None
Table 43. — For the nummation of long-period tides — Continued. 609 Baj^of series Mf MSf 1231 1232 1233 1234 1235 1236 :237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247' 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 '275 1276 1277 1278 1279 1280 1281 1282 1285 1286 J287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 129S 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 3 4 4^ 5 6 13 14 >5 16 17 l8«- 18 19 23 9 9<- 13 14 15 16 17 18 23«- 23 I3«- 13 14 15 16 17 16 17 l8«- 18 19 20 23 23<- 3*- 3 4 5 6 7<- 7 12 12<- 13 14 15 I6«- 16 17 18 19 3 4 5<- 5 6 7 9<- 9 12 13 14 I4«- 15 16 17 19 16 I7<- 17 18 19 20 23 IK- 3 4 5 6 7 7<- 14 14*- 15 16 17 18 19 2CX- 3<- 3 4 5 6 13 14 15 16 >7 I7«- 18 19 23<- 23 3 3 4 I 5 2 6«- 3 6 3*- 7 4 8 S 9 6 10 Dayoi series 131 1 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1320 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 I3« 1367 1368 1369 1370 1371 1372 1373 1374 Mf 23 4 4<- 5 6 7 13 14 15 16 17 l3 19 I9<- 2<- 3 4 5 6 7 8 9 9«- 13 14 15 l6<- :6 17 18 19 23<- MSf 1375 7<- 1376 8 1377 9 1378 10 1379 II 1380 12 I18I 13 I3S2 14 1383 I4<- 1384 15 1385 16 1386 17 1387 18 -1388 19 1389 20 1390 21 12 13 14 15 16 l6<- 17 18 19 20<- 20 23 0«- 5<- 6 7 8 9^ I3<- 13 14 15 16 17 18 l8<- 19 20 3 4 5 6 7 7«- 10 ll<- 13 14 15 16 l6<- 17 18 19 23 0*- 13<- 14 15 16 17 18 19 9*- 9 13 14 15 i5 i6<- 17 23 23<- 3 4 5<- 5 6 7 13 14 15 16 17 18 19 I9<- 20 21 2<- 3 7 8<- Day of Mf series 1391 2I<- 1392 22 1393 23 1394 1395 I 1 1396 2 1397 3 1398 4<- 1399 4 1400 5 1401 6 1402 7. Tablb 44. 22 00 29 25 36 57 44 39 52 36 60 59 70 12 ft a 2-4 00 7 30 15 02 22 37 30 17 38 06 46 08 54 32 63 31 73 44 2-5 00 7 42 15 26 23 13 31 08 39 13 47 36 S6 25 66 03 77 22 2-6 00 7 53 15 48 23 49 31 57 40 18 49 01 58 16 6833 81 05 27 00 8 04 16 II 24 23 32 45 41 22 50 24 60 08 71 03 84 54 2-8 00 8 15 16 32 24 56 33 32 42 22 51 45 61 54 73 32 88 51 2-9 00 8 25 16 S3 25 29 34 17 •43 25 53 05 6339 76 00 92 56 3'o 00 f 35 17 14 26 00 35 01 44 22 54 22 65 23 78 26 97 II 3"i CX5 8 44 17 33 26 31 35 44 45 21 55 38 67 05 80 52 lOI 36 3-2 00 8 54 17 52 27 01 36 26 46 17 56 53 68 45 83 16 106 15 3 '3 00 9 03 18 II 27 30 37 06 47 12 5805 70 23 85 39 III 11 3-4 00 9 12 18 30 27 56 37 46 48 05 59 16 71 58 87 59 116 25 3-5 00 9 21 18 47 28 26 38 25 48 57 60 25 73 32 90 19 122 05 3-6 00 9 29 19 05 28 53 39 01 49 47 61 30 75 04 92 35 128 19 37 00 9 38 19 22 29 19 39 39 50 37 62 38 76 34 94 49 135 21 3-8 CX) 9 46 19 38 29 44 40 14 51 25 63 42 78 02 97 01 143 37 3 9 00 9 53 19 54 30 09 40 49 52 12 64 44 79 27 99 10 154 19 4-0 00 10 01 20 09 30 33 41 23 52 58 65 45 80 50 loi 16 180 00 HW Phase. 360° 350° 340° 330° 320° 310° 300° 290° 280° 270° LW Phase. 180 170 160 ■5° 1^0 130 120 no 100 90 *I. e., the phase of the diurnal wa%'e (B) at the time of HWor I,W of the semidiurnal wave (A). HW phase = (time of HW ol A - time of HW of £} b, LW phase = (time of LW ot A - time of HW of B) b. If one of the speeds be somewhat variable, the resultant times and heights will be given more accurately by keeping the phases between - 90° and + 90°. If they do not fall within these limits, this and the following table may be entered with the phases: HW phase = (time of LW oi A - time of LW of B) b, LW phase = (time of HW of ^ - time of LW oi B) b : the resultant heights must, however, have their signs changed. For tropic tides HW phase = n^ + %A°-B°,h^ phase = nir ±^+ y^ A° - B°, n being an integer. (See §§ 5, 20, Part III.) REPORT FOR 1897 — PART II. APPENDIX NO. 9. a semidiurnal wave due to a diurnal wave. [The amplitude of the semidiurnal wave is taken as unity.] 611 HW Phase. tW Phase. 260° 250° ■ 240° 230° 220° 210° 200° 190° iSo° I,W Phase. 80 70 60 50 40 3° 20 10 When the top argument is used the tabular values are positive; when the bottom argument, they are negative. To express the acceleration in time divide by a, the speed of the semidiurnal component. To find the acceleration when b is not exactly equal to % a, multiply the tabular values by 2b This acceleration is directly expressed in time by multiplying the tabular values by 2b Table 17 is a graphic form of this table. Rollet de I'lsle has given in the Annales Hydrographique for 1896 (p. 248) a graphic table serving ^e pur- pose of Tables 17, 18, or Tables 44, 45, and which he calls an abacus. It is really the inverse of Tables 17, 18. 612 UNITED STATES COAST' AND GEODETIC SURVEY. 3600 35°° 340° 330° 320° 310° 300° 290° 280° 270° I,W Phase. 180° 170° 160° 150° 140° 130° 120° 110° 100° ^° * See footnote, preceding table. For high waters use the tabular values as given; but for low waters, alter their signs. REPORT FOR 1897 — PART 11. APPENDIX NO. 9. composed of a diurnal and semidiurnal wave. [The amplitude of the semidiurnal wave is taken as unity.] 613 HW Phase. LW Phase. LW Phase. 