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Address to Society of Construction Lawyers FCL by Justice David A R Williams QC, Dubai, Sunday 20 March 2011
Home » Judges' Addresses » Address to Society of Construction Lawyers FCL by Justice David A R Williams QC, Dubai, Sunday 20 March 2011
Judges' Addresses
It is my privilege to deliver a brief opening address. I am an international arbitrator and part time Justice at the DIFC Courts. I understand the focus today will be on various aspects of the 4th Edition, FIDIC Contract. In my international arbitration practice I have encountered the FIDIC Contract on several occasions but I certainly do not qualify as a FIDIC expert and it would be perilous for me to venture any opinions on it.
Therefore in my brief opening remarks I propose to mention a couple of matters of current debate in the international arbitration field which may be of some interest to you. I shall also touch briefly on aspects of the role of the DIFC Courts.
In recent years concerns have been expressed about the actual or perceived lack of independence of many expert witnesses. Expert witnesses are frequently seen as unduly partisan. Recently Dr Klaus Sachs, a leading international arbitrator from Germany has presented an article entitled "Experts, Neutrals or Advocates — A New Approach to Expert Evidence1.
Dr Sachs introduces his paper by saying that in respect of party appointed experts "one increasingly often hears complains that there evidence is of little value because it advocates too much in the interests of the party presenting it. The result is that the Tribunals then find themselves faced with deciding between opinions of opposing experts who have arrived at diametrically opposite and hardly reconcilable conclusions".
Dr Sachs' concept, which he has apparently successfully used in practice, is as follows. The parties agree that there will be no party appointed experts. They submit a list of experts in the appropriate disciplines to the Tribunal. The Tribunal then selects the experts from the list. At that point, the chosen expert becomes the Tribunal expert although paid for by the parties. Under this scheme the parties are not excluded from the appointment process.
There have been similar concerns expressed about party-appointed arbitrators especially in international investment arbitrators. Professor Jan Paulsson, a leading international arbitrator, published an inaugural lecture at the University of Miami in April of 2010 entitled "Moral Hazard in International Dispute Resolution"2. He put forward in his address a more radical proposal, namely that the office of party appointed arbitrators should be entirely eliminated in the field of international arbitration and that all arbitrators should be appointed by arbitral institutions. This proposal has received a mixed reception. It has been pointed out that it is in conflict with the overriding principle of party autonomy. Moreover, experienced arbitrators have noted that where dispute resolution clauses call for the parties, or two party-appointed arbitrators, to appoint the Chairman it is commonly observed that they try very hard to avoid the default mechanism which is the utilisation of an institution to appoint the Chairman. This is said to demonstrate a lack of confidence in institutions in respect of appointments.
A related matter of recent discussion is that concerning dissenting opinions in international investment arbitration. A leading arbitrator, Dr Albert Jan van den Berg has conducted an analysis of all dissenting opinions by party-appointed arbitrators in investment arbitration.3 Dr van den Berg points out that dissenting opinions appear to have become an accepted practice in international arbitration. He criticises this trend and points out that his study reveals that virtually all dissenting opinions have been issued by the party-appointed arbitrator whose appointor was the losing party in the arbitration. Since nearly 100% of the dissents favour the party that appointed the dissenter there are obvious concerns about neutrality.
Turning briefly to the DIFC Courts most of you will know that the DIFC Courts operates within a defined geographic area in Dubai and that it is a creature of Dubai statutes. It may fairly be described as a commercial court. It aims for expedition and efficiency. Case management is an essential feature of the DIFC Courts Rules. The Court is under a duty to give directions to ensure that trials proceed quickly and efficiently. To that extent the objectives of the Court are very similar to the objectives of the participants in this conference who are involved in dispute resolution in the construction industry.
The following passages from a judgment, which I gave in the Court last year, underscore this point. The case involved the flagrant disobeyance of a procedural timetable by one of the parties and in my judgment I said this:
"52. As to the need for expedition in commercial litigation, I respectfully endorse the observation of Heydon J. of the High Court of Australia in Aon Risk Services Australia Ltd v Australian National University [2009] H.C.A. 27 at [137]:
"The character of the litigation commenced. The litigation thus commenced was commercial litigation. While in general it is now seen as desirable that most types of litigation be dealt with expeditiously, it is commonly seen as especially desirable for commercial litigation. Its claims to expedition may be less than those of proceedings involving, for example, extraordinary prejudice to children; or the abduction of children; or a risk that a party will lose livelihood, business or home, or otherwise suffer irreparable loss or extraordinary hardship, unless there is a speedy trial. But commercial litigation does have significant claims to expedition. Those claims rest on the idea that a failure to resolve commercial disputes speedily is injurious to commerce, and hence injurious to the public interest.
As Rogers J stated in Collins v Mead:
'For example, if banks are unable to collect overdue loans from borrowers speedily, if small traders can not recover monies owed to them speedily the commercial life of the [c]ommunity is detrimentally [a]ffected. The consequences of delay in the hearing of a commercial dispute … will impact not just on the two or three persons or companies who are the immediate parties, but may have an effect on the creditors of the business, on employees, and perhaps on other traders unrelated to the immediate dispute.'
Commercial life depends on the timely and just payment of money. Prosperity depends on the velocity of its circulation. Those who claim to be entitled to money should know, as soon as possible, whether they will be paid. Those against whom the entitlement is asserted should know, as soon as possible, whether they will have to pay. In each case that is because it is important that both the claimants and those resisting claims are able to order their affairs. How they order their affairs affects how their creditors, their debtors, their suppliers, their customers, their employees, and, in the case of companies, their actual and potential shareholders, order their affairs. The courts are thus an important aspect of the institutional framework of commerce. The efficiency or inefficiency of the courts has a bearing on the health or sickness of commerce."
54. This court must indicate as clearly as possible that the cavalier disregard of timetable orders will not be tolerated. For it to allow such disregard would undermine the aims of case management and defeat the statutory objectives of the RDC, which include ensuring that cases are dealt with expeditiously and fairly (r.1.5(4) of the RDC), saving expense (r.1.5(2) of the RDC) and ensuring that the parties are on an equal footing (r.1.5(1) of the RDC). It is also abundantly clear that the parties have a duty to assist the court in furthering the statutory objectives of the RDC (r.1.7 of the RDC) and that lawyers are under a separate duty to behave co-operatively (para 13 of DIFC Courts' Code of Professional Conduct for Legal Practitioners). In addition, the conduct of litigation is not merely a matter for the parties. The claims of other litigants and the public interest in achieving the most efficient use of resources must also be taken into account. The court must avoid "the waste of public resources and the inefficiency occasioned by the need to revisit interlocutory processes".
The DIFC Courts has dealt with some construction cases. It is also likely to have a supportive role in relation to arbitration which is almost always the ultimate dispute resolution mechanism in construction contracts. In this respect, like any modern Court, its role in arbitration matters is not an interventionist role but a supportive role. The DIFC Courts under DIFC arbitration law is authorised to assist the arbitral process in the following areas:
• Article 3 — Stay of Proceedings
• Article 5 — Interim Measures of Protection
• Article 17 — Appointment of Arbitrators
• Article 19 — Challenge Procedure
• Article 20 — Termination of Mandate of Arbitrator
• Article 23 — Decision on Jurisdiction of the Arbitral Tribunal
• Article 34 — Court Assistance in Taking Evidence
• Article 41 — Applications to Set Aside Arbitration Awards
• Article 42 — 44 — Recognition and Enforcement of Awards
I congratulate the organisers for bringing together experts from several disciplines to discuss the interlocking construction contract issues by reference to FIDIC 4th Edition. The quality of the speakers should ensure a valuable educational experience for all participants in this fundamentally important area.


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Odval na Ostravici nebo také Odval Rudná je ostravský odval, který v minulosti sloužil především pro potřeby klopení žhavé strusky z vysokých pecí Nové huti. V současnosti je využíván vzácně.

Lokalizace
Odval o rozloze 3,5 ha se nachází v městském obvodu Slezská Ostrava. Prostor odvalu je ze severu ohraničen Rudnou ulicí, z východu Frýdeckou ulicí, z jihu železniční vlečkou Nová huť - Vítkovické železárny a ze západu řekou Ostravicí. Před výstavbou Rudné ulice se odval nacházel i v prostoru pod touto nynější ulicí a severně od ní směrem ke Kunčičkám.

Historie
Výstavba odvalu byla zahájena v roce 1952 a 27. listopadu téhož roku zde bylo zahájeno klopení strusky. Odval byl vybudován pouze jako provizorní řešení do doby zprovoznění definitivního odvalu Lihovarská. K tomu došlo v roce 1958 a od té doby byl odval používán jen sporadicky. V současnosti je odval využíván pouze v případě nemožnosti klopení na haldě Lihovarská.

Reference

Haldy v Ostravě
Hory a kopce v okrese Ostrava-město
Nová huť
Kunčice (Ostrava)

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This Week, The Oregon Quilt Project Officially Commenced In Beautiful Sisters, Oregon - Home Of The Stitchin' Post And TheSisters Outdoor Quilt Show.
The image of Oregon Quilt Shows was uploaded at July 30, 2018 at 5:16 pm. This post is posted at the Quilt category. Oregon Quilt Shows is tagged with Oregon Quilt Shows, Oregon, Quilt, Shows..
Blinds are one of many significant components in a room. Oregon Quilt Shows ready to block the sunshine is too brilliant about the outside and about the other hand is also able to address the main space whilst not obvious in the outside. So excellent blackout function till there's scarcely a room that had a screen without any drapes.
To make a harmonious mixture of decor of the room through the selection of suitable curtains, we should be watchful while in the combination and complement of colors, designs, as well as the curtain resources using the idea of house and the shape and size of the window itself. Not just that, the election blackout should also be used to paint the surfaces the contrast isn't it and also like the curtains have a coloring that's not in harmony together with the shade of the wall paint, the result will look strange?
Drapes than valuable with regards to function, may also be treated being an element of decor that will beautify the room. These objects can be combined with forms and types along with the concept of the room of windows to be able present a separate bedroom decoration and in the future together.


