d$ is generated by a fresh breeze ($U_d = 10 m/s,$ or 5 Beafort, 2.47 m $H^{(1/3)}$), then the first severe state $S_{s1}$ may be thought generated by a high wind (7 Beaufort, 5.5 m $H^{(1/3)}$). The second severe state $S_{s2}$ occurs under conditions between gale and severe gale (8--9 Beaufort, 9.9m $H^{(1/3)}$). These extremely harsh conditions represent an energy density more than 16 times that of the design spectrum, and may be expected to challenge the device design.

While the increase in significant wave-height between the design spectrum $S_d$ and the severe case $S_{s2}$ may seem dramatic, there is no doubt that such conditions will be encountered within the operational life of a WEC. For example, while deep water conditions for the Eastern Mediterranean off Israel's coasts may see significant wave heights greater than 2 m only 6 \% of the time, and wave heights in summer rarely exceed 1--1.5 m, nevertheless storms with $H^{(1/3)}$ in excess of 5 m occur almost yearly. The 10-year return period significant wave height is nearly 7 m, which clearly falls  within the expected operational life of a converter.

From a pure survivability standpoint, it is immediate only that the smallest converter A1 is not viable. In particular, the very small damping of this configuration (see Table \ref{table:dimensional power capture}), while allowing for efficient power capture from the roll mode, also leads to overly large displacements even for design conditions. With survivability as the central aim of design, larger structures will necessarily fare better, though the differences between devices D, E, and F are in practice rather small. While other authors (e.g.\ Maisondieu \cite{Maisondieu_2015} or Brown et al \cite{brown2010towards}) have investigated survivability of WECs, they have been forced to do so without reference to the hydrodynamics and actual displacements of a floating device, but rather purely based on estimations of the incident wave power.

\subsection{Grading WEC sizes}
\label{subsec:grading}

We shall now make a preliminary attempt to sum up the results of the preceding sections. The intricacies of WEC economics, as well as the many factors which are outside the scope of the present study, such as moorings, specifics of the PTO, control strategies, power conversion and transmission, and other environmental factors from seasonal variability to extreme events, will need to be taken into account for a fuller analysis. In addition, WEC cost will not be considered, and is likely to impact significantly the ultimate design.

\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{Comparison_Abs_Powers}
\caption{Comparison of power absorbed (in Watt) by devices A1 through H under the four conditions -- the design monochromatic wave (dotted line), the design spectrum $S_d$ (solid line), the first severe spectrum $S_{s1}$ (dashed line), and the second severe spectrum $S_{s2}$ (dash-dotted line), where $U_d = 10$ m/s.}
\label{fig: comparison abs power}
\end{figure}

Figure \ref{fig: comparison abs power} compares the dimensional absorbed power for the nine devices considered under different conditions. Once again, we point out that the monochromatic case is essentially artificial, and included only for illustrative purposes. The lessons to be drawn from this comparison will likely change as wave-power technology matures. While current oscillating-body devices may be rather small, and situated in shallow water with the intention of keeping costs down, future developments will naturally lead to a move into the more powerful wave-regimes further offshore (see Stiassnie et al \cite{Stiassnie2015} for a discussion).

As an example, while there is a 15 \% reduction in absorbed power between Case E and Case H under the design spectrum, the corresponding increase in absorbed power for severe case $S_{s2}$ is upwards of 60 \%. The fact that, off the Eastern Mediterranean Coast, some 45 \% of average wave power comes during storm events that occur only 5 \% of the time indicates the utility of the larger design \cite{Kroszynski1979}.  This is compounded by the increase in potential survivability of the larger devices as indicated in the previous section.
Depending on the variability of the wave-energy resource, more or less weight may ultimately be given to each of the considerations just outlined.  The fact that the larger devices exhibit smaller relative motions may also be a benefit for their reliability, in terms of limiting loading during normal operation. Ultimately, an effort will have to be made to weigh the additional cost of a larger device against the increase in survivability. Both of these in turn will need to be weighed against the potential of continuing operation during high-energy events, while sustaining a slight performance decrease for low-energy sea states.