260° 80° 250° 70° 240° 60° 230° 50° 220° 40° 210° 30° 200° 20° 190° 10° 180° 0° ! 4^2 t when b is not exactly equal to J^ a, mean value = i + (tabular value — i ) -^ • Table iS is a graphic form of this table. The above column of mean values may be compared with expression (29), Part III, and with the last column of Table 21. 614 ■UNITED STATES COAST AND GEODETIC SURVEY. Table 46. — Hyperholic fitnciiona. u V sinh u cosh u tanh M In degrees. In degrees. 9 = tan V = sec V = sin V e" £~~" O'OO O'OOOOOO O'OOOO 0-0000 0-000 0-0000 I -0000 O'OOOO I'OOOO I'OOOO 0'02 1-1459156 0-0200 1-1458 I -145 0-0200 1-0002 0-0200 I '0202 0-9802 o'04 2-291831 0-0400 2-2912 2-288 0-0400 1-0008 0-0400 I 'O408 0.9608 o-o6 3-437747 0-0600 3-4357 3-428 0-0600 1-0018 0-0599 i'o6i8 0-9418 o-o8 4-58366 0-0799 4-5788 4-561 0-0801 1-0032 0-0798 1-0833 0-9231 O'lO 5-72958 0-0998 5-720 5-693 0.1002 1-0050 0-0997 1-1052 0-9048 0'12 6-87549 0-II97 6-859 6-811 0-1203 1-0072 0-1194 1-1275 0-8869 0-I4 8-02I4I 0-1395 7-995 7-917 0-1405 1-0098 0-1391 1-1503 0-8694 o-i6 9-16732 0-1593 9-128 9-0H 0-1607 1-0128 0-1586 1-1735 0-8521 o-i8 10-3132 0-1790 10-258 10-100 O-1810 1-0162 0-1781 1-1972 0-8353 0'20 11-4592 0-1987 11-384 11-167 0-2013 1-0201 0-1974 1-2214 0-8187 0'22 12-6051 0-2183 12-505 12-216 0-2218 1-0243 0-2165 1-2461 0-8025 0'24 13-7510 0-2377 13-621 13-254 0-2423 1-0289 0-2355 1-2712 0-7866 0'26 14-8969 0-2571 14-732 14-271 0-2629 1-0340 0-2543 1-2969 0-7711 0-28 16-0428 0-2764 15-837 15-265 0-2837 1-0395 0-2729 1-3231 0-7558 0-30 17-1887 0-2956 16-937 16-245 0-3045 1-0453 0-2913 1-3499 0-7408 0-32 18-3346 0-3147 18-030 17-197 0-3255 1-0516 0-3095 1-3771 0-7261 o'34 19-4806 0-3336 19-116 18-134 0-3466 1-0584 0-3275 1-4049 0-7II8 0-36 20-6265 0-3525 20-195 19-045 0-3678 1-0655 0-3452 1-4333 0-6977 0-38 21-7724 0-3712 21-267 19-935 0-3892 1-0731 0-3627 1-4623 0-6839 0-40 22-9183 0-3894 22-331 20-801 0-4108 I -081 1 0-3799 I '4918 0-6703 0-42 24-0642 0-4082 23-386 21-648 0-4325 1 -0895 0-3969 I '5220 0-6570 0-44 25-2101 0-4264 24-434 22-470 0-4543 1 -0984 0-4136 1-5527 0-6440 0-46 26-3561 o-/\i\/\(> 25-473 23-275 0-4764 1-1077 0-4301 1-5841 0-6313. 0-48 27-5020 0-4626 26-503 24-045 0-4986 1-1174 0-4462 i-6i6i 0-6188 0-50 28-6479 0-4804 27-524 24-803 0-5211 1-1276 0-4621 1-6487 0-6065 0-52 29-7938 0-4980 28-535 25-533 0-5438 1-1383 o'4777 1-6820 0-5945 0-54 30-9397 0-5155 29-537 26-245 0-5666 1-1494 0-4930 1-7160 0-5827 0-56 32-0856 0-5328 30-529 26-930 0-5897 1-1609 0-5080 1-7507 0-5712 0-58 33-2316 0-5500 31-511 27-595 0-6131 1-1730 0-5227 1-7860 0-5599 o'6o 34-3775 0-5669 32-483 28-237 0-6367 1-1855 0-5370 1-8221 0-5488 0-62 35-5234 0-5837 33-444 28-861 0-6605 1-1984 0-5511 1-8589 0-5379 0-64 36-6693 0-6003 34-395 29-462 0-6846 1-2119 0-5649 1-8965 0-5273 0-66 37-8152 0-6167 35-336 30-045 0-7090 1-2258 0-5784 1-9348 0-5169 0-68 38-9611 0-6329 36-265 30-604 0-7336 1-2402 0-5915 1-9739 0-5066 070 40-1070 0-6489 37-183 31-149 0-7586 1-2552 0-6044 2-0138 0-4966 072 41-2530 0-6648 38-091 31-670 0-7838 1-2706 0-6169 2-0544 0-4868 074 42-3989 0-6804 38-587 32-174 0-8094 1-2865 0-6291 2-0959 0-4771 076 43-5448 0-6958 39-872 32-663 0-8353 1-3030 0-6411 2-1383 0-4677 078 44-6907 0-7III 40-746 33-132 0-8615 1-3199 0-6527 2-1815 0-4584 o-8o 45-8366 0-7261 41-608 33-587 0-8881 1-3374 0-6640 2-2255 0-4493 0-82 46-9825 0-7412 42-460 34-025 0-9150 1-3555 0-6751 2-2705 0-4404 0-84 48-1285 0-7557 43-299 34-446 0-9423 1-3740 0-6858 2-3164 0-4317 0-85 49-2744 0.7702 44-128 34-848 0-9700 1-3932 0-6963 2-3632 0-4232 0-88 50-4203 0-7844 44-944 35-238 0-9981 1-4128 0-7064 2-4109 0-4148 o-go 51-5662 0-7985 45-750 35-613 1-0265 1-4331 0-7163 2-4596 0-4066 o"92 52-7121 0-8123 46-544 35-976 1-0554 1-4539 0-7259 2-5093 0-3985 0-94 53-8580 0-8260 47-326 36-323 1-0847 1-4753 0-7352 2-5600 0-3906 o"96 55-0039 0-8394 48-097 36-660 . 1-II44 1-4973 0-7443 2-6117 0-3829 0-98 56-1499 0-8528 48-857 36-983 1-1446 1-5199 0-7531 2-6645 0-3753 I'OO 57-2958 0-8658 49-605 37-293 1-1752 1-5431 0-7616 2-7183 0-3679 I -02 58-4417 0-8787 50-343 37-593 1-2063 1-5669 0-7699 2-7732 0-36059 I -04 59-5876 0-8913 51-069 37-880 1-2379 1-5913 0-7779 2-8292 0-35345 I -06 60-7335 0-9038 51-783 38-158 1-2700 1-6164 0-7857 2-8864 0-34646 I -08 61-8794 0-9160 52-485 38-423 1-3025 1-6421 0-7932 2 '9447 0-33960 i-io 63-0254 0-9281 53-178 38-677 1-3356 1-6685 0-8005 3'oo42 0-33287 I-I2 64-1713 0-9400 53-860 38-924 1-3693 1-6956 0-8076 3-0649 . 