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The Observation of an Atheist Historian: What Makes Christianity Stand Out Among World Religions
January 10, 2020 May 20, 2021 kevingdrendel
The radical quality of the love of Jesus stands out over and above all other examples. I have written on this before (the Christian expression of the Golden Rule compared to other religions). Most other world religions express some concept of the Golden Rule, but not in the way that Jesus did.
Other world religions state the Golden Rule in a limited way, such as not doing things to others that you would not want them to do to you. It's the idea of refraining from doing evil. Under that concept of the Golden Rule, we simply need to avoid doing evil to our neighbors. There is no compulsion to do good to them. Ignoring your neighbor would be perfectly acceptable on this less golden iteration of the principal.
Most major world religions do not express the Golden Rule positively, as Jesus did: do unto others what you would have them do unto you. In this statement of the principal, doing unto others is an affirmative duty. Simply refraining from doing them evil is not the concept of the Golden Rule expressed by Jesus.
Jesus made this clear in the parable of the Good Samaritan. The parable begins with a man who was robbed and left injured on the road. A priest and Levite (the priestly cast of Judaism) walked by the man on the other side of the road, ignoring him, while a Samaritan (an outcast to Jews) crossed the road to tend to the injured man. The good Samaritan was the example of the person who demonstrated love for a "neighbor" because he didn't just ignore the injured man lying in the road. The idea of the Golden Rule that Jesus expressed includes an affirmative duty to do good.
To be fair, some religions come close to an affirmative expression of the Golden Rule, which I affirm in the previous blog piece, but there is one additional expression of the Golden Rule that stands alone: that is the concept of loving even our enemies and doing good to those who intend evil toward us.
I think of these things as I pause from listening to Douglas Murray in a discussion with Esther Riley on the Unbelievable? podcast with Justin Brierley, the host. (See Douglas Murray and Esther O'Reilly – Christian Atheism and the search for identity. The video is embedded below.)
Douglas Murray, an atheist and openly gay man, makes the observation that most Christian tenets can be found in other cultures, save one: that is the principal that of loving and forgiving even our enemies. Loving and forgiving our enemies is the ultimate statement of the Golden Rule.
Even when we have enemies who intend to do us harm, and even when they actually do us harm, Jesus says, "Forgive them." The conversation got into some recent examples of that expression of love and forgiveness that I will explore.
The first example involved the white, female police officer who came home to her apartment building after a long shift, entered into what she thought was her apartment, and, finding a man there, she shot him. She claimed that she was tired after her shift, went to the wrong floor, right apartment number, by mistake, and she thought it was her apartment.
Her first instinct upon seeing a man in the apartment was to draw her gun and shoot. What made this case such a public spectacle was the fact that the young man in her apartment was black, and her first instinct was to shoot him.
It was his apartment. Botham Jean was shot in his own living room because the instinctive assumption of a white cop (to put it bluntly) was that he was dangerous.
Of course, the case reveals on public display the insidious nature of latent racism. Would she have reacted the same way if she found a young white man sitting on the coach?
As if that racial element to the story was not enough to draw national attention, what happened at the trial catapulted the story into the worldwide spotlight. After the trial was concluded, and the verdict of "guilty" was pronounced, the victim's brother stood up in court and asked the judge to do the unthinkable: he asked for permission to give the white, racist offender a hug. Words don't do this act of extreme love and forgiveness justice: here is the video:
Botham Jean's brother demonstrated this extreme act of love and forgiveness because of his Christian faith.
This is the Golden Rule as Jesus expressed it in action.
The judge in the case was an African-American woman. When the convicted former cop asked the judge if she thought God would forgive her, the judge responded by also giving her a hug and handing her a Bible.
The act of the judge sparked anger, as if it diminished or white-washed the ugly racial quality of the crime. The judge responded this way:
"Following my own convictions, I could not refuse that woman a hug. I would not," Kemp told the Associated Press. "And I don't understand the anger. And I guess I could say if you profess religious beliefs and you are going to follow them, I would hope that they not be situational and limited to one race only."
(See Judge defends giving Amber Guyger hug and a Bible, saying she 'could not refuse') Her statement echoes the parable of the Good Samaritan.
A primary point of the Good Samaritan story is that the Samaritan, not the priest or the Levite, was good because he did the right thing (not because of who he was or wasn't).
There is a form of color-blindness that ignores (to our shame and detriment) the reality of real injustice and the insidious nature of latent racism that still exists in our society. This very case is an example of the reason why we shouldn't use color-blindness to cover up the vestiges of racism that still exist and need to be exposed. When the gut reaction of a white person is to shoot a black person first and ask questions later, we know we have a problem!
But these very sensitive and ugly points to the story shouldn't detract from the beautiful and authentic demonstration of love and forgiveness from Botham Jean's brother and the judge that was inspired by their Christian faith.
Another example comes to mind that, unfortunately, has the same racial overtones. A young white supremacist walked into a black church in Charleston, NC and shot up the congregation on a Sunday morning, killing nine people and wounding many others. It was a horrific crime that was motivated by the offender's open racism, and the offender wasn't the least bit sorry or repentant for what he did.
Very shortly after the incident, relatives of the slain church members asked the judge for permission to address the unrepentant shooter directly. When they were given permission, one by one, they offered him forgiveness, even as they expressed the pain of their loss, and though he remained unrepentant. (See 'I forgive you.' Relatives of Charleston church shooting victims address Dylann Roof)
This is a radical example of individual forgiveness and corporate forgiveness inspired by the words of Jesus that we are to love and forgive even our enemies. The only act that might be more extreme would be the act of giving up one's life for another, even for an enemy, which Jesus did by going to his own death on the cross where he said, "Father forgive them…." (Luke 22:33-34), as he was being crucified and the soldiers cast lots to divide up his clothing.
The discussion between Douglas Murray and Esther Riley focused on the social commentary that followed both of these demonstrations of forgiveness. Many people were angry that the public outcry against racism and the demonstration of justice was overshadowed by the story of the forgiveness shown by the victims' families.
People wanted to focus on the racism and meting out the justice, but Murray and Riley opined that the most poignant aspect of both stories was not the racism or the demonstration of justice, but the radical love and forgiveness inspired by Christian Faith that was perfectly modeled by what Jesus said and did in his own life.
While the story could be summarized as another incident of the oppressed always finding themselves in the position of victim and having to forgive their oppressors, the example of the African American judge provides a welcome twist. She did not stand in the place of the oppressed, but in the place of the ultimate authority – the symbol of justice and judgment.
Her demonstration of forgiveness and offer of redemption is also symbolic of God, our Judge, who offers all of us forgiveness and redemption … if we will accept it.
While I don't want to discount the history of oppression of African-Americans in the United States, we shouldn't miss the poignancy of the radical nature of Christian love and forgiveness that emerged in these circumstances. These stories are, in a sense, the Good Samaritan parables of our day, but they aren't just parables; they are all the more poignant because they are real-life examples.
I don't want to stretch the analogy too far, but I think we can stretch it a little bit further. These two very clear examples of the radical nature of distinctly Christian love are similar to the Good Samaritan parable in the aspect of racism. While Samaritans were kin to the Jews of the first century, they were viewed as second class citizens. Upstanding Jews of that time didn't mix with Samaritans, and Samaritans didn't have the privileges that Jews had. In both the parable and the modern stories, the people considered to be "lesser" in some way were the heroes. They were the ones demonstrating God's love.
Another thing I don't want to miss in all of this is the significance of Douglas Murray making the observation that Christianity stands apart from all other religions in this radical concept of loving and forgiving even our enemies. The significance of Douglas Murray acknowledging this point is that he is an atheist. Yet, he acknowledges the ethical eminence of this notion of radical love and forgiveness that is uniquely Christian.
The discussion between Douglas Murray, Esther Riley (who is unapologetically Christian) with Justin Brierley is linked in its entirety below of you are interested:
Postscript: I will add this postscript. The radical love and forgiveness that is uniquely Christian in construct, is not merely an ethical principal. It demonstrated by Jesus, who Christians believe was the exact embodiment of God, the Creator of the universe, in human form. Jesus shows us that God is, in His very nature, love, and invites us into relationship with Him.
Let me provide one more example of the radical expression of love. This one is involves a woman who I have heard speak personally. Her name is Jeanne Bishop. I will link to a Ted Talk below for her story.
She is a former prosecuting attorney who worked out of the Cook County State's Attorney Office. Her sister, who was pregnant, and her brother-in-law were murdered in their own home when they walked in on an intruder.
It was a violent and senseless crime committed by a young high school student who entered the home with a glass cutter and sat, waiting for his victims to show up. He handcuffed Jeanne's brother-in-law, and shot him point blank in the head while his wife, Nancy, watched, despite pleas to give him money to leave. Then he shot Nancy in the stomach and left her to bleed to death on the basement floor with her lifeless baby inside her. (See Jeanne Bishop recalls sister's Winnetka murder 25 years later)
"[T]he coroner's report estimated Nancy lived for another 10-15 minutes. Nancy crawled to a metal shelf, and using her own blood, left a simple message. It was a heart, followed by the letter 'U.'
"Nancy then crawled over to Richard's body, and died next to her husband."
The prosecutor, whose profession often results in a certain jaded mentality toward criminals, found herself in the dilemma of the Christian who is called to forgive even her enemies. Through some very difficult personal struggle, she was able to do that. Her story, the one I heard her tell, is about the way the process worked in her life.
After some time and much difficulty, she was able to say that she forgave the young man in her heart, but she wanted to have nothing to do with him. Over time, she realized this was not enough, that Christ urges us to do more, that God was gently urging her to do more.
Motivated by her faith, and what she believed God was urging her to do, she met with the offender in prison and forgave him face-to-face. She came to know him as a person. Through the experience of doing the hard thing, forgiving him in person and getting to know him, she was eventually motivated to change her profession from being a public prosecutor to being a public defender of people like him.
This is what the love of Jesus, through the Holy Spirit's intimate work in our individual hearts, can accomplish. He (not an ethical principal) works within us to change us from the inside out. Only then can such a dramatic act as forgiving a killer of one's own sister (pregnant with child) and her husband can happen. And not just that, the once prosecutor became a compassionate public defender when she understood in a way that few could possibly understand the humanity of a criminal.
These acts of uniquely Christian love and forgiveness are truly supernatural. We are not capable of them in and of ourselves. Only by the power of a transcendent God who experienced the darkness of evil in human flesh Himself are we able to rise above it.
We still have racial barriers, but not just racial barriers. We have ethnic, gender, national, sexual, political, socioeconomic, worldview, emotional and other barriers that wall people out from each other. Jesus breaks down walls, not as an ethical principal or even a force: Jesus himself is our peace (Ephesians 2:14). He brings light into our darkness because He is our light. (John 8:12) It isn't an ethical principal that makes the difference; it is the Person of Jesus who desires to dwell in us through the Holy Spirit who makes all the difference.
Posted in Christian, Current Events, Faith, Forgiveness, Justice, Love, MercyTagged Botham Jean, Douglas Murray, Esther Riley, forgiveness, God is love, Golden Rule, Jeanne Bishop, love, love your enemy, Unbelievable!
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Over 12,000 tourists visited Langkawi from Sept 16-22, says exco
Friday, 24 Sep 2021 07:48 PM MYT
Passengers boarding the AirAsia flight to Langkawi at the Sultan Azlan Shah Airport in Ipoh September 17, 2021. State Tourism, Arts and Culture, Youth and Sports, Entrepreneur Development and Cooperatives Committee chairman Mohd Firdaus Ahmad said 8,587 tourists came to Langkawi by air, while 4,020 used the ferry service. ― Bernama pic
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ALOR SETAR, Sept 24 — A total of 12,607 tourists have visited Langkawi since the reopening of the tourism industry on the resort island under the tourism bubble pilot project from September 16 up till Wednesday.
State Tourism, Arts and Culture, Youth and Sports, Entrepreneur Development and Cooperatives Committee chairman Mohd Firdaus Ahmad said the number involved 11,853 domestic tourists and 754 foreign tourists.
"There were 8,587 tourists who came to Langkawi by air, while 4,020 used the ferry service," he said at a press conference after visiting the Kuala Kedah Ferry Terminal here today.
He said the Ministry of Tourism, Arts and Culture had previously targeted about 30,000 tourists to visit Langkawi this month, but it is not likely to be achieved due to the strict standard operating procedures imposed on tourists.
Mohd Firdaus stressed, however, that the mandatory Covid-19 early detection test conducted on tourists was important to control the spread of the disease among those who visit Langkawi.
"Maybe now the number of tourists has decreased a little but in November and December, we expect the number to go up, as the school holidays would have begun at the time," he said.
Meanwhile, state Agriculture and Agro-Resources, Transport and Primary Industries Committee chairman Azman Nasrudin, who was also present, said the number of ferry trips to Langkawi would be increased if there was a higher demand from tourists.
"Currently, there are only two ferry trips a day, one from Kuala Kedah to Langkawi Island and one trip from Langkawi Island to Kuala Kedah. If the number of tourists increases, we will ask the Ministry of Transport to allow more ferry trips," he said. — Bernama
Sarawak to continue with plans to open trade, tourism offices in Brunei, Kalimantan, says state minister
UN: World tourism won't return to pre-pandemic levels until 2024
Sarawak Tourism Board: Arab tourism players keen to promote state
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Chris Eckman BANDCAMP
A portal for my work as a musician, record producer and soundtrack composer. You will find concert listings and updates for my various projects. Thanks for stopping by.
Early Snow (from "Where the Spirit Rests") 6:03
Atmospheres.1: "Pisatelj in mesto" 1:50
Theme: Hidden Memories of Angela Vode 1:46
Theme: "Pistelj in Mesto" 2:20
German Record Critics Prize (Preis der deutschen Schallplattenkritik)
This is one I didn't see coming. Though I have known this for a couple of days, today I can announce that my solo album "Where the Spirit Rests" is one of the ten winners of the 2021 "German Record Critics Prize" (Preis der deutschen Schallplattenkritik). This long running juried prize honors a cross-section of styles and genres and this year the selected records include a Mahler symphony and the live album from the incredible Angel Bat Dawid & Tha Brothahood.
https://www.schallplattenkritik.de/.../die-jahrespreise-2021
Bandcamp presale & video for new solo album "Where the Spirit Rests"
Today we launched the pre-sale for my new solo album via Bandcamp. The album will be available via Glitterhouse (Europe) and Drums & Wires (North America). It will be out on June 4th. You can order the LP / CD / download here: https://bit.ly/33tXhp
Here's the video for the album's title track: https://bit.ly/3vMLGOE
The images for the video were generously loaned to me by Robin Mellway. I found his gorgeous super 8 films of the north american west on Youtube. I am very thankful that he allowed me to include his work in the video as I think there is a strong connection between the song and these images.
NEW SOLO SONG: "Early Snow"
Here is the image video for my new song "Early Snow":
https://www.youtube.com/watch?v=2eLvPeDtBfc
This was the first song I finished writing for the upcoming solo album - almost exactly a year ago. Alastair McNeill joins me on analog synth and production.
The album "Where the Spirit Rests" is out on June 4th. It's available for pre-order via Glitterhouse mail order and "Early Snow" can be streamed at the usual places: https://linktr.ee/ChrisEckman
2018 Album Releases
After a few years of not making any new music, things got a little bit carried away. The 2018 discography:
Dirtmusic - Bu Bir Rya
The Strange - Echo Chamber
Distance, Light & Sky - Gold Coast
Katalena - Človek ni zver (producer)
Chloe Mons - Hôtel de l'univers (producer)
A big thanks to all the stand up, hyper-talented people I worked with on these. I feel damn lucky to have been included.
Bassekou Kouyate & Ngoni Ba: BA POWER
I produced the latest album for Bassekou Kouyate & Ngoni Ba. We recorded it in Bamako, Mali in November and I mixed it at Studio Zuma in Ljubljana.
Distance, Light & Sky debut album is out!
Distance, Light & Sky is a new band I have with Chantal Acda & Eric Thielemans. The album was recorded and mixed by Phill Brown (Talk Talk/ Beth Gibbons/ Bob Marley) and was just released by Glitterhouse Records. We will be touring in April. Check out the opening track "Son" at Soundcloud:
https://soundcloud.com/glitterhouse/01-son
Glitterhouse releases Walkabouts reissues
Deluxe Reissues of their 1996 and 1997 classics with re-worked and extended artwork plus a bonus CD containing five unreleased tracks of that period. Comes in a multifold digipak with 16 page booklet and as a two-LP set (180 gram heavyweight vinyl, gatefold sleeve) plus two CDs! Also available is a boxed set with both albums plus the live album from the 1997 orchestra tour: Bruxelles.
http://label.glitterhouse.com/releases.php?show=197
Interview in Abus Dangereux (FR)
Interview about "Harney County" and the new Dirtmusic album in the French magazine: Abus Dangereux
"Artists On Albums" piece for DOA
I wrote an piece for the DOA website about the influence of the Eno/Byrne album "My Life in the Bush Of Ghosts" on my listening and musical path.
http://www.adequacy.net/2014/04/artists-on-albums-aoa40-chris-eckman-on-my-life-in-the-bush-of-ghosts/
Saturday, October 15, 2022 Sat, Oct 15, 2022 Chris Eckman (trio) Club il Giardino, Lugagnano, IT Club il Giardino, Lugagnano, IT
Friday, October 14, 2022 Fri, Oct 14, 2022 Chris Eckman (trio) Circolo Il Progresso, Florence, IT Circolo Il Progresso, Florence, IT
Thursday, October 13, 2022 Thu, Oct 13, 2022 Chris Eckman (trio) TBA, Rome, IT TBA, Rome, IT
Wednesday, October 12, 2022 Wed, Oct 12, 2022 Chris Eckman (trio) Germi luogo di contaminazione, Milan, IT Germi luogo di contaminazione, Milan, IT
Tuesday, October 11, 2022 Tue, Oct 11, 2022 Chris Eckman (trio) Raindogs House , Savona, IT Raindogs House , Savona, IT
Sunday, October 9, 2022 Sun, Oct 9, 2022 Chris Eckman (trio) Freakout Club , Bologna, IT Freakout Club , Bologna, IT
Saturday, June 4, 2022 Sat, Jun 4, 2022 Chris Eckman (solo) + Jim White Cactus Club, Brugge, BE Cactus Club, Brugge, BE
Friday, June 3, 2022 Fri, Jun 3, 2022 Chris Eckman (solo) Muziekgieterij, Maastricht, NL Muziekgieterij, Maastricht, NL
Thursday, June 2, 2022 Thu, Jun 2, 2022 Chris Eckman (solo) Ancienne Belgique , Brussels, BE Ancienne Belgique , Brussels, BE
Wednesday, June 1, 2022 Wed, Jun 1, 2022 Chris Eckman (solo) Rock Lobster, Antwerp, BE Rock Lobster, Antwerp, BE
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J. R. Smith Biography, Age, Height, Weight, Family, Wiki & More
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J. R. Smith
9 September,1985 (Monday)
Freehold Borough, Monmouth, New Jersey
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J. R. Smith was born on 09-09-1985 in Freehold Borough, Monmouth in the state of New Jersey, United States. He is an American Basketball Player.
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Luperina rotroui är en fjärilsart som beskrevs av Rothschild 1920. Luperina rotroui ingår i släktet Luperina och familjen nattflyn. Inga underarter finns listade i Catalogue of Life.