\section{Conclusions}
\label{sec:conclusions}

We have investigated in detail the hydrodynamics of a model WEC consisting of two floating, axisymmetric cylinders connected at their upper and lower perimeters by a continuously distributed damper -- allowing power capture from heave and roll modes. While other authors have studied various aspects of the problem of floating cylinders, the present work has addresses for the first time the problem of a twin cylinder WEC allowed to move in three degrees of freedom. The inclusion of a floating, submerged cylinder as a mechanical reference for power extraction makes this design viable in deep water. With further development of the wave energy industry, it may be anticipated that WECs will follow wind turbines in moving further offshore, making such self-reacting devices more and more relevant \cite{Stiassnie2015}.

Our design procedure initially focused on optimizing device behavior for a damper of constant characteristic in monochromatic waves. At the outset, the heave-only case was considered, presenting a simple situation where a single device (characterized by a size parameter $q$ and a damping parameter $C$), coinciding with the resonant maximum of a freely floating body, outperformed all others. Allowing the device also to sway and roll was seen to introduce additional complexity, and a differentiation was observed between devices operating preferentially in roll/sway and those operating preferentially in heave.

Despite the multiplicity of possible designs when the device is allowed to undergo heave, sway, and roll motions, the monochromatic case presents a clear picture from the standpoint of power absorption: the device closest to heave resonance is found to perform best. This conclusion is an artifact of the idealization represented by the monochromatic theory -- a fact established by the subsequent investigation of WEC performance under an irregular sea.

For our design purposes, a Pierson-Moskowitz spectrum, characterized by wind speed, was chosen to evaluate the designs obtained from the monochromatic case. Under this spectrum, the maxima of absorbed power were found to shift markedly with respect to the monochromatic case, reflecting the potential for detuning to increase captured power in WEC design. Larger values of absorbed power under the design spectrum were found for devices slightly larger and slightly smaller than the monochromatic optimum, raising the question of how to determine device sizing in light of other criteria.

To this end, we have devised some example metrics for grading the sizes of our twin-cylinder WEC. We note that wave energy presents particular difficulties in many respects. While a fixed offshore structure may be designed for survival with very high safety factors, this is inappropriate for oscillating body WECs; by their nature, they must undergo the largest possible motions in order to extract energy. At the same time, device loading should be minimized to avoid fatigue and failure. Taking into account the fact that WECs may be expected to be operational for on the order of 25 years (see Starling \cite{EMEC2009}), and it becomes clear that survival is a paramount issue.  We have presented an example approach to quantitatively evaluate the competing aims of survivability and power extraction within the framework of our floating twin-cylinder device.

To a certain extent all renewable energy technologies, WECs more than most, cannot control their operating conditions, but must work within their environment, subject to the resulting fluctuations of the resource. It must be expected that, like wind turbines, oscillating body WECs will be designed with a ``survival mode", when normal operation cease, and the device changes its characteristics in order to avoid extreme loads. (We might note that overtopping WECs or OWCs (see Section \ref{sec:Introduction}), due to a different working principle and resulting size, will likely have a very different survivability analysis than oscillating body designs.) This may mean increasing the damping, altering the water plane area or mass (see Stallard et al \cite{Stallard2009}), or other approaches (see Coe and Neary \cite{Coe2014}). Due to the nascent state of commercial wave-energy technology, it is difficult to offer concrete design recommendations based on the results for floating twin-cylinders. Our discussion does bear out the fact that a slight over-engineering may be preferable, given the large relative contribution of infrequent, high-energy events to the annual energy budget at many sites, and the demands of survival and robustness. We believe these results to be applicable more broadly to oscillating-body converters, constrained in size as they are by the incident wavelength, indicated by the striking similarities in performance between our twin-cylinder configuration and a single bottom-referenced cylinder.

While many topics of critical importance have not been touched upon -- from moorings to electricity transmission to specifics of power take-off -- we hope to have presented a rather comprehensive picture of the design of a floating twin-cylinder WEC in deep water. There is considerable room for future work, e.g taking into account seasonal resource variability, extreme wave statistics, PTO control, and device cost, as well as detailed studies of reliability under regular operation, where it is hoped that some of the gaps left by this work will be filled in.