0-32628 I-I4 65-3172 0-9518 54-531 39-160 1-4035 1-7233 0-8144 3'i268 0-31982 ri5 66-4631 0-9632 55-189 39-387 1-4382 1-7517 0-8210 3-1899 0-31349 i-i8 67-6090 0-9745 55-837 39-607 1-4735 1-7808 0-8275 3-2544 0-30728 1-20 68-7549 0-9857 56-476 59-817 1-5095 1-8107 0-8337 3-3201 0-30II9 B = the angle at the center of the hyperbola made by any secant line and the transverse axis of the hyperbola. u = twice the area of the hyperbolic rector thus determined, the length of the semiaxis being unity. tan0 = tanh m. w = an auxiliary angle called the gudermanian* such that the equations of the hyperbola are x = sec v,y = tan v. * For representations of this angle and for further particulars concerning hyperbolic functions see Chapter IV, by James McMahon, in Merriman and Woodward's Higher Mathematics; and Hoiiel, Recueil de Formules et de Tables numfirique. Newman and Glaisher have tabulated «-'' and c* in the Transactions of the Cambridge Phil. Soc, Vol. 13 C1883), III. REPORT FOR 1897 — PART II. APPENDIX NO. 9. Table 46. — Hype^'hoUc fanctiona — Continued. 615 u V e sinh M cosh u tanh u In degrees. [n degrees. =:tan V =:sec V = sin V ^ £~" 1-22 69-9009 0-9967 57-103 40-023 1-5460 1-8412 0-8397 3-3872 0-29523 1-24 71-0468 1-0074 57-721 40-215 1-5831 1-8725 0-8455 3-4556 0-28938 1-26 72-1927 I -0180 58-328 40-401 1 -6209 1-9045 0-8511 3-5254 0-28365 1-28 73 '3386 1-0284 58-925 40-582 1-6593 1-9373 0-8565 3-5966 0-27804 i-30 74'4845 1-0387 59-511 40-753 1-6984 1-9709 0-8617 3-6693 0-27253 1-32 75-6304 1-0490 60-087 40-920 1-7381 2-0053 0-8668 3-7434 0-26714 1-34 76-7763 1-0586 60-654 41-080 1-7786 2-0404 0-8717 3-8190 0-26185 1-36 77-9223 1-0684 61-212 41-232 1-8198 2-0764 0-8764 3-8962 025666 1-3'^ 79-0682 1-0779 61-758 41-380 1-8617 2-II32 0-8810 3-9749 0-25158 1-40 80-2141 1-0873 62-295 41-523 1-9043 2-1509 0-8854 4-0552 0-24660 1-42 81-3600 1-0965 62-823 41-657 1-9477 2-1894 0-8896 4-1371 0-24171 1-44 82-5059 I -1055 63-343 41-788 I-9919 2-2288 0-8937 4-2207 0-23693 1-46 83-6518 I-II45 63-851 41-915 2-0369 2-2691 0-8977 4-3060 0-23224 1-48 84-7978 1-1231 64-351 42-034 2-0827 2-3103 0-9015 43929 0-22764 1-50 85-9437 1-1317 64-843 42-148 2-1293 2-3524 0-9051 4-4817 0-22313 1-52 87-0896 1-1402 65-327 42-261 2-1768 2-3955 0-9087 4-5722 0.21871 1-54 88-2355 1-1484 65-800 42-370 2-2251 2-4395 0-9121 4-6646 0-21438 1-56 89-3814 1-1566 66-265 42-473 2-2743 2-4845 0-9154 4-7588 0-21014 , 1-58 90-5273 1-1646 66-728 42-571 2-3245 2-5305 0-9186 4-8550 0-20598 i-6o 91-6732 1-1724 67-171 42-668 2-3756 2-5775 0-9217 4-9530 0-20190 1-62 92-8192 1-1800 67-612 42-756 2-4276 2-6255 0-9246 5-0531 0-19790 1-64 93-9651 1-1876 68-045 42-846 2-4806 2-6746 0-9275 5-1552 0-19398 1-66 95-1110 I -1953 68-469 42-930 2-5346 2-7247 0-9302 5-2593 0-19014 1-68 96-2569 1-2023 68-885 43-013 2-5896 2-7760 0-9329 5-3656 0-18637 1-70 97-4028 1-2094 69-294 43-090 2-6456 2-8283 0-9354 5-4739 0-18268 1-72 98-5487 1-2164 69-696 43-166. 2-7027 2-8818 0-9379 5-5845 0-17907 1-74 99-6947 1-2233 70-091 43-233 2-7609 2-9364 0-9402 5-6973 0-17552 1-76 100-8406 1-2300 70-476 43-303 2-8202 2-9922 0-9425 5-8124 0-17204 1-78 101-9865 1-2366 70-856 43-373 2-8806 3-0492 0-9447 5-9299 0-16864 1-80 103-1324 1-2432 71-228 43-433 2-9422 3-1075 0-9468 6-0496 0-16530 1-82 104-2783 1-2495 71-593 43-497 3-0049 3-1669 0-9488 6-1719 0-16203 1-84 105-4242 1-2559 71-952 43-556 3-0689 3-2277 0-9508 6-2965 0-15882 1-86 106-5702 1-2619 72-303 43-615 3-1340 3-2897 0-9527 6,-4237 0-15567 1-88 107-7161 1-2680 72-649 43-666 3-2005 3-3530 0-9545 6-5535 0-15259 1-90 108-8620 1-2739 72-987 43-720 3-2682 3-4177 0.9562 6-6859 0-14957 1-92 110-0079 1-2797 73-319 43-770 3-3372 3-4838 0-9579 6-8210 0-14661 I '94 111-1538 1-2854 73-645 43-816 3-4075 3-5512 0.9595 6-9588 0-14370 1-96 112-2997 1-2910 73-966 43-864 3-4792 3-6201 0-9611 7-0993 0-14086 ' 1-98 113-4456 1-2964 74-274 43-910 3-5523 3-6904 0-9626 7-2427 0-13807 2-00 1 14-5916 1-3017 74-584 43-950 3-6269 3-7622 0-9640 7-3891 .0-13534 2-02 115-7375 ■ 1-3070 74-886 43-993 3-7028 3-8355 0-9654 7-5383 0-13266 2-04' 116-8834 ' 1-3122 75-183 44-032 3-7803 3-9103 0-9667 7-6906 0-13003 2-06 118-0293 1-3173 75-472 44-070 3-8593 3-9867 O-968Q 7-8460 0-12745 2-08 119-1752 1-3222 75-758 44-108 3-9398 4-0647 0-9693 8-0045 0-12493 2-10 1 20-321 1 1-3271 76-037 44-145 4-0219 4-1443 0-9705 8-1662 0-12246 2-12 121 -4671 1-3319 • 76-311 44-177 4-1055 4-2256 0-9716 8-3311 0-12003 2-14 122-6130 1-3365 76-578 44-208 4-1909 4-3085 0-9727 8-4994 0-11765 2-16 123-7589 I -341 2 76-843 44-239 4-2779 4-3932 0-9737 8-6711 0-11533 2-18 124-9048 1-3457 77-102 44-270 4-3666 4-4797 0-9748 8-8463 0-11304 2-20 126-0507 I -3501 77-354 44-297 4-4571 4-5679 0-9757 9-0250 0-11080 2-22 127-1966 1-3544 77-603 44-327 4-5494 4-6580 0-9767- 9-2073 0-10861 2-24 128-3425 1-3587 77-848 44-352 4-6434 4-7499 0-9776 9-3933 0-10646 2-26 129-4885 1-3628 78-084 44-378 4-7394. Table 46. — Hyperbolio functions — Continued. « V sinh K cosh w tanh M In degrees. In degrees. = ta.nv = sec V = sin V 3-00 171-8873 1-4713 84-301 44-861 10-0179 10-0677 0-9951 20-0855 0-04979 3-IO 177-6169 1^808 84-841 44-883 11-0765 II-1215 0-9959 22-1980 0-04505 3-20 183-3465 1-4894 85-331 44-906 12-2459 12-2866 0-9967 24-5325 0-04076 3-30 189-0761 1-4971 85-775 44-925 13-5379 13-5748 0-9973 27-1126 0-03688 