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text:
\section{Introduction and analogy between waves in Rayleigh beams and couple-stress elastic materials}
\label{sec01}

Bloch waves in structured media have received a significant attention in models of photonic and phononic crystals, that encompass problems of electro-magnetism, optics, acoustics and more recently elasticity \cite{Yablonovitch1987,John1987,Kushwaha1993,Sigalas1993}.
An important feature of elastic waves, even in an isotropic case, is the presence of two types of waves linked to dilatation and shear, respectively.

More recently, there was a significant interest generated by studies of micropolar media and couple-stress materials
(see, for example, \cite{Morini2013b,Engelbrecht2013,Zisis2014,Gourgiotis2014}).
In particular, the Mindlin's approach \cite{Mindlin1964} leads to additional higher-order derivatives in the governing equations. Engelbrecht et al.\ \cite{Engelbrecht2005} follow Mindlin's interpretation of a micro-structure as a polycrystal, whose micro-elements are taken as deformable cells. In the limit when the cells are rigid, their approach would lead to the Cosserat model. It was noted that the higher-order model leads to the novel dispersion properties of waves supported by such a micro-structured medium.
We also note a formal equivalence between constrained Cosserat model and the couple-stress model of structured media (see, for example, \cite{Koiter1964,Mindlin1962}), and a further analogy with the theory of Rayleigh beams which accounts for rotational inertia, which will be discussed in the text below.

When rotational inertia is neglected in analysis of flexural waves, this is fully appropriate in the long wave approximations and is well adopted to the Euler-Bernoulli beam theory. Dispersion and filtering of elastic waves in structured prestressed Euler--Bernoulli beams were considered in \cite{Gei2009},
which includes analysis of band-gap shift, defect-induced annihilation and localized modes.
It is also known that the couple-stress effects are neglected in the classical models of linear elasticity.

It appears that there is a mathematical underlying framework, which applies both to the couple-stress approach in elasticity as well as flexural waves in beam models accounting for the effects of rotational inertia.

The Rayleigh beam theory is used here to account for the rotary motion of beam elements. This approach also allows for the description of flexural waves  at high frequency ranges \cite{Graff1991}.
In the Rayleigh beam theory, the assumptions regarding the geometry of the deformation and the material properties remain unchanged, so that the rotation
$\phi$ of the cross sections is not an independent parameter, but it is {\em constrained} to the transverse displacement $v$ by the relation $\phi = v'$.
However, in writing the equations of motion, both the translational inertia and the rotational inertia of beam elements are taken into account, so that
\begin{equation}
V' = \beta v + \rho A \ddot{v}, \quad M' = V - P v' - \rho I \ddot{v}',
\end{equation}
where $V$ is the internal shear force, $M$ the internal bending moment, $\beta$ the stiffness of a Winkler type elastic foundation, $P$ the prestress,
$\rho$ the mass density, $A$ the area of the cross-section, and $I$ the area moment of inertia of the cross-section.

Combining the equations of motion with the {\em constitutive} equation $M = -EI v''$, where $EI$ is the bending stiffness, we obtain the governing equation
for the transverse motion $v$ of a homogeneous prestressed Rayleigh beam resting on an elastic foundation of the Winkler type
\begin{equation}
\label{ray}
EI\, v'''' - P\, v'' + \rho A\, \ddot{v} - \rho I\, \ddot{v}'' + \beta v = 0.
\end{equation}

We note that this physical problem has a formal mathematical analogy with the problem of shear wave propagation in couple-stress elastic materials.
In fact, the dynamic equation governing the anti-plane motion in a couple-stress elastic material is given by \cite{Mishuris2012}
\begin{equation}
\label{cstress}
G \ell^2\, \Delta^2 w - 2G\, \Delta w + 2\rho\, \ddot{w} - \frac{J}{2}\, \Delta \ddot{w} = 0,
\end{equation}
where $w$ is the out-of-plane displacement; the shear modulus $G$ and the mass density $\rho$ are the classical {\em macroscopic} material parameters, whilst the characteristic length
$\ell$ and the micro-rotational inertia $J$ are the generalized parameters connected with the {\em microstructure}.
Clearly, equation (\ref{cstress}) is the two-dimensional analogue of the Rayleigh beam equation (\ref{ray}) with $\beta = 0$.

Both equations (\ref{ray}) and (\ref{cstress}) are not  classical wave equations, because of the fourth-order terms. In the case of the Rayleigh beam, the coefficients near fourth-order terms include the bending stiffness $EI$ and the rotational inertia $\rho I$, showing that they are essentially linked to the geometrical properties of the structural element, in particular the area moment of inertia of the cross-section.
On the other hand, in the case of a couple-stress material, the fourth-order terms are proportional to the microstructural parameters $G\ell^2$ and $J$. In both cases, these higher-order terms are responsible for the dispersive character of wave propagation.

In several studies on wave propagation in generalized continua \cite{Mindlin1964,Gourgiotis2013,Morini2014}, it has been shown that elastic waves can be made non-dispersive for special values of the rotational inertia. In particular, for shear waves in couple stress materials described by eq.\ (\ref{cstress}), this value is given by $J = 2 \rho \ell^2$ \cite{Mishuris2012}. By analogy, we obtain that flexural waves in a Rayleigh beam are non-dispersive for a special value of the prestress, namely $P = EA$.