\section*{Acknowledgements}
This research was supported by the Israel Science Foundation (Grant 464/13).

redpajama_set_name: RedPajamaArXiv


Jip & Janneke, the most famous toddler heroes of Holland. Not many people grew up without their adventures. The combination of relatable stories of Annie M.G. Schmidt with the black and white illustrations by Fiep Westendorp is still a great success.
The two silhouettes made their first entrance in "Het Parool" in 1952; which back then started as a dutch resistance newspaper against the nazi's in World War II. In the very first story, Jip got a new neighbour; a little girl next door named Janneke. The two get along very well and together they experience the most fun adventures, with dog Takkie and cat Siepie. The stories of Jip & Janneke appeared every week as a short episode on the children's page of Het Parool. After every year they bundled the new episodes into a book, which was published by "de Arbeiderspers".
Over the years, Jip & Janneke have been through some changes. Fiep Westendorp decided, for instance, that the necks and tall bodies don't really contribute to the drawing. She decided to refine the illustrations. It resulted in how we know Jip & Janneke nowadays.
The classic Jip & Janneke book by Annie M.G. Schmidt and Fiep Westendorp as we know it didn't appear until 1977 when it came out as a collection bundle. Today we have reached the 47th edition of the book and it is still one of the best sold children's books of Holland.


Justin Morelli Blog Tagged "Press"
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A distinctive top quality gift! A beautiful basket filled with 2 boxes of hearty gourmet crackers, 2 types of award-winning hand crafted aged cheddar, award-winning gourmet antipasto, deluxe mixed nuts, honey roasted peanuts, shortbread cookies, Cafe Françoise mix, chocolate mint sticks, aromatic apple cider, and chocolate covered almonds.
This Christmas basket is garnished in the Festive Season, wrapped in cello and topped with a beautiful hand tied bow.


Spill control and prevention plans are critical to employee safety and protecting the environment. OSHA and the EPA have many regulations, like the Spill Prevention Control and Countermeasure (SPCC) Plan to help protect employees and the environment.
An SPCC plan is a facility-specific, comprehensive plan outlining both the containment and countermeasures implemented to prevent a spill from occurring and a series of response procedures in the event that one does.
For more on the SPCC plan and regulation, see the EPA's Facility Owner/Operator Guide.
Accidents happen and when a spill occurs, efficient and safe clean-up is critical. To make sure your employees and the environment is protected, download our Spill Procedure Guide.
Brady SPC provides some of the most sophisticated absorbent and spill containment solutions in the industry.
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text:
Unemployment rate in the U.S. broadcasting industry 2001-2018
The statistic above presents unemployment data for the U.S. broadcasting industry from 2001 to 2018. In 2018, 2.7 percent of all private wage and salary workers in the broadcasting industry were unemployed.
private wage and salary workers
* Broadcasting includes radio and television broadcasting and providers of cable and other subscription programming.
Leading broadcasting groups in Luxembourg 2017, by market share
TV- and radio license fee yields in Sweden 2015-2016, by broadcasting company
Ranking of programming and broadcasting companies in the Nordics 2019, by employees
Ranking of broadcasting companies in the Nordic countries 2019, by turnover
Ranking of programming and broadcasting companies in the Nordics 2019, by profit
Grupo Radio Centro revenue 2017-2018
Industry revenue of »programming and broadcasting activities« in Italy 2011-2023
Industry revenue of »programming and broadcasting activities« in the Netherlands 2011-2023
Industry revenue of »programming and broadcasting activities« in Denmark 2011-2023
Industry revenue of »sound recording and music publishing activities« in Finland 2011-2023
Leading broadcasting groups in Luxembourg in 2017, by TV daily audience market share
TV- and radio license fee yields in Sweden in 2015 and 2016, by broadcasting company (in million SEK)
Ranking of companies in the programming and broadcasting industry in the Nordic countries as of March 2019, by number of employees
Ranking of companies in the programming and broadcasting industry in the Nordic countries as of March 2019, by turnover (in million euros)
Ranking of companies in the programming and broadcasting industry in the Nordic countries as of March 2019, by net profit (in 1,000 euros)
Revenue generated by Grupo Radio Centro in 2017 and 2018 (in billion Mexican pesos)
Industry revenue of »programming and broadcasting activities« in Italy from 2011 to 2023 (in million U.S. Dollars)
Industry revenue of »programming and broadcasting activities« in the Netherlands from 2011 to 2023 (in million U.S. Dollars)
Industry revenue of »programming and broadcasting activities« in Denmark from 2011 to 2023 (in million U.S. Dollars)
Industry revenue of »sound recording and music publishing activities« in Finland from 2011 to 2023 (in million U.S. Dollars)