3 "40 194-8057 1-5041 86-177 44-936 14-9654 14-9987 0-9978 29-9641 0-03337 3-50 200-5352 1-5104 86-541 44-948 16-5426 16-5728 0-9982 33-1155 0-03020 3-60 206-2648 1-5162 86-870 44-961 18-2854 18-3128 0-9985 36-5982 0-02732 370 211-9944 1-5214 87-168 44-966 20-2113 20-2360 0-9988 40-4473 0-02472 3-80 217-7240 1-5261 87-445 44-971 22-3394 22-3618 0-9990 44-7012 0-02237 3'9o 223-4535 1-5303 87-681 44-975 24-6911 24-7113 0-9992 49-4024 0-02024 4-00 229-1831 1-5342 87-901 44-980 27-2899 27-3082 0-9993 54-5981 0-01832 5-00 286-4789 1-5573 89-227 44-989 74-202 74-208 0-9999 148-41 0-006738 6-00 3437747 1-5658 89-716 44-993 201-71 201-72 0-9999 403-43 0-002479 7-CX3 401-0705 1-5690 89-895 45-000 548-35 548-35 I -0000 1096-6 0-000912 8-00 458-3662 1-5701 89-960 45-000 1490-5 1490-5 i-oooo 2981 -0 O-O00335 9-00 5i5'662o 1-5705 89-986 45-000 4051-6 4051-6 I -0000 8103 -I 0-000123 10 -CX) 572-9578 1-5706 89-995 45-000 IIOI3-2 11013-2 I-oooo 22026-5 0-000045 00 00 1-5708 90-000 45-000 00 CO I'OOOO CO slnh M = - , cosh 7 ■ cosh u e" + e-' Table 47.^ — Period of a wave. lycngth of wave in feet W- Depth of water {h). I 10 100 I 000 10 000 100 000 I 000  1762947 17629473 lO 0-442 1-398 5-922 55-789 557-51 5575 -I 5575 1 557508 5575085 lOO 0-442 1-398 4-419 18-726 176-41 1762-9 17629 176295 1762947 1 ooo 0-442 1-398 4-419 13-975 59-218 557-89 5575-1 55751 557508 ' lo ooo 0-442 1-398 4-419 13-975 44-192 187-26 1764-1 17629 176295 loo ooo 0-442 1-398 4-419 13-975 44-192 139-75 592-18 5579 55751 The period ( r ) of a wave is determined by the equation ,_27CX I . ■21th _ 0-1953 ^ r-=ii^/tanhi4^,= tanh 6-283185-=?- where^ is taken equal to 32-1722 feet per second, as in this table; or r2_ 0-195373^^ j^ tanh 6-283185-' if ^ IS taken equal to 32-16. EEPORT FOR 1897 — PART II. APPENDIX NO. 9. Table iS.— Wave velocity. 617 Uepth of water! i (A)- Length of wave in feet (A). I 10 100 I 000 10 000 100 000 I 000 000 10 000 000 100 000 000 Infinite. i Feet. I i IO lOO  22-63 Ft./ sec. 5-672 17-92 53-40 71-56 71-56 71-56 Ft./ sec 5-672 17-94 56-68 168-9 226-3 226-3 Ft./ sec. 5-672 17-94 56-72 179-2 534-0 715-6 Ft./ sec. 5-672 17-94 56-72 179-4 566-8 1689 Ft./ sec. 5-672 17-94 56-72 179.4 567-2 1793 Ft./ sec. 5-672 17-94 56-72 179-4 567-2 1794 A./ sec. 5-672 17-94 56-72 179-4 567-2 1794 The wave velocity, i. e., velocity of propagation is A/r. Tables 47 and 48 are adapted from Airy's Tides and Waves. Table 49. — Jiatio of vertical to horizontal axes of elliptic orbits of water particles. y 2.1 Ratio of y 2nl. Ratio of y 2^^ Ratio of ! K A axes. A A axes. A A axes. 0-00 0-0000 0-0000 o-io 0-6283 0-5568 0-40 2-5133 0-9869 o-oi 0-0628 0-0627 0-12 0-7540 0-6375 0-50 3-1416 0-9962 0-02 0-1257 0-1250 0-14 0-8796 0-7062 0-60 3-7699 0-9989 0-03 0-1885 0-1863 0-16 1-0053 0-7638 0-70 4-3982 0-9995 0-04 0-2513 0-2461 0-18 1-1310 0-8113 080 5-0265 0-9999 0-05 0-3142 0-3042 0-20 1-2566 0-8501 90 5-6549 0-9999 0-06 0-3770 0-3601 I -00 6-2832 0-9999 0-07 0-4398 0-4134 0-25 1-5708 0-9171 0-08 0-5027 0-4642 0-30 1-8850 0-9549 10-00 62-8319 I -0000 0-09 0-5655 0-5120 0-35 2-1991 0-9757 00 00 I -0000 The maxitium horizontal displacement at the bottom (where jy^o) being A, and 2A, the distance between the i bci of the elliptic orbits, the maximum displacements for other depths are: y ~ X = ^ cosh ly^A cosh 2 tt ^, y y = A sinh ly =A sinh 2 ?r ^ ; .-. -^ = tanh2!r4^, X A' i tvhjch is the ratio tabulated above. 618 UNITED STATES COAST AND GEODETIC SURVEY. Table 50. —Propagation of a free tide wave along a uniform channel. Depths. Velocity of propagation. Wave length, Time required to travel Difference in phase of tide wave Knots, or Fath- oms. Feet. Feet per second. nautical miles, per hour. statute miles per hour. statute miles. I foot. I naut. mile I Stat. mile. per statute mile. per foot. J. A. A. Radians. O O'OOO 0-000 0-000 0-00 [0-00 [o-oooo I 5-672 3-358 3-867 48-03 0-1763 0-2978 0-2586 7-4953 14196 2478 2 8 -022 4-750 5-469 67-93 0-1247 0-2105 0-1828 5-2996 10037 1752 3 9-824 5-817 6-698 83-20 o-joiS 0-1719 0-1493 4-3269 08195 •1430 4 1 1 '344 6-717 - 7-735 96-07 0-08818 0-1489 0-1293 3-7472 07097 1239 5 12-683 7-510 8-648 107-41 0-07886 0-1332 0-1156 3-3517 06348 1 108 I 6 13-894 8-226 9-473 117-66 0-07199 0-1216 0-1056 3-0597 05795 lOIl 7 15-007 8-886 10-232 127-09 0-06662 0-1125 0-09775 2-8327 05365 0936 8 16-043 9-499 10-938 135-86 006234 0-1053 0-09141 2-6498 05019 0876 9 17-016 10-075 11 -602 144-10 0-05877 0-09921 0-08621 2-4983 04732 0826 10 17-937 10620 12-230 151-90 0-05574 0-09416 0-08177 2-3700 04489 0783 II 18-812 11-139 12-826 159-31 0-05316 0-08985 0-07794 2-2597 04280 0747 2 12 19-649 11-634 13-397 166-40 0-05089 0-08598 0-07463 2-163S 04098 0715 13 20-451 12-109 13-944 173-19 0-04890 0-08258 0-07174 2-0785 03937 0687 14 21-223 12-566 14-470 179-73 0-04713 0-07955 0-06911 2-0030 03794 0662 15 21-968 13-007 14-978 186-04 0-04552 0-07686 0-06676 1-9351 03665 0640 16 22-688 13-434 15-469 192-14 0-04407 0-07446 0-06464 1-8737 03549 0619 17 23-387 13-847 15-945 198-05 0-04275 0-07220 0-06270 1-8177 03443 0601 3 18 24-065 14-249 16-408 203-79 0-04156 0-07018 0-06094 1-7664 03345 0584 4 24 27-787 16-453 18-946 235-32 0-03598 0-06079 0-05277 1-5298 02897 0506 5 30 31-067 18-395 21-182 263-09 0-03219 0-05435 0-04721 1-3683 02591 0452 6 36 34-032 20-151 23-204 288-21 0-02939 0-04963 0-04310 1-2491 02366 0413 7 42 36-759 21-765 25-063 311-30 0-02720 0-04593 0-03990 1-1564 02190 0382. 