The key features characterising a dynamic response of a Rayleigh beam on the Winkler foundation are linked to the presence of exponential and quasi-polynomial terms in the representation of flexural displacements. In particular,  exponential terms account for the high gradient regions, and of course they can be seen in the representation of dynamic Green's functions.
For the purpose of illustration, we show in Fig.~\ref{fig00} the graphs of flexural displacements $v(x)$ produced by the time-harmonic point force applied to the Rayleigh beam at the origin.
These represent the special resonant cases when the solution has a linear growth at infinity; the oscillation in Fig.~\ref{fig00}b is produced as a result of a compressive prestress.

\begin{figure}[!htcb]
\centering
\includegraphics[width=15cm]{fig00.pdf}
\caption{\footnotesize Flexural vibrations produced by a time-harmonic point force applied at the origin to a Rayleigh beam on the Winkler foundation in the special resonant cases when the solutions have a linear growth at infinity (see Section \ref{sec04}, formulae (\ref{GF40}) and (\ref{GF41})). We plot several profiles captured at different times within the period of the oscillatory force. A beam with circular cross section is assumed having the following properties: Young modulus 210 GPa, mass density 7.85$\cdot$10$^3$ kg/m$^3$, foundation stiffness 2.64 MPa, radius of the cross section 0.01 m. The angular frequency is 1034.4 s$^{-1}$. Transversal displacements $v(x)$ are normalised with respect to the maximum value at the origin $v_\text{max}(0)$, whereas the longitudinal position $x$ is normalised with respect to the radius of the cross-section $a$. (a) Tensile prestress equal to 1.32 kN. (b) Compressive prestress equal to -1.19 kN. }
\label{fig00}
\end{figure}

The structure of the paper is as follows. Section \ref{sec02} presents a detailed account of dispersion properties of flexural waves in a homogeneous prestressed Rayleigh beam supported by a Winkler foundation.
In Section \ref{sec03} we consider a periodic multi-phase medium, with Bloch waves representing quasi-periodic solutions of the equation of motion. Analysis of band gaps and filtering properties is included in this section. Section \ref{sec04} includes analytical closed form representations for dynamic Green's functions and their derivatives (concentrated couples), that describe the full range of admissible localised and propagating wave forms. Finally, in Section \ref{sec05}, we construct a special class of so-called quasi-periodic Green's functions, required to study a dynamic response of periodic systems of masses placed along the Rayleigh beam; this also includes periodic system of bodies with given rotational inertia. The coupling between the rotational action and transverse motion of masses appears to be important, which is discussed in that section.
The analysis is generic and our systematic study is applicable to flexural systems such as plates and shells, especially for the cases where rotational inertia appears to provide a significant contribution.

\section{Dispersion properties of a homogeneous prestressed Rayleigh beam on an elastic foundation}
\label{sec02}

The dispersion relation for a homogeneous prestressed Rayleigh beam, governed by \eq{ray}, is obtained by assuming that a sinusoidal signal $v = \xi \exp i(kx - \omega t)$ is propagating in the beam, which gives
\begin{equation}
\label{disp}
\omega = \sqrt{\frac{k^2r^2(k^2r^2 + \overline{P}) + \overline{B}}{R(k^2r^2 + 1)}},
\end{equation}
where $r$ is the radius of inertia of the beam cross-section
\begin{equation}
r = \sqrt{\frac{I}{A}},
\end{equation}
$\overline{P}$ and $\overline{B}$ are the dimensionless prestress and foundation stiffness, respectively,
\begin{equation}
\overline{P} = \frac{Pr^2}{EI} = \frac{P}{EA}, \quad
\overline{B} = \frac{\beta r^4}{EI} = \frac{\beta I}{EA^2},
\end{equation}
and $R$ is the normalized inertia term, having the dimension of a squared time
\begin{equation}
R = \frac{\rho r^2}{E} = \frac{\rho I}{EA}.
\end{equation}
Correspondingly, the phase velocity $c$ and the group velocity $c_g$ are given by
\begin{equation}
c = \frac{r}{\sqrt{R}} \sqrt{\frac{k^2r^2 (k^2r^2 + \overline{P}) + \overline{B}}{k^2r^2(k^2r^2 + 1)}}, \quad
c_g = \frac{r^2}{R} \frac{k^4r^4 + 2k^2r^2 + \overline{P} - \overline{B}}{c(k^2r^2 + 1)^2}.
\end{equation}

\subsection{Asymptotics limits and the action of prestress}

The solution of the dispersion equation (\ref{disp}) has the following properties:
\begin{enumerate}
\item In the long-wavelength limit, $kr \to 0$, the angular frequency behaves as
\begin{equation}
\omega = \sqrt{\frac{\overline{B}}{R}} + \frac{\overline{P} -\overline{B}}{2\sqrt{R\overline{B}}} k^2r^2 + O(k^4r^4).
\end{equation}
\item In the short-wavelength limit, $kr \to \infty$, the angular frequency behaves as
\begin{equation}
\omega = \frac{1}{\sqrt{R}} kr + \frac{\overline{P} - 1}{2\sqrt{R}(kr)} + O(k^{-3}r^{-3}).
\end{equation}
\item For a compressive prestress, $\overline{P}$ < 0, (or tensile prestress, satisfying $\overline{P} < \overline{B}$) the angular frequency displays a minimum $R\omega_{\text{min}}^2 = 2\sqrt{1 + \overline{B} - \overline{P}} + \overline{P} - 2$ for $k^2r^2 = \sqrt{1 + \overline{B} - \overline{P}} - 1$. Consequently, buckling occurs when both phase and group velocities are zero, $c = 0$ and $c_g = 0$. These conditions are met for a compressive prestress $\overline{P}$ satisfying $\overline{P}_{\text{buckl}} = -2\sqrt{\overline{B}}$ with $k_{\text{buckl}}^2r^2 = \sqrt{\overline{B}}$.\footnote{
The buckling load can be obtained also from the static equation $EI v'''' - P v'' + \beta v = 0$, which admits a non-trivial and bounded solution only for compressive prestress, $\overline{P} < 0$, and non-negative determinant of the characteristic equation, $\Delta = \overline{P}^2 - 4\overline{B} \geq 0$.
}
\end{enumerate}

\subsection{Dispersion diagrams for Rayleigh beam versus Euler-Bernoulli beam}

Three diagrams in Fig.~\ref{fig01} include the graphs of normalised frequency, phase and group velocities of flexural waves as functions of $kr$ for a homogeneous prestressed Rayleigh beam and $\overline{B} = 0$.
Curves are plotted for five values of normalised prestress $\overline{P} = \{0,0.2,0.5,1,1.5\}$. Dispersion diagrams for a homogeneous prestressed Euler--Bernoulli beam are also plotted for comparison (dashed lines).
In the absence of the elastic foundation, the Rayleigh and the Euler-Bernoulli beams deliver a similar dynamic response in the low frequency regimes. However, as the frequencies rise, the behaviour of elastic waves propagating along the Rayleigh and the Euler-Bernoulli beams is notably different. In contrast with the Euler--Bernoulli beam, where dispersive flexural waves show quadratic growth of frequency as a function of $k r$, the waves in the Rayleigh beam become asymptotically non-dispersive in the high-frequency regimes, as their group and phase velocities tend to a constant as $k r  \to \infty$. The action of prestress is especially important in the low frequency regime, as the higher positive prestress leads to an increase in the phase and group velocities of flexural waves. Contribution from prestress appears to be small at higher frequencies.

\begin{figure}[!htcb]
\centering
\includegraphics[width=16cm]{fig01.pdf}
\caption{\footnotesize Dispersion diagrams for a homogeneous prestressed Rayleigh beam. (a) Normalised angular frequency $\sqrt{R}\omega$ vs.\ normalised wave number $k r$; (b) Normalised phase velocity $\sqrt{R} c / r$ vs.\ normalised wave number $k r$; (c) Normalised group velocity $\sqrt{R} c_g / r$ vs.\ normalised wave number $k r$. Curves are plotted for five values of prestress $\overline{P} = \{0,0.2,0.5,1,1.5\}$. Dispersion diagrams for a homogeneous prestressed Euler--Bernoulli beam are also plotted for comparison (dashed lines). Note that for a Rayleigh beam and for prestress $\overline{P} = 1$ waves are non-dispersive.}
\label{fig01}
\end{figure}

Similarly, in Fig.~\ref{fig02}, we show the graphs of normalised frequency, phase and group velocities for elastic configurations that include the Winkler foundation ($\overline{B} = 0.01$). Curves are plotted for six values of normalised prestress $\overline{P} = \{-0.199,0,0.2,0.5,1,1.5\}$. Although it does not produce significant change at high frequencies, the changes observed in the low frequency regimes are dramatic. It is important to note non-monotonic behaviour of $\sqrt{R}\omega$ as a function of $kr$ both for the Rayleigh and the Euler--Bernoulli beams; the local minimum point, where the group velocity is zero corresponds to a standing wave supported by the action of compressive prestress. In particular, when the magnitude of the compressive prestress reaches the level, such that the point of minimum corresponds to the zero frequency, we observe buckling of the beam - in this very special case both Rayleigh and Euler--Bernoulli beams respond in the same manner.
Note that the lower dispersion curve in Fig.~\ref{fig02}a corresponds to compressive prestress $\overline{P} = -0.199$ and the beam is near the onset of buckling.
For the chosen value of foundation stiffness $\overline{B} = 0.01$, the buckling prestress is $\overline{P}_{\text{buckl}} = -0.2$. As $\overline{P} \to \overline{P}_{\text{buckl}}$, both phase and group velocities tend to zero for the same value of the wave number, $kr = 0.01^{1/4}$, corresponding to a buckled configuration with wavelength $\lambda/r = 2\pi/(kr) = 19.8692$.

\begin{figure}[!htcb]
\centering
\includegraphics[width=16cm]{fig02.pdf}
\caption{\footnotesize Dispersion diagrams for a homogeneous prestressed Rayleigh beam on an elastic foundation ($\overline{B} = 0.01$). (a) Normalised angular frequency $\sqrt{R}\omega$ vs.\ normalised wave number $k r$; (b) Normalised phase velocity $\sqrt{R} c / r$ vs.\ normalised wave number $k r$; (c) Normalised group velocity $\sqrt{R} c_g / r$ vs.\ normalised wave number $k r$. Curves are plotted for six values of prestress $\overline{P} = \{-0.199,0,0.2,0.5,1,1.5\}$. Dispersion diagrams for a homogeneous prestressed Euler--Bernoulli beam are also plotted for comparison (dashed lines). Note that for compressive prestress $\overline{P} = -0.199$ the beam is near the onset of buckling, $\overline{P}_{\text{buckl}} = -0.2$.}
\label{fig02}
\end{figure}

\section{Dispersion and band-gaps for a structured Rayleigh beam}
\label{sec03}

In this section we analyse the dispersion properties of Bloch-Floquet flexural waves within a prestressed structured periodic Rayleigh beam on an elastic foundation of the Winkler type.
Analytical estimates are derived  for the boundaries of stop bands, i.e. the intervals of  frequencies at which the waves are evanescent. Exponentially localised waveforms occur at these frequencies when a point force is applied to the Rayleigh beam, which highlights the importance of our results.