Being a thought-led creative agency with a name like Philosophy is a big ask. But it doesn't mean we waste valuable time and money debating for months on end.
Instead, we ensure our solutions are grounded in research, practical processes and creativity that make a difference for the right audience. It also means that we make it our business to understand new developments not only in our own industry but, most importantly, in yours.


text:
from ..registry_tools import iso_register
from .core import UnitedStates

@iso_register('US-WI')
class Wisconsin(UnitedStates):
    """Wisconsin"""
    include_columbus_day = False
    include_federal_presidents_day = False
    include_christmas_eve = True
    FIXED_HOLIDAYS = UnitedStates.FIXED_HOLIDAYS + (
        (12, 31, "New Years Eve"),
    )

redpajama_set_name: RedPajamaGithub


text:
Q: Django max_upload_size gets ignored I have this code but for some reason the file size gets ignored, even if I set this directly to ('max_upload_size', 5242880) at formatChecker.py the value seems to get ignored after the upload has happened.
settings.py
MAX_UPLOAD_SIZE = "5242880"

formatChecker.py
from django.db.models import FileField
from django.forms import forms
from django.template.defaultfilters import filesizeformat
from django.utils.translation import ugettext_lazy as _
from myproject.settings import MAX_UPLOAD_SIZE

class ContentTypeRestrictedFileField(FileField):
    """
    Same as FileField, but you can specify:
        * content_types - list containing allowed content_types. Example: ['application/pdf', 'image/jpeg']
        * max_upload_size - a number indicating the maximum file size allowed for upload.
            2.5MB - 2621440
            5MB - 5242880
            10MB - 10485760
            20MB - 20971520
            50MB - 5242880
            100MB 104857600
            250MB - 214958080
            500MB - 429916160
    """

    def __init__(self, *args, **kwargs):
        self.content_types = kwargs.pop('content_types', [])

        super(ContentTypeRestrictedFileField, self).__init__(*args, **kwargs)

    def clean(self, *args, **kwargs):
        data = super(ContentTypeRestrictedFileField, self).clean(*args, **kwargs)

        file = data.file
        try:
            content_type = file.content_type
            if content_type in self.content_types:
                if file._size > int(MAX_UPLOAD_SIZE):
                    raise forms.ValidationError(_('Please keep filesize under %s. Current filesize %s') % (
                        filesizeformat(MAX_UPLOAD_SIZE), filesizeformat(file._size)))
            else:
                raise forms.ValidationError(_('Filetype not supported.'))
        except AttributeError:
            pass
        return data

models.py
...
class Post(models.Model):
    postattachment = ContentTypeRestrictedFileField(
      blank=True,
      null=True,
      upload_to=get_file_path_user_uploads,
      content_types=['application/pdf',
                     'application/zip',
                     'application/x-rar-compressed',
                     'application/x-tar',
                     'image/gif',
                     'image/jpeg',
                     'image/png',
                     'image/svg+xml',
                     ]
     )
...

Any idea why that problem occurs?
Did I forget something here?

A: add MAX_UPLOAD_SIZE = "5242880" in setting.py
then in views file
from django.conf import settings
file._size > settings.MAX_UPLOAD_SIZE

or file._size > int(settings.MAX_UPLOAD_SIZE)
in init method, it pop two keys, so it doesn't exists
    def __init__(self, *args, **kwargs):
       self.content_types = kwargs.pop('content_types', [])
       self.max_upload_size = kwargs.pop('max_upload_size',[])

so remove these line
self.content_types = kwargs.pop('content_types', [])
self.max_upload_size = kwargs.pop('max_upload_size', [])

redpajama_set_name: RedPajamaStackExchange