8 48 39-297 23-268 26-794 332-79 0-02545 0-04297 0-03733 1-0818 02049 0358 9 54 41-681 24-680 28-419 352-98 0-02399 0-04052. If r=the periodic time (= 12-4206012 solar hours or^%, lunar day for the tide -wave), then X, or wave-length, = r V^ feet = — -_ Vgh miles. Difference in phase = ^-^r—. A. The nautical mile is taken as 6080 feet. TEEASURY DBPARTMBlSrT UNITED STATES COAST AND GEODETIC SURVEY W. W. DUFFIELD StrPEEmTENDEKT PHYSICAL HYDROGRAPHY MANUAL OF TIDES FAJEIT III By ROLLIN" -A.. UA-RlilS APPENDIX No. 7— REPORT FOR 1894 WASHINGTON aOVEBNMENT PRINTINa OFFICE 1896 TEEASURT DEPAETMENT UNITED STATES COAST AND GEODETIC SURVEY W. W. DUFFIELD SUPEEINTENDENT PHYSICAL HYDROGRAPHY MANUAL OF TIDES P^ET III By ROLLIN A.. HA-RRIS APPENDIX No. 7— REPORT FOR 1894 WASHINGTON aOVERNMENT PRINTING OFFICE 1895 APPENDIX No. 7—1894. M^J^NJJJ^Ju OF TIDES. PART III. SOME CONNECTIONS BETWEEN HARMONIC AND NONHARMONIC QUANTITIES, INCLUDING APPLICATIONS TO THE REDUCTION AND PREDICTION OF TIDES. Submitted for publication December 22, 1894. 125 PREFACE TO PART III. The following pages have been prepared for the purpose of supplying some of the immediate wants of the Tidal Division. They constitute Part III of a proposed manual of tides, the plan of which may be outlined thus : Part I. History of tides, including old methods of treatment. Part II. Tidal observation, equilibrium theory, and harmonic analysis. Part III. Connections between harmonic and nonharmonic quantities. Part IV. Tidal theory and astronomical quantities deduced from tidal observations. Part V. Tidal currents. Tables. Auxiliary tables and a collection of tidal constants, including cotidal charts. As no considerable portion of the material constituting Part III is elsewhere available, it has seemed desirable to publish it in advance of Parts I and II, whose subject-matter has been largely treated in various publications. The object of this paper is, in a general way, indicated by its title; but it may not be out of place to comment here upon certain of its aims and features. Owing to a lack of reasonably precise definitions and adequate modes of reduction, the nonharmonic constants (i. e. those referring to high and low water) have too often been of little service where an accurate and satisfactory knowledge of the tide was desired. It is true that quite accurate predictions have been made, based upon the tables resulting from nonharmonic discussions; but it is to be observed (1) that the tables so used were designed chiefly for particular stations and so were of little or no use generally; (2) that long series of observations were necessary for the construction of such tables; (3) that most nonharmonic discussions have been confined to stations where the diurnal ii' equality is so small as to give no serious trouble. The present attempt to overcome these objections provides an approximate method of analyzing a single month's observations upon high and low water for obtaining both the harmonic and the nonharmonic constants, regardless of the amount of diurnal inequality, and without dififlcult com- putations. It also provides tables for the analysis and prediction of tides, designed for general use, regardless of the location of the station. Little attention has heretofore been given to the classification of tides for the purpose of selecting the best possible port of reference for a given subordinate station, proximity in geo- graphic position having been the sole criterion with few exceptions. Now, it is believed, are given for the first time such harmonic and nonharmonic criteria as will readily show how the types of tide at different stations compare. In this connection are given rules for referring one station to another. The chapter entitled "Prediction of Tides" describes a proposed tide-predicting machine combining the merits of the Thomson and the Ferrel machines; a general graphic process enaljjling one (by means of a set of curves constructed once for all from the harmonic constants of the given station) to predict tides without a machine and without computation, the amount of work involved depending upon the accuracy desired; also an approximate arithmetical method for predicting high and low waters from certain tidal constants. 