For each of the phases $m$ ($m = 1,...,N$)  within the unit cell of the periodic structure, Fig.~\ref{fig03a}, the time-harmonic transverse displacement $v_m$ satisfies the following ordinary differential equation
\begin{equation}
\label{gov}
E_m I_m v_m'''' - (P - \rho_m I_m \omega^2) v_m'' + (\beta - \rho_m A_m \omega^2) v_m = 0.
\end{equation}
The internal bending moment $M_m$ and the internal shear force $V_m$ are given by
\begin{equation}
M_m = -E_m I_m v_m'', \quad V_m = -E_m I_m v_m''' + (P - \rho_m I_m \omega^2) v_m',
\end{equation}
respectively.

\begin{figure}[!htcb]
\centering
\includegraphics[width=8cm]{fig_unit-cell.pdf}
\caption{\footnotesize Unit cell.}
\label{fig03a}
\end{figure}

The flexural displacements are sought in the form
\begin{equation}
\label{form}
v_m = C^{(m)} \exp(i k^{(m)} x),
\end{equation}
The substitution of (\ref{form}) into (\ref{gov}) yields the following equations for the circular frequency $\omega$
\begin{equation}
\label{quartic}
(k^{(m)} r_m)^4 + (\overline{P}_m - R_m\omega^2) (k^{(m)} r_m)^2 + \overline{B}_m -
R_m\omega^2 = 0,
\end{equation}
where we have introduced the following dimensionless parameters
\begin{equation}
\overline{P}_m = \frac{Pr_m^2}{E_mI_m}, \quad
\overline{B}_m = \frac{\beta r_m^4}{E_mI_m},
\end{equation}
and the normalized quantity
\begin{equation}
R_m = \frac{\rho_m A_m r_m^4}{E_mI_m}.
\end{equation}
The quartic (\ref{quartic}) admits the following four roots
\begin{equation}
\label{roots}
k^{(m)}_{1,2,3,4} = \pm \frac{1}{r_m} \sqrt{-\frac{\overline{P}_m - R_m\omega^2}{2}
\pm \sqrt{\frac{(\overline{P}_m - R_m\omega^2)^2}{4} + (R_m\omega^2 - \overline{B}_m)}}.
\end{equation}
Consequently, the general solution for the transverse displacement $v_m$ within each phase is given by
\begin{equation}
v_m(x) = \sum_{p=1}^{4} C_p^{(m)} \exp(ik_p^{(m)}x),
\end{equation}
The $4N$ constants $C_p^{(m)}$ can be obtained by imposing the junction conditions at each internal interface of the
elementary block ($m = 1, ..., N-1$), namely continuity of displacement, rotation, bending moment, and the shear force,
\begin{equation}
\label{eq1}
v_m(x_m) = v_{m+1}(x_m),
\end{equation}
\begin{equation}
v_m'(x_m) = v_{m+1}'(x_m),
\end{equation}
\begin{equation}
-E_mI_m v_m''(x_m) = -E_{m+1}I_{m+1} v_{m+1}''(x_m),
\end{equation}
\begin{equation}
\label{eq2}
-E_m I_m v_m'''(x_m) + (P - \rho_m I_m \omega^2) v_m'(x_m) =
-E_{m+1} I_{m+1} v_{m+1}'''(x_m) + (P - \rho_{m+1} I_{m+1} \omega^2) v_{m+1}'(x_m).
\end{equation}
The remaining four equations follow from enforcing the Bloch-Floquet conditions at the boundary of the elementary block,
\begin{equation}
\label{eq3}
v_1(0) = v_N(d) \exp(-iKd),
\end{equation}
\begin{equation}
v_1'(0) = v_N'(d) \exp(-iKd),
\end{equation}
\begin{equation}
-E_1 I_1 v_1''(0) = -E_N I_N v_N''(d) \exp(-iKd),
\end{equation}
\begin{equation}
\label{eq4}
-E_1 I_1 v_1'''(0) + (P - \rho_1 I_1 \omega^2)v_1'(0) =
[-E_N I_N v_N'''(d) + (P - \rho_N I_N \omega^2)v_N'(d)] \exp(-iKd),
\end{equation}
where $K$ is the Bloch parameter. Equations (\ref{eq1})--(\ref{eq2}) and (\ref{eq3})--(\ref{eq4})
provide a homogeneous linear system for the $4N$ unknown constants $C_p^{(m)}$, $p =
1,...,4$, $m = 1,...,N$, so that the vanishing of the determinant of the associated matrix yields the
dispersion equation for the structured Rayleigh beam.

\paragraph{Analytical estimates of band gap positions.}

A simple formula can be obtained for the estimation of band gap positions by considering homogeneous beams with physical characteristics equal to each of the constituents of the periodic beam.

For each of these homogeneous beams, the dispersion curve is unique and is given by eq.\ (\ref{disp}).
Now, we shift the dispersion curve by multiples of $2\pi$, so that the dispersion diagram becomes artificially periodic, see Fig.~\ref{fig03}a. The intersections of the shifted curves are given by
\begin{equation}
\label{intersec}
\omega_m^n = \sqrt{\frac{(\frac{n\pi}{d})^2r_m^2[(\frac{n\pi}{d})^2r_m^2 + \overline{P}_m] + \overline{B}_m}{R_m[(\frac{n\pi}{d})^2r_m^2 + 1]}}, \quad n = 0,1,2,3,...
\end{equation}
Note that for even values of $n$, we obtain the intersection points occurring at $K d = 0$, whereas for odd values of $n$, intersections occur at $K d = \pi$.

The formula (\ref{intersec}) can be used to estimate the position of the band gaps, as it is shown in Fig.~\ref{fig03} for the case of two constituents, having the same stiffness, $E_1 = E_2$, and cross-section, $A_1 = A_2$, $I_1 = I_2$, but different mass density (with contrast $\rho_2/\rho_1 = 0.8$  and $0.6$ in Fig.~\ref{fig03}b and \ref{fig03}c, respectively). In this example, the normalized prestress is $\overline{P}_1 = \overline{P}_2 = 0.1$ and the normalized Winkler stiffness $\overline{B}_1 = \overline{B}_2 = 0.0001$. The unit cell is chosen such that $r/d = 0.015$ and $l_1 = l_2 = d/2$.

It is clear from the figure that, as the contrast in mass density increases, band gaps open up and, at the same time, they move upward. The predicted intersections for homogeneous beams having either mass density $\rho_1$ and $\rho_2$ also open up. These intersections are indicated in the figure by horizontal segments. Note that the segment corresponding to phase 1 remains fixed, while the segment corresponding to phase 2 move upward.
By taking the midpoint between two corresponding horizontal segments, it is possible to obtain a good prediction of the band gap positions, especially for small contrast.

\begin{figure}[!htcb]
\centering
\includegraphics[width=12cm]{fig03j.png}
\caption{\footnotesize Dispersion diagrams, representing a normalised circular  frequency $\sqrt{R_1}\omega$ vs.\ Bloch parameter
$K d$, for a Rayleigh beam on an elastic foundation with piecewise constant mass density ($E_1 = E_2$, $A_1 = A_2$, $I_1 = I_2$, $\overline{P}_1 = \overline{P}_2 = 0.1$, $\overline{B}_1 = \overline{B}_2 = 0.0001$, $r/d = 0.015$, $l_1 = l_2 = d/2$): (a) Homogeneous beam $\rho_1 = \rho_2$ (for the homogeneous beam the dispersion diagram has been made artificially periodic by shifting the curve by multiples of $2\pi$); (b) Periodic beam with density contrast $\rho_2/\rho_1 = 0.8$; (c) Periodic beam with density contrast $\rho_2/\rho_1 = 0.6$. Horizontal segments indicate the predicted intersections $\omega_m^n$, see eq.\ (\ref{intersec}), for homogeneous beams with mass density either $\rho_1$ or $\rho_2$. By taking the midpoint between the horizontal segments, a good prediction of the band gap positions is obtained, especially for small contrast.}
\label{fig03}
\end{figure}

\newpage

\section{Green's function for a Rayleigh beam. Localised and propagating wave forms}
\label{sec04}

In this section, we first construct the time-harmonic Green's function associated to a concentrated unit force applied to a infinite, homogeneous, prestressed Rayleigh beam on an elastic foundation. Also, we derive the solution associated to a concentrated couple.

On the basis of the fundamental solutions, we analyse localised wave forms (produced by a rigid body attached to a Rayleigh beam) as well as propagating modes.

\subsection{Concentrated force}

The time-harmonic Green's function $\ensuremath{\mathcal{G}}_F(x,t) = G_F(x) \exp(-i\omega t)$ corresponding to a transverse concentrated unit force placed at the origin $x = 0$ of a homogeneous Rayleigh beam (assumed positive when oriented as the displacement $v$) solves the equation
\begin{equation}
\label{delta}
G_F''''(x) - 2b\, G_F''(x) + c\, G_F(x) = \frac{\delta(x)}{EI},
\end{equation}
where $\delta(x)$ is the Dirac delta function and
\begin{equation}
b = \frac{\overline{P}}{2r^2} - \frac{R}{2r^2}\omega^2, \quad
c = \frac{\overline{B}}{r^4} - \frac{R}{r^4}\omega^2.
\end{equation}
Assuming that $c \neq 0$ and $b^2 - c \neq 0$, a solution of the corresponding homogeneous equation can be sought in the form $C \exp(ikx)$, which leads to the characteristic equation
\begin{equation}
\label{eqcar}
k^4 + 2b\, k^2 + c = 0.
\end{equation}

\paragraph{Stop band Green's function.}
Assuming a \lq forbidden' frequency $\omega$ within a band-gap, the four roots of the characteristic equation (\ref{eqcar}) possess non-zero imaginary parts, corresponding to a localised (or evanescent) mode.
We distinguish between two cases:
\begin{itemize}

\item Case 1:  Low frequency and high tensile prestress ($c \geq 0$, $b > 0$, $b^2 - c > 0$).

The four roots are purely imaginary and admit the following representation
\begin{equation}
k = \pm i k_{I_{1,2}}, \quad k_{I_{1,2}} = \sqrt{b \pm \sqrt{b^2 - c}}.
\end{equation}
In this case, the Green's function takes the form
\begin{equation}
G_F(x) = -\frac{1}{2EI(k_{I_1}^2 - k_{I_2}^2)} \left\{ \frac{1}{k_{I_1}}e^{-k_{I_1}|x|} -
\frac{1}{k_{I_2}}e^{-k_{I_2}|x|} \right\},
\end{equation}
which corresponds to a monotonic decaying mode, without oscillations.

\item Case 2:  Low frequency and low prestress (tensile or compressive) ($c > 0$, $b^2 - c < 0$).

The four roots are complex and admit the following representation
\begin{equation}
k = \pm (k_R \pm i k_{I}), \quad k_{R} = \sqrt{\frac{\sqrt{c} - b}{2}}, \quad
k_{I} = \sqrt{\frac{\sqrt{c} + b}{2}}.
\end{equation}
In this case, the Green's function takes the form
\begin{equation}
G_F(x) = \frac{e^{-k_I |x|}}{4EI(k_R^2 + k_I^2)}\left\{ \frac{1}{k_I}\cos(k_R|x|) +
\frac{1}{k_R}\sin(k_R|x|) \right\},
\end{equation}
which corresponds to a oscillating decaying mode.

\end{itemize}

\paragraph{Pass band Green's function.}
Assuming a frequency $\omega$ within a pass band, the characteristic equation (\ref{eqcar}) always admits real roots corresponding to a propagating wave. We distinguish between two cases:

\begin{itemize}

\item Case 3: Low frequency and high compressive prestress ($c \geq 0$, $b^2 - c > 0$, $b < 0$).

The four roots are real and admit the following representation
\begin{equation}
k = \pm k_{R_{1,2}}, \quad k_{R_{1,2}} = \sqrt{-b \pm \sqrt{b^2 - c}}.
\end{equation}
In this case the Green's function takes the form
\begin{equation}
G_F(x) = \frac{i}{2EI (k_{R_1}^2 - k_{R_2}^2)} \left\{ \frac{1}{k_{R_1}} e^{ik_{R_1}|x|} - \frac{1}{k_{R_2}} e^{ik_{R_2}|x|} \right\},
\end{equation}
which corresponds to two waves travelling away from the source ({\em outgoing waves}: the {\em Sommerfeld radiation condition} states that the wave must travel from the point of application of the concentrated force to infinity).