127 128 UNITED STATES COAST AND GEODETIC SUEVEY. By correlating harmonic and nonharmouic quantities a better insight into tides is obtained than from the consideration of either alone. In this way observable phenomena may be inferred from harmonic constants, inconsistencies may be detected, and missing quantities supplied; also particular values of nonharmonic quantities may be reduced to their mean values, and conversely. §§ 46, 47 are given mainly for the purpose of furnishing a key to the study of Perrel's non- harmonic analyses. These discussions, covering years of observations, throw much light upon the nature of tidal inequalities. The appended tables, already alluded to, are based principally upon those of Baird and Darwin. The considerable computation involved in their preparation has been performed by members of the Tidal Division. Their scope differs somewhat from that of the paper accompany- ing them. This remark applies especially to Tables 1 to 13, which are designed for the harmonic analysis of tides from hourly ordinates, a subject to be treated in Part II. I have to acknowledge much assistance from Mr. L. P. Shidy, whose long experience in tidal work has rendered his advice particularly valuable, and whose cooperation has enabled the work to go on. [MANUAL OF TIDES.] CONTENTS OF PART III. Chapter I. PROPERTIES OF COMPOUND WAVE HAVING A PREDOMINATING COMPONENT. Pagfc Displacement in time and height of the maxima and minima of the predominating component 131 Approximate average values of resultant heights , 132 Combination of two components of equal periods I33 Combination of two components whose periods are as 1 : 2 I34 Accurate and approximate values of resultant heights for special cases compared 133, 135 Restrictions in the application of approximate rules I35 Effect of components Vvith variable amplitudes I35 Speed of a wave the two components of which have speeds or periods nearly equal 135 Effect of small components having speeds commensurable with that of the predomiuating one 136 Conditions for a predominating component I37 Chapter II. COMPUTATION OF NONHARMONIO QUANTITIES FROM HARMONIC TIDAL CONSTANTS. Definitions ". X39 Computation of mean lunitidal intervals 140 Computation of mean range of tide 140 Computation of hourly heights of the M tide 142 Computation of range and speed of diurnal wave and range of tide near times of moon's extreme declination. 142 Computation of ages of tidal inequalities I43 Computation of spring, neap, perigean, and apogean ranges 144 Computation of quantities connected with extreme declinational tides 145 Computation of average range of diurnal wave and of great and small diurnal ranges of tide 147 Computation of average height inequalities; application to planes of reference 147 Computation of half-tide level 148 Tides chiefly diurnal 148 Chapter III. REDUCTIONS OP OBSERVATIONS MADE UPON HIGH AND LOW WATERS. Object '. 149 First reduction I49 Mean sea level and long-period tides 150 Eeduction of spring and neap tides I53 Eeduction of parallax tides 157 Eeduction of declinational tides.... 160 "Datum planes 167 Ferrel's expressions for tidal inequalities 168 Chapter IV. TO RRDUCB results TO THEIR MEAN VALUES. Effect of the longitude of the moon's node 173 Approximate relations used in reducing quantities to their mean values 173 S, Ex. 8, pt. 2 9 129 ' 130 TTNITED STATES COAST AND GEODETIC SUEVEY. Chapter V. ON THE CLASSIFICATION OP TIDES. Page. Harmonic and nonhannonio criteria 175 Predictions made for a principal station adapted to a subordinate station 176 •Constants inferred by comparison of tidal characteristics 177 Cotidal lines I79 Chaptei: VI. PREDICTION 01' TIDES. British tide-predicting machines 180 Ferrel's tide-predicting machine..... 181 A proposed tide-predicting machine 181 A graphic method of predicting tides 183 An approximate method of predicting tides by computation 186 TABLES. 1. The principal harmonic components, with their speeds, coefficients, etc 190 2. Dependence of component speeds upon certain astronomical quantities 194 3. EquUibrmm arguments (Vg + u) at the midnight preceding January 1 of each year, from 1850 to 1950, for the meridian of Greenwich, together with the elements used in computing them 195 4. For adapting the uniformly varying portion ( Fq) of the equilibrium arguments of Table 3 to Greenwich midnight, beginning any day throughout the year 205 5. For adapting the uniformly varying portion ( Vo) of the equilibrium arguments of Table 3 to local midnight for any degree of west longitude 207 6. Values of N, I, and P for Greenwich midnight, beginning each month, from 1850 to 1949 209 7. Values of /, v, |, v' , and 2 v ', corresponding to each half degree ot N 214 8 Values of iJ for completing n for Lj 217 9. Values of Q for completing u for Mi 218 10. Factors F and / for reduction and prediction of tides, computed for the middle of each year, or for July 2 at Greenwich mean noon for common years, and at preceding midnight for leap years 219 11. Values of log R' for obtaining the factor Fof L2 from that of Mj 236 12. Values of log Q' for obtaining the factor F of Mi from that of O, 237 13. Factors F and/, corresponding to every tenth of a degree of I, for reduction and prediction of tides 238 14. Factors {F) for reducing Mn, Ki -I- 0., and X to mean values 247 15. Acceleration in H W and L W of a wave, due to another wave of equal speed 248 16. Height of H W and L W for a tide composed of two waves of equal speed 248 17. Acceleration in H W and L W of a semidiurnal wave, due to a diurnal wave 250 18. Height of H W and L W for a tide composed of a diurnal and semidiurnal wave 250 19. High and low water inequalities in height 250 20. Great tropic range and its duration 251 21. Eifects of various tidal components upon the mean semirange of tide 252 22. Value of i Mn when Mi = 1 253 23. Value of M.2 when i Mn = 1 253 24. PROPERTIES OF A COMPOUND WAVE HAVING A PREDOMINATING COMPONENT. 1. The equation of a compound wave may be written y = A cos {at + a) + B cos {bt + (3) + G cos {ct+ y)+ . . . (1) where A, a, a refer to the predominating component (i. e. the one which determines the number and approximate times of the maximum and minimum values of y) and where B, b, /?; G, c, y; . . . refer to the subordinate components. The values of t rendering y a maximum or a minimum are roots of the derived equation Aa sin {at + a) + Bb sin {bt + /?) + (7c sin {ct + y) + . . . =0. (2) Any root of (2) which causes y to become a maximum may be written t = ^±JL _ ^ _ « , (3) a a a and a minimum (2 ?t + 1) TT ic a ., * = a ¥ ~ a' ^^> where n is an integer or zero and v, to, quantities usually small in comparison with tt. v/a (or w/») is the amount by which the time of any maximum (or minimum) of the compound wave precedes the corresponding time of the predominating component. 2. Approximate expressions for v, w, and y. When V is small and b not many times greater than a, we may assume cos ~ V = cos V (5) a ^ sin - » = — sin v, (6) a a ^ ' and so for cos — v and sin - v. When b differs little from a these are approximations even if v be not so small. Of course the same approximations are true when w replaces v. From equations (2), (3), and (4) we have Aa sin 2 n TT + Bb sin -(2n7r+j^— a J+Gcsiji~(2n7r+jr-aJ+ . .. tan v = a j~-- - ^- ^ ^ ^r , (7) Aa^ cos 2 n 7t+ Bb'^cos -(2n7i:+^ /3—:x] + Gc^ cos ^{^2n7r+—y— a j+ . .. Aasin2'n+iyr+Bbsin-(2n+l7r+j^-aJ + Gcsin^(^2n+l7i+^y-aJ+ . .. tan w=a FT""^"" a \ c / a \ ' (^) Aa'cos2n+l7r+Bb^cos-{^2n+l7t+j/3-aJ + Gc'cos—[2n+l7r+ — /3—aJ+ • ■ ■ 131 132 UNITED STATES COAST AND GEODETIC SURVEY, and the corresponding maximum and minimum values of y are y^\A<iQs2nn-\-B coB^(2nn -\-^fi-a\-if.G c.os-('2n7i-[-'^ y-a\-{- . . .1 cos ?; + ^Asm2n7r + B^^sm^-('2n7r + ^/J-a'\+0Um-('iin7t + -y-a\+ . . .1 sin tJ, (9) y = 1 J.cos2M4-l;r+ B cos -(2n+\.7t+j^ fi—aj + cos^(2 n+1 n+^y—aj + . . .Icosw + yAsm2n+l7c+B-sm-(2n+l7i+'^^-a^+C^(Ms^('2n+l 7i+~y-a)+ . . .'\smw.(lO) 3. To find the approximate average value of the maxima of a compound wave when the speeds of the components are independent of one another. Since cos -» = 1 - ^^ + A.taTi^ V 2 2-4 (11) equation (9) may be written y = n _ 1^^ + . . ^ Ta cos 2 n TT + Boosif2n 71 + pJ-a\ + Gcos^-(^2n7r+^y-a\+ . . .1 + tan * Tl - *^^ + . . .1 [Asiu2n7t + B-sm^f2n7r + ^6-a^+C-sm-f2n7t + -y-a\+ .. L a, a\ b J a a\ c J It is required to find an expression for the average value of y; that is, the value of •] ■ (12) ao « = (13) a, b, c, . . . being independent of one another. The part of (13) which is independent of v reduces to A. The part of (13) which contains tan V to iirst power is 00 ^ / ; JLa sin 2 m ;r + Bb sin-f 2 wtt + j/?— aj+ (7c sin -("2 n n + -y — a\ . . . \ •^^Aa" cos 2 n n + BV" cos- (2 nn ■ir'^ ft - a^ + Cc^ cos ^(^2 w tt + ^^ -«} + (14) The approximate value of the denominator of (14) is Aa^. The numerator consists of products and squares of its terms; the average value of any such product is zero, while for the squares we have, since the average value of (sine)^ = i, ^ [W b'' + G"" c^ + . . .]. Thus (14) is approximately equal to 2 ia^ t^' ^' + ^' '' + • • •)• (15) In like manner the part of (13) containing tan v to the second power is approximately equal to _ 1 [£2 6»+ 0^c^+ . . .]