\item Case 4: High frequency and any prestress ($c < 0$, $b^2 - c > 0$).

The four roots are real and imaginary, and admit the following representation
\begin{alignat}{2}
k &= \pm k_R, & \quad k_R &= \sqrt{-b + \sqrt{b^2 - c}}, \label{eq37} \\
k &= \pm i k_I, & \quad k_I &= \sqrt{b + \sqrt{b^2 - c}}. \label{eq38}
\end{alignat}
In this case the Green's function takes the form
\begin{equation}
G_F(x) = \frac{1}{2EI (k_R^2 + k_I^2)} \left\{ \frac{i}{k_R} e^{ik_R |x|} - \frac{1}{k_I} e^{-k_I |x|} \right\},
\end{equation}
which corresponds to a evanescent wave superimposed to an {\em outgoing} travelling wave.

\end{itemize}

The four regimes (Cases 1--4) are shown in Fig.~\ref{cases} in the plane $\overline{P}$ -- $\sqrt{R}\, \omega$ for foundation stiffness $\overline{B} = 0.01$. Localised wave forms are observed for frequencies within a band gap (below $\sqrt{R}\, \omega_\text{min}$, reported with a dashed line): Case 1 corresponds to a monotonic decay (without oscillations); Case 2 corresponds to an oscillatory decay. On the other hand, propagating modes are observed for frequencies within a pass band: Case 3 corresponds to two travelling waves; Case 4 corresponds to an evanescent wave superimposed to a travelling wave.

\begin{figure}[!htcb]
\centering
\includegraphics[width=6.5cm]{cases.pdf}
\caption{\footnotesize Map of four dynamic regimes for an infinite, homogeneous, prestressed Rayleigh beam on an elastic foundation in the plane $\overline{P}$ -- $\sqrt{R}\, \omega$ ($\overline{B} = 0.01$). Localised modes correspond to a frequency within a band gap (below $\sqrt{R}\, \omega_\text{min}$, reported with a dashed line): monotonic decay without oscillations (Case 1); decay with oscillations (Case 2). Propagating modes correspond to a frequency within a pass band: two travelling waves (Case 3); evanescent wave superimposed to a travelling wave (Case 4).}
\label{cases}
\end{figure}

\paragraph{Special resonant cases giving rise to linear or quasi-linear growth at infinity.}

Special cases are obtained for the choice of dynamic parameters for which $c = 0$ or $b^2 - c = 0$. In these cases, the characteristic equation (\ref{eqcar}) admits double roots, resulting in the linear or quasi-linear growth at infinity, as shown in Fig.~\ref{fig00} in the introduction.

\begin{itemize}

\item Special case $c = 0$, $b > 0$

\begin{equation}
G_F(x) = -\frac{1}{2EI k_I^3} \left\{ e^{-k_I |x|} + k_I |x| \right\}, \label{GF40}
\end{equation}
where $k_I = \sqrt{2b}$.

\item Special case $c = 0$, $b < 0$

\begin{equation}
G_F(x) = \frac{1}{2EI k_I^3} \left\{ i e^{i k_R |x|} + k_R |x| \right\}, \label{GF41}
\end{equation}
where $k_R = \sqrt{-2b}$.

\item Special case $b^2 - c = 0$, $b < 0$

\begin{equation}
G_F(x) = -\frac{1}{4EI k_R^3} e^{i k_R |x|} \left\{ i  + k_R |x| \right\}, \label{GF42}
\end{equation}
where $k_R = \sqrt{-b}$.

\end{itemize}

\subsection{Concentrated couple}

The time-harmonic function $\ensuremath{\mathcal{G}}_C(x,t) = G_C(x) \exp(-i\omega t)$ corresponding to a concentrated unit couple (assumed positive when oriented as the rotation $v'$) placed at the origin $x = 0$ of a homogeneous Rayleigh beam is the solution of the following equation
\begin{equation}
\label{deltap}
G_C''''(x) - 2b\, G_C''(x) + c\, G_C(x) = -\frac{\delta'(x)}{EI}.
\end{equation}
Note that the function $G_C(x)$ has the property $G_C(x) = -G_F'(x)$. For the four cases introduced above, the function $G_C$ takes the form
\begin{itemize}

\item Case 1:  Low frequency and high tensile prestress ($c \geq 0$, $b > 0$, $b^2 - c > 0$).

\begin{equation}
G_C(x) = -\frac{\mathop{\mathrm{sign}}(x)}{2EI(k_{I_1}^2 - k_{I_2}^2)} \left\{ e^{-k_{I_1}|x|} - e^{-k_{I_2}|x|} \right\},
\end{equation}
which corresponds to a monotonic decaying mode, without oscillations.

\item Case 2:  Low frequency and low prestress (tensile or compressive) ($c > 0$, $b^2 - c < 0$).

\begin{equation}
G_C(x) = \frac{e^{-k_I |x|}}{4EI k_R k_I} \sin(k_R x),
\end{equation}
which corresponds to a oscillating decaying mode.

\item Case 3: Low frequency and high compressive prestress ($c \geq 0$, $b^2 - c > 0$, $b < 0$).

\begin{equation}
G_C(x) = \frac{\mathop{\mathrm{sign}}(x)}{2EI (k_{R_1}^2 - k_{R_2}^2)} \left\{ e^{ik_{R_1}|x|} - e^{ik_{R_2}|x|} \right\}.
\end{equation}
which corresponds to two outgoing travelling waves.

\item Case 4: High frequency and any prestress ($c < 0$, $b^2 - c > 0$).

\begin{equation}
G_C(x) = \mathop{\mathrm{sign}}(x) \frac{1}{2EI (k_R^2 + k_I^2)} \left\{ e^{ik_R |x|} - e^{-k_I |x|} \right\}.
\end{equation}
which corresponds to a evanescent wave superimposed to an {\em outgoing} travelling wave.

\end{itemize}

\subsection{Localised wave forms}

We now consider a rigid body, having the mass $M$ and the moment of inertia $I_M$, attached to the beam at the origin and vibrating at a frequency $\omega$ within a band gap, see Fig.~\ref{mass}a. The effect of the rigid body is replaced by the inertial force $M \ddot{v}(0) = -M \omega^2 v(0)$ and the inertial couple $I_M \ddot{\phi}(0) = -I_M \omega^2 v'(0)$, see Fig.~\ref{mass}b.

\begin{figure}[!htcb]
\centering
\includegraphics[width=13cm]{mass2.pdf}
\caption{\footnotesize (a) A rigid body having mass $M$ and moment of inertia $I_M$ attached to a Rayleigh beam at $x = 0$. (b) Inertial force $M \ddot{v}(0)$ and inertial couple $I_M \ddot{\phi}(0)$ transmitted by the body to the beam. (c) Translational localised mode corresponding to $v(0) \neq 0$ and $v'(0) = 0$. (d) Rotational localised mode corresponding to $v(0) = 0$ and $v'(0) \neq 0$.}
\label{mass}
\end{figure}

The transverse displacement $v(x)$ is given in terms of Green's functions as follows
\begin{equation}
\label{emme1}
v(x) = M \omega^2 v(0) G_F(x) + I_M \omega^2 v'(0) G_C(x).
\end{equation}
By taking the derivative of the previous equation, we get
\begin{equation}
\label{emme2}
v'(x) = M \omega^2 v(0) G_F'(x) + I_M \omega^2 v'(0) G_C'(x).
\end{equation}
Eqs. (\ref{emme1}) and (\ref{emme2}) can then be evaluated at $x = 0$, which leads to the system
\begin{equation}
\left\{
\begin{aligned}
v(0) &= M \omega^2 v(0) G_F(0) + I_M \omega^2 v'(0) G_C(0), \\[3mm]
v'(0) &= M \omega^2 v(0) G_F'(0) + I_M \omega^2 v'(0) G_C'(0).
\end{aligned}
\right.
\end{equation}
For the Rayleigh beam, we have $G_F'(0) = G_C(0) = 0$, so that the system decouples and we obtain the
equations
\begin{equation}
\label{system}
M = \frac{1}{\omega^2 G_F(0)}, \quad
I_M = \frac{1}{\omega^2 G_C'(0)}.
\end{equation}
We note that two independent localised modes are identified, which will be referred to as a {\em translational mode} and a {\em rotational mode}, see Fig.~\ref{mass}c and \ref{mass}d, respectively. The {\em translational localised mode} is obtained by setting $v(0) \neq 0$ and $v'(0) = 0$; in this case the rotation at the origin is zero, so that the moment of inertia $I_M$ is not involved and the motion is sustained only by the mass $M$ given by the first equation in (\ref{system}). The {\em rotational localised mode} is obtained by setting $v(0) = 0$ and $v'(0) \neq 0$; in this case the transverse displacement at the origin is zero, so that the mass $M$ is not involved and the motion is sustained only by the moment of inertia $I_M$ given by the second equation in (\ref{system}).

Examples of translational and rotational localised modes are shown in Fig.~\ref{figloc2}a and \ref{figloc2}b, respectively, for normalised frequency $\sqrt{R}\, \omega = 0.05$, foundation stiffness $\overline{B} = 0.01$, and different values of prestress $\overline{P} = \{ -0.16, 0, 0.2 \}$. The value of normalised mass $\overline{M} = M / (2\rho A r)$ (for translational mode) and normalised moment of inertia $\overline{I}_M = I_M / (2\rho I r)$ (for rotational mode) needed to sustain the motion for each value of prestress is indicated in the figure.

\begin{figure}[!htcb]
\centering
\includegraphics[width=15cm]{figloc2.pdf}
\caption{\footnotesize Amplitude of localised vibration modes along the $x$-axis of the beam for normalised frequency $\sqrt{R}\, \omega = 0.05$, foundation stiffness $\overline{B} = 0.01$, and different values of prestress $\overline{P} = \{ -0.16, 0, 0.2 \}$. (a) Translational localised mode, the amplitude is normalised by the value at the origin. The mass $\overline{M}$ is located at $x = 0$. Note that the rotation at the origin is zero for this mode and thus the moment of inertia $\overline{I}_M$ is not involved. (b) Rotational localised mode, the amplitude is normalised by $r$ and the value of rotation at the origin. The rigid body having moment of inertia $\overline{I}_M$ is located at $x = 0$. Note that the transverse displacement at the origin is zero for this mode and thus the mass $\overline{M}$ is not involved.}
\label{figloc2}
\end{figure}

\noindent
Explicit formulae can be obtained for the mass $M$ and the moment of inertia $I_M$ associated to the translational and rotational localised mode, namely
\begin{equation}
\begin{aligned}
M &= \frac{4EI k_I(k_R^2 + k_I^2)}{\omega^2} = \frac{2EI}{r^3 \omega^2}
\sqrt{\overline{B} - R \omega^2} \sqrt{\overline{P} - R\omega^2 +
2\sqrt{\overline{B} - R\omega^2}}, \\[3mm]
I_M &= \frac{4EI k_I}{\omega^2} = \frac{2EI}{r \omega^2} \sqrt{\overline{P} - R\omega^2 +
2\sqrt{\overline{B} - R\omega^2}}.
\end{aligned}
\end{equation}
The dependency of the dimensionless frequency $\sqrt{R}\, \omega$ upon the prestress $\overline{P}$ for $\overline{B} = 0.01$ and three values of $\overline{M}$ (translational mode) and three values of $\overline{I}_M$ (rotational mode) is reported in Fig.~\ref{figloc}a and \ref{figloc}b, respectively.

\begin{figure}[!htcb]
\centering
\includegraphics[width=15cm]{figloc.pdf}
\caption{\footnotesize Normalised circular frequency $\sqrt{R}\, \omega$ associated to a body, having dimensionless mass $\overline{M}$ and dimensionless moment of inertia $\overline{I}_M$, attached to an infinite, homogeneous, prestressed Rayleigh beam on an elastic foundation as a function of prestress $\overline{P}$ ($\overline{B} = 0.01$). The upper boundary of the band gap $\sqrt{R}\, \omega_\text{min}$ is reported as a dashed line. (a) Translational localised mode corresponding to $v(0) \neq 0$, $v'(0) = 0$, sustained by the mass $\overline{M}$. (b) Rotational localised mode corresponding to $v(0) = 0$, $v'(0) \neq 0$, sustained by the moment of inertia $\overline{I}_M$.}
\label{figloc}
\end{figure}

\subsection{Propagating wave forms}

Propagating wave forms for the case of high frequency and different values of prestress (Case 4) and the case of low frequency and high compressive prestress (Case 3) are shown in Fig.~\ref{propagm}.

These solutions are bounded at infinity, which is different from the special resonant cases of Green's function for the Rayleigh beam on the Winkler foundation, as shown in Fig.~\ref{fig00} in the introduction, where the neighbourhood of the origin was an effective hinge dominated by an exponentially localised wave form, whereas the linear growth was present at infinity.
In parts (a) and (b), we show the Green's function (point force solution) and its derivative (concentrated couple) for the case when the solution includes a propagating wave as well as an evanescent term. This is reflected in the higher amplitude at the origin, as observed in the Fig.~\ref{propagm}a.
On the contrary, the parts (c) and (d) include the Green's function and its derivative for the choice of dynamic parameters giving rise to two propagating waves, without an evanescent term present. The wave pattern is typical for interference of two periodic solutions, as expected in this case.

The pass band Green's function will be used in the next section in order to construct the quasi-periodic solutions corresponding to a multi-mass system in a Rayleigh beam.

\begin{figure}[!htcb]
\centering
\includegraphics[width=15cm]{propagm.pdf}
\caption{\footnotesize Amplitudes of propagating modes along the $x$-axis of the beam for Winkler stiffness $\overline{B} = 0.01$. Propagating  mode induced by a concentrated force (a), and a concentrated couple (b), placed at the origin for normalised frequency $\sqrt{R}\, \omega = 0.15$ (high-frequency, Case 4) and different values of prestress $\overline{P} = \{ -0.16, 0, 0.2 \}$. Propagating  mode induced by a concentrated force (c), and a concentrated couple (d), placed at the origin for normalised frequency $\sqrt{R}\, \omega = 0.08$ (low-frequency and high compressive prestress, Case 3) and different values of prestress $\overline{P} = \{ -0.18, -0.15, -0.12 \}$.}
\label{propagm}
\end{figure}

\clearpage

\section{Quasi-periodic Greens function. Dispersion of Bloch waves for a multi-mass system in a Rayleigh beam}
\label{sec05}

The results of Section \ref{sec04} are extended here to the cases of periodic multi-mass systems. An infinite Rayleigh beam is taken together with a periodic array of identical bodies possessing mass $M$ and the moment of inertia $I_M$. Instead of a single source or a localised couple, discussed in the earlier section \ref{sec04}, we now have a periodic array of point forces and point moments accordingly.
For the purpose of analysis of Bloch waves in the periodic structured Rayleigh beam, we need the quasi-periodic Green's function.

Assuming that $G_F(x; \omega)$ is a pass-band Green's function for a single point force applied to the origin at the Rayleigh beam, the quasi-periodic Green's function $g_F(x; \omega, K)$ representing a periodically distributed sources generating a propagating wave, has the form
\begin{equation}
g_F(x; \omega, K) = \sum_{n=-\infty}^{\infty} G_F(x - nd; \omega) e^{i K nd},
\end{equation}
where $d$ stands for the period of the structure, and in the computations below we assume $d = 1$.

It is also noted that the function $g_F$ satisfies the differential equation:
\begin{equation}
g_F'''' - 2b\, g_F'' + c\, g_F = \sum_{n=-\infty}^{\infty} \frac{\delta(x - nd)}{EI} e^{i K nd},
\end{equation}
with delta function terms being periodically distributed along the infinite beam, and the phase-shift factors being applied accordingly to the forcing terms.

Along the same lines, a function $g_C(x; \omega, K)$ is introduced as the solution of the equation
\begin{equation}
g_C'''' - 2b\, g_C'' + c\, g_C = -\sum_{n=-\infty}^{\infty} \frac{\delta'(x - nd)}{EI} e^{i K nd},
\end{equation}
which represents an infinite array of periodically distributed concentrated moments subjected to the
Floquet phase shift accordingly. We note that $g_C$ can be written in terms of the single couple Green's function $G_C(x;\omega)$ as follows
\begin{equation}
g_C(x; \omega, K) = \sum_{n=-\infty}^{\infty} G_C(x - nd; \omega) e^{i K nd}.
\end{equation}
It is also straightforward to see that
\begin{equation}
g_C(x; \omega, K) = -g_F'(x; \omega, K).
\end{equation}
The flexural displacement $v(x)$ in the Rayleigh beam, subjected a action of periodically distributed forces and moments, satisfies the following equations
\begin{equation}
v(x) = m \omega^2 v(0) g_F(x; \omega, K) + I_M \omega^2 v'(0) g_C(x; \omega, K)
\end{equation}
and
\begin{equation}
v'(x) = m \omega^2 v(0) g_F'(x; \omega, K) + I_M \omega^2 v'(0) g_C'(x; \omega, K).
\end{equation}
Evaluation of $v$ and $v'$ at the origin leads to the homogeneous system of linear algebraic equations
\begin{equation}
\begin{bmatrix}
1 - M \omega^2 g_F(0;\omega, K) & -I_M \omega^2 g_C(0;\omega, K) \\
- M \omega^2 g_F'(0;\omega, K)  & 1 - I_M \omega^2 g_C'(0;\omega, K)
\end{bmatrix}
\begin{bmatrix}
v(0) \\
v'(0)
\end{bmatrix} =
\begin{bmatrix}
0 \\
0
\end{bmatrix}.
\end{equation}
This system has non-trivial solutions if and only if the determinant of its matrix is zero, i.e.
\begin{equation}
\label{disp2}
1 - \omega^2 (Mg_F(0;\omega, K) + I_M g_C'(0;\omega, K)) + MI_M \omega^4 (g_F(0;\omega, K)g_C'(0;\omega, K) -g_F'(0;\omega, K)g_C(0;\omega, K)) = 0.
\end{equation}
The above equation (\ref{disp2}) defines $\omega$ as an implicit function of $K$ and hence it is the dispersion equation for Bloch-Floquet waves in the structured Rayleigh beam.

We note that the nodal values of the quasi-periodic Green's functions $g_F(0;\omega, K)$ and $g_C(0;\omega, K)$, used in \eq{disp2}, are evaluated in the explicit form,
\begin{align}
g_F(0;\omega, K) &= \frac{1}{2EI k_R k_I (k_R^2 + k_I^2)}
\left(
k_R \frac{\sinh(k_I d)}{\cos(K d) - \cosh(k_I d)} -
k_I \frac{\sin(k_R d)}{\cos(K d) - \cos(k_R d)}
\right), \label{GF} \\[3mm]
g_C(0;\omega, K) &= -g_F'(0;\omega, K) = \frac{i \sin(K d)}{2EI (k_R^2 + k_I^2)} \cdot
\left(
\frac{1}{\cos(K d) - \cos(k_R d)} -
\frac{1}{\cos(K d) - \cosh(k_I d)}
\right), \label{GF1} \\[3mm]
g_C'(0;\omega, K) &= -g_F''(0;\omega, K) = -\frac{1}{2EI (k_R^2 + k_I^2)}
\left(
k_R \frac{\sin(k_R d)}{\cos(K d) - \cos(k_R d)} +
k_I \frac{\sinh(k_I d)}{\cos(K d) - \cosh(k_I d)}
\right), \label{GF2}
\end{align}
where $k_R$ and $k_I$ are defined in (\ref{eq37}) and (\ref{eq38}). With reference to \eq{GF}--\eq{GF2}, we note that all the coefficients of the equation \eq{disp2} are real, and the dispersion diagrams (see Fig.~\ref{figQPj}) can be constructed by obtaining real roots $K$ of \eq{disp2} for given values of the radian frequency $\omega$. It also worthwhile saying that the left-hand side of \eq{disp2} is a rational function of $Z = \cos(Kd)$.

Furthermore, the dispersion equation \eq{disp2} is solved
to demonstrate the dispersion properties of Bloch waves in a structured Rayleigh beam.

Fig.~\ref{figQPj} incorporates four examples, all showing band gaps within the low frequency range.
In particular, the diagram (a) of this figure corresponds to the case when point masses, without rotational inertia, are distributed periodically along the length of the beam, whereas the diagram (b) describes the case when the mass is formally taken as zero and only rotational inertia is taken into account. While for low frequencies the diagram (a) is similar to the one observed in the Euler--Bernoulli structured beam, the second case (b) appears to be important for the effects of the rotational inertia, and in particular, it shows waves of small group velocity and wide band gaps in the low frequency range.

In the next two diagrams (c) and (d) of Fig.~\ref{figQPj}, we show the combined effects of translational and rotational inertia produced by bodies placed periodically along the length of the Rayleigh beam.
The comparison of these computations shows interesting localisation effects associated with the rotational inertia. With the increase of the rotational inertia of periodically distributed solids, the dispersion curves become flatter, which implies reduction in the absolute value of the group velocity. This effect is especially pronounced in the high frequency range. An observation of special importance is in the change of sign of the group velocity for the second band, as follows from the comparison of diagrams (c) and (d) for $\overline{I}_M=0.5$ and  $\overline{I}_M=2$, and also the change of the sign of curvature of the dispersion curve in the second band.

\begin{figure}[!htcb]
\centering
\includegraphics[width=15cm]{figQP.pdf}
\caption{\footnotesize Dispersion diagrams for a multi-mass system in a Rayleigh beam (for foundation stiffness $\overline{B} = 0.01$, null prestress $\overline{P} = 0$ and unit cell size $d = r$). $\overline{M} = M / (2\rho A r)$ and $\overline{I}_M = I_M / (2\rho I r)$ are the dimensionless mass and moment of inertia of the bodies periodically distributed along the beam.}
\label{figQPj}
\end{figure}

\newpage

\section{Conclusion}
\label{sec06}

The paper has addressed an important class of problems that arise for flexural Bloch waves in elastic systems with
rotational inertia and prestress, in the presence of a Winkler foundation. This has been done for so-called Rayleigh beams and structured Rayleigh beams. We have also highlighted a direct analogy with the models of couple-stress materials; although physical origins of the two problems are different, the mathematical underlay is identical and hence the mathematical study of the dynamic response of structured Rayleigh beams also brings a valuable insight into the study of shear waves in Cosserat type composites.

The dispersion properties of waves in the Rayleigh beam appear to be very different from those in Euler beams in the high frequency regimes. This has been demonstrated analytically. The combined action of rotational inertia and prestress has been highlighted. The low frequency bandgaps have been created via introduction of the term that corresponds to an elastic foundation of the Winkler type.

Defect modes and localisation have been studied through analysis of stop band and pass band dynamic Green's functions. Quasi-periodic Green's functions for translational and rotational inertia have been constructed, and the corresponding dispersion equations are written in closed form, which allows for effective and accurate solution and analysis.