. (16) EEPOET FOE 1894— PAET II. 133 The sum of (15) and (16) being ^,[B^b^+G^c^+ . . .1, • (17) it follows that The average value of the maxima of a compound wave is, approximately, '^+A^^i^'^'+^'''+ • • • J (18) provided a,b,c, . . . are each incapable of expression in terms of the others, and the quantity added to A is relatively small. This also represents the average depression of the minima ; or, it is the average amplitude or semirange of the compound wave. When there are tixed relations between the speeds of the components, the speeds of no two being commensurable with each other, formula (18) may still give the semirange of the compound wave, while the average elevation of the maxima or depression of the minima may not be given with sufiQcieut accuracy. For, any alteration in the average range on account of such a relation implies an alteration in the average of the maxima or minima or both, while the average of the maxima or minima may be altered without necessitating an alteration in the average range. 4. A wave composed of two simple waves of equal periods. Let its equation be y = A cos at + B cos {at + p). (19) . , ^ BsinyS . ■ . - tan a< = tan v = tan w = ^ ^ ^ cos d - (^^) 2/ = ± V A' + B^ + 2AB cos p- (21) The average value of this radical, supposing fJ to take all values from zero to tt*, is — r VA' + B^ + 2 AB cos p d p. ~ Jo (22) This integral represents the average value of the third side of a triangle two of whose sides are the fixed lengths A, B, while the included angle (or its supplement /?) varies uniformly. The integral VA^+W+2ABc6sp d p (^^) X' represents one-half the periphery of an ellipse whose semiaxis major is A + 5, and whose semi- axis minor is A — B. W; A hi By writing 2 cp for ft and sin Oiov 2 J -^ ^, this integral becomes 2 {A + B) times the integral m (sin d, (p) which is tabulated by Legendre in his Traits des Fonctions EUiptiques, Table IX. However the average value of y be computed, the result will serve as a test of the accuracy of formula (18) for this special case, even when B is not small in comparison with A. Compare the values in the last column of Table 16 with column 9, Table 21, where the special form of (18), ^ + 4i-^l^''''5orA + ^, (24). has been tabulated, * So far as obtaining an average value is concerned, it is indifferent whether we regard the speeds as equal and vary /S, or regard the speeds as differing by an infinitesimal and make /3 constant or zero. In the latter case the radical may be written V A' + B^ + 2 AB COB CO where co^ (a -^ h) 1. 134 UNITED STATES COAST AND GEODETIC SUEVEY. It will be seen that the agreement is very close when B is comparatively small; when B = A the discrepancy amounts to a trifle over '2 per cent of A or B. It follows from this that if a and b denote the semiaxes of an ellipse, the periphery of the same is, as is well known, approximately To find when yS' is small. 1 r^' , J, j VA^ + B^ + 2ABcosp d ^ Upon replacing cos /3 by 1 - 2 sin^ ^, the radical may be written A + B- which becomes 2ABsm^^ A-yB A + B-flA^ ^ 2 A+B ftl2 when /? is small. The average value of ^ when /i varies from to /?' is ^; this gives for the required result /^" ^^ (25) A + B- 6 A + B In like manner the value of 1 r^ + p' P -\-B'^ + 2ABco^fid p, which is the average value of the radical between ft = 7t and ft=7t + fd', becomes, when yS' is small ft"' AB A-B + & A-B' (26) 5. A wave composed of two simple waves, the period of one being twice that of the other. Let its equation be y =Aco&at + Bcos,{ht+ ft). (27) .-. i:Asi.xi.^v -\- B t&n^v C0& ft - B siw ft = (28) where v = — at and b = ^a. The angle J v can be found by the following graphic process : Fig. 1 Lay off ft from the horizontal line, figure 1, and let B be laid off in the direction thus determined. Draw an arc to the radius 4 A; move this radius until the heavy lines are equal. The angle so determined is ^ v. It is convenient to work upon cross-section paper, using a thread for the radius iA. REPORT FOR 1894— PART II. 135 The values of v are sliown ou Plate I (Table 17) for each value of p. The corresponding values of y are given ou Plate II (Table 18). Formula (18) now becomes whose values are given in column 15, Table 21. The values in this column, compared with those obtainable from Table 18, afford a means of testing formula (18) for the special case here considered, even when B is greater thaii A. 6. If the formula expresses the average amplitude of a wave composed of the simple waves A and B, even when B is not small in comparison with A, then the formula ^ + ro 1^' ^' + ^' ''^ (3i> expresses the average amplitude of a wave composed of the simple waves A, B, and G, provided c differs from b by an infinitesimal and B- + C of (31) is not greater than B^ of (30). The amplitude of B and G combined is VW +C^+2BGcos (h ~ c) t. (32) Replacing B'' of (30) by the square of this expression, the average value of B'' becomes W + GK In like manner, if formula (31) apply to three waves, the formula ^ + iTa> [^' ^' + OU"^ + D' d?] (33) will apply to four waves, provided c and d differ from 6, and so from each other, by an infinitesimal and B^+ G'^ + D^ of (33) is not greater than B^ + (7^ of (31) or W of (30). So one can proceed for any number of simple waves. 