The analysis presented in this work is generic and is straightforward to extend to flexural systems such as plates and shells, in the cases where rotational inertia effects appear to be important. This is likely to be applicable to the frame-type flexural systems as well as high contrast systems, modelling dynamic shields of elastic waves (also known as ''invisibility cloaks'') and earthquake protection systems.

\vspace{6mm}
{\bf Acknowledgements}. AP would like to acknowledge financial support from the
European Union's Seventh Framework Programme FP7/2007-2013/ under REA grant
agreement number PCIG13-GA-2013-618375-MeMic.
AM gratefully acknowledges financial support from the
European Union's Seventh Framework Programme FP7/2007-2013/ under REA grant
agreement number PITN-GA-2013-606878-CERMAT2.

\bibliographystyle{jabbrv_unsrt}
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\end{document}

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A lot of the time when you read about a female scientist, there's an underlying tone of surprise that they chose a career in science. Either that, or the emphasis is not on their great minds and their achievements, but their appearance, or love life... Or their stroganoff recipe.
To challenge these very unscientific assumptions, Twitter user Daurmith has written a series of short biographies of male scientists as if they were women. The tweets were originally written in Spanish, but translated by popular demand.


Welcome to dulcimerevents.com! Our goal is to make it easy to locate and submit mountain dulcimer and hammered dulcimer events, past and present. This isn't just for festivals, workshops, and concerts. It's also for recurring events like jams and club meetings. If there's dulcimer going on, post your event!
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Chidwick, Ann. (1994). New beginnings: A remarriage preparation program manual . Milwaukee, WI: Families International, Inc.
Chidwick, Ann. (1994). New beginnings: A remarriage preparation workbook . Milwaukee, WI: Families International, Inc.
Family Life Committee.(1987). Beginnings . Wahroonga, Australia: South Pacific Division of Seventh-day Adventists.
Flowers, Ron & Watts, Kit. (1985). Preparing for marriage . Department of Family Ministries, General Conference of Seventh-day Adventists, 12501 Old Columbia Pike, Silver Spring, MD.
Mace, David R. (1984). Getting ready for marriage . Nashville, TN: Abingdon Press.
Parrott, Les & Leslie. (1995). Saving your marriage before it starts . Grand Rapids, MI: Zondervan Publishing House.
A marriage curriculum for engaged, about-to-be engaged, and the newly married. Includes two videos with eight half-hour sessions, separate men's and women's workbooks full of exercises and self-tests to help couples apply what they learn.
Prokopchak, Steve & Mary. (1994). Called together: A marriage prepartion workbook . Nashville, TN: Thomas Nelson Publishers.
White, Ernest & White, James. (1992). Counsel for the nearly and newly married: Leader's guide . Nashville, TN: Convention Press.
White, Ernest & White, James. (1992). Counsel for the nearly and newly married: Couple's guide . Nashville, TN: Convention Press.
Wright, H. Norman. (1977). Premarital counseling . Chicago, IL: Moody Press.
Wright, H. Norman. (1997). Before you say "I do: A marriage preparation manual for couples . Eugene, OR: Harvest House Publishers.
Wright, H. Norman. (1997). So you're getting married. Ventura, CA: Regal Books. ISBN: 0830717722.
Wright, H. Norman. (2003). Before you say "I do" devotional. Eugene, OR: Harvest House Publishers. ISBN: 0736909222.


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Becky Bauer -- October 2006 Issue
It Will Affect You - The Tragedy of Commercial Fishing
After many years of debate, the dogged determination of environmental groups, and scientific studies, the cause of many large-scale disasters—killing floods and landslides, starvation from lack of topsoil in which to grow food, air and water degradation—is now recognized as a threat to global security. This threat is the clear cutting of forests throughout the world.
I am certain you are wondering what clear cutting has to do with commercial fishing. A recent study conducted by Dr. Les Watling of the Darling Marine Center at the University of Maine states that commercial bottom trawling and dredging destroys 150 times more sea bottom than clear cutting destroys forests per year. 150 x 16m = 2400m acres of sea bottom per year; 2,400,000,000 acres, a number so large as to be almost incomprehensible.
In 1966 as I was flying across the Amazon at sunrise, a kindly Pakistani scientist invited me to join him at the plane's windowed hatch. For as far as the eye could see, the Amazon River wound its way through a tree canopy that was so thick, no ground could be seen. The Amazon Forest back then sheltered plants and animals, many yet to be discovered, plants with medical miracles in their leaves and roots, animals and insects that had existed for millions of years, and billions of trees that filtered carbon dioxide to produce clean air, with root systems that filtered pollutants from water and prevented floods, and with leaves that helped to keep the earth's surface cool. The Amazon Forest was so large that it even created its own weather.
The Amazon River, winding over 4,000 miles from Peru to the Atlantic Ocean was the largest source of freshwater in the world, depositing millions of cubic feet of water per second into the Atlantic where it spread to distant nations.
Forty years after I watched that spectacular sunrise over the relatively unspoiled Amazon, studies by Brazilian scientists indicate that about six million acres of Amazon rain forest disappear each year to clear cutting. When I look at recent satellite images of the once lush Amazon forest, I see desolate land stretching miles and miles from the river's banks, a wasteland slashed with roadways, where nothing can live. The river now polluted with heavy metals, cyanide, mercury, trash, and sewage still empties its millions of cubic feet of water per second into the Atlantic where it is carried thousands of miles.
Just as clear cutting has destroyed forests, commercial bottom trawling and dredging is destroying acres of sea bottom yet to be explored. These acres surely contain undiscovered species, may hold the key to our energy crises, and sustain a pyramid of life on which all life depends. Not only are sea bottoms and coral reefs destroyed but so, too, are entire schools of fish, along with the innocuous sounding "bycatch," hundreds of tons of sea turtles, dolphin, whale, sea horses, mollusks, sharks, even sea birds.
What happens when a species of marine life goes missing? When an indigenous fisherman cannot find the lobster he is hired to catch in order to feed his family because a commercial fisherman and his American partners smuggled millions of dollars of illegal Caribbean lobster into the U.S., what can he do? Like so many, he may be forced to leave his family to seek work in town, or perhaps he makes his way north, hoping to cross into the United States.
What happens to the people living and working in a beautiful resort town when tourists stop coming because it is no longer safe to go into the water? This has already happened. In Recife, Brazil, where the beach is filled with travelers as well as locals, people no longer go into the water. Bull and Tiger sharks have taken over the coastal waters and have struck swimmers and surfers 44 times in the past 10 years. Why?
Scientists were called in because the sharks have always been there and were not troublesome. The same sharks in Cuban waters pose little threat and actually interact with divers without incident. So why is it different in Recife? The mangroves south of town that sheltered both the sharks and their prey were destroyed when they were filled in so that a port could be built. The fringing reef along Recife's beaches was damaged by fishing and pollution. And, over-fishing by commercial enterprises took all the fish that remained both in and off-shore. Videos of the sea bottom along the coast of Recife show a dead reef bereft of any fish life with gaps large enough to allow the Bull and Tiger sharks to enter once-protected swimming waters where people now have been maimed and killed. The tourism industry in Recife realizes that tourists will stop coming and yet there is little they can do to protect their livelihoods.
This series in the coming months will explore in depth commercial fishing as it exits today. The bounty of the seas is not limitless; however, it is sustainable if those in authority in countries around the world realize that seas are not static nor is the life that lives within. When a commercial trawler takes an entire population of grouper off one country's shores, another country along the line will suffer the consequences. Sea life is not stationary; it migrates with the tides, the moon, and the seasons. We are all connected by the seas.


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You are here: Home / Archives for former Secretary of State Madeleine Albright
Veteran Fulton Police chief to retire in December
October 29, 2021 By Brian Hauswirth
The longtime Fulton police chief will retire in December, after more than 39 years of law enforcement in mid-Missouri. Steve Myers has served as Fulton's chief for more than 19 years. He was named to the post in 2002 and implemented community policing,…
Filed Under: 93.9 The Eagle, body cameras, community policing, former Secretary of State Madeleine Albright, former Vice President Dick Cheney, Fulton Police Chief Steve Myers, Homepage, Mid-Missouri News, Montgomery County Sheriff, Syndicated, U.S. Sen. Bernie Sanders


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Playing the odds: Will tricks help you win Powerball?
Emily Chappell
EChappell@dmg.gannett.com
What would you do if you won the Powerball?
It's a question that's swirling through nearly everyone's conversations across the United States, with the Powerball reaching record-breaking numbers. The big win is up to $1.5 billion right now, higher than it's been before.
It's causing people to race out and buy tickets, all while planning just what they'd do with the money.
But American Statistical Association's Executive Director Ronald Wasserstein said it's probably best not to start planning what to do with those winnings quite yet.
"Some human being will win eventually," Wasserstein said. "It just isn't going to be you."
Maryland Lottery looking for winning Berlin ticket
Currently, the chance of winning is one in 292 million, he said. And even if people try different tricks to win, it won't matter.
Go anywhere on the Internet, he said, and it won't be hard to find tips to improve your odds.
Some people suggest picking certain numbers that have historically been more selected, others say avoid using birthdays, some swear by letting the machine choose the numbers.
No matter how badly people want to believe, Wasserstein said these suggestions won't make a difference.
Every set of Powerball numbers has the same chance of being selected, he added. Picking some as basic as one, two, three, four, five and six as the Powerball number, has the same odds as doing something more elaborate.
The only thing to consider is that if by very small chance those numbers won, Wasserstein said, the winner would have to split the winnings with everyone else who was that unoriginal.
That said, it's not worth worrying about, he added.
"The only way that you improve your chances of winning costs you extra money, and doesn't help you a lot," Wasserstein said.
Buying 10 tickets instead of one theoretically increases the probability of winning by factor of 10, he said.
But it won't make a difference.
"That's still unimaginably, impossibly, ridiculously small," he added.
Md. couple matches all but one Powerball number, wins $50k
Even buying 200 tickets, which Wasserstein said increases the chances to one in 1.5 million, isn't going to help much. People are "still virtually guaranteed to lose," he said, and they'll be out $400.
"You really don't increase your chances of winning in a way that amounts to much of anything," he added.
The lottery can be confusing to people, he said, because people hold on to the mindset that someone will win eventually. And the fact that someone will win, makes people hopeful that person will be them.
In the end, Wasserstein said he'll be the dose of reality — there aren't really ways to improve the odds, and the odds are slim.
"There's no magic to picking the numbers," he said.
Maryland Lottery Director Gordon Medenica said this is one of the "most exciting things to happen" in lottery, with the Powerball as high as it is now.
"This is uncharted territory," Medenica added.
The one thing he recommends to anyone playing is to sign their ticket the minute they buy it.
And, Medenica said, make sure to check all the numbers.
There are "lots of prizes" besides the top prize worth winning, he said.
Brian Sherry, the clerk working at Assawoman Ale Shoppe, said they've been somewhat busy selling Powerball tickets, though this past weekend was busier than the week's been.
Friday night after many people got paid, and before Saturday's drawing, they saw a lot of people, Sherry added.
He expects a bigger turnout once people are out of work Tuesday evening.
The Ocean City store only recently got Powerball about a month ago, he said.
Of the people buying tickets, Sherry hasn't noticed people trying to use any sort of system. Instead, he said, most people who have come in have let the machine pick.
Regardless, Sherry said, they've seen the country-wide hype trickle into the resort town, with a lot of non-regulars playing the lottery.
But, he said, with this kind of jackpot, he's not surprised.
"For that kind of money, I don't blame them," he said.
On Twitter @EmilyChappell13