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R. and
P. R7 Benwell, 1970: PSCM hdices in syn(ptic climatology and 1ong-rangefecasthíg. Met. Mag., Lond. 99, 232.Murray, R. anR.PW. Lewis, 1966: Sorne
aspects of the synopie climatology of the British Jles as measure' ' simple}ndices. Met. rag., Lond. 95, 193.
Naris, J., 1964: Seasonal pe .tence and
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of European b)ckin
during 1958-1960. Tellus 16, 394.Namias, J., 1965: Sb6rt-period c1im c
fluctuations. Science247, 696.
erkins, H. T., 197e: Inertial oscillations in
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n of Major Baltic
Inflow-gods Hole Oceanogr. Instn, 155,-(unpblished manuscript).
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the bottdì fauna in the eastøfn Baltic in i959. Ann. b ., Copenh. 16A1959), 86.
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'62: TJ distribution ofbottom fauna in t eastern Baltic in 1960. Ann. biol., Copenh. 960), 93.
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ARCHIEF
lab.
y. Scheepsbouwkunde
Technische Hogeschool
Deift
Dr obert R. Dic westoft, Suffolk,
Sonderdruck aus der Deutschen Hydrographisehen Zeitschrift, Band 26, 1973, Heft 3
/
j
A Close Approximation to the Exact Theory of Water Waves
By
B. V. Korvin-Kroukovsky
UDC 551.466.2 + Theory
Summary. The simple formulation of wave properties, which was stated by Lord Rayleigh [1876], prior to the expansion of it in series, is examined. The application of it to waves of varying steepness yielded wave profiles, which are found to be identical with the ones obtained by the use of G. G. Stokes' [1849, 1880] series of 8-th order. Deviations from the, usually postulated, uniform air pressure on water surface are found to be negligibly small. The simple form of the expressions, and the resultant ease of their application, makes them useful in problems arising in marine engineering.
Eine starke Approximation zur exakten Theorie der Wasserwellen (Zusammenfassung). Unter-sucht wurden die von Lord Ray1 e igh [1876] aufgestellten einfachen Formeln für Welleneigen-schaften, ehe sie in Reihen entwickelt wurden. Es wird festgestellt, daß ihre Anwendung auf Wellen unterschiedlicher Steilheit Wellenprofilformen ergibt, die mit denen der Stokesschen Reihen-entwicklung bis zur 8. Ordnung (G. G. Stokes [1849, 1880]) identisch sind. Abweichungen von einem gleichmäßig verteilten Luftdruck, der gewöhnlich zugrundegelegt wird, erweisen sich als unwesentlich. Die Formelausdrücke sind wegen ihrer einfachen Form und der daraus resultierenden einfachen Anwendung für den Gebrauch bei meerestechriischen Aufgaben geeignet.
Une approximation serrée de la théorie exacte des vagues de la mer (Résumé). On examine l'expression simple des propriétés des ondes qui a été établie, antérieurement à son développement en série, par Lord Ray1 e i g h [1876]. Son application à des vagues d'escarpement varié donne des profils de vagues qui se trouvent identiques à ceux obtenus en utilisant les séries du 8ème ordre de G. G. Stokes [1849, 1880]. On a trouvé que les écarts de la prcssion de l'air sur la surface de l'eau, pression généralement considérée comme uniforme, étaient négligeables. La forme 8imple des expressions et les facilités qui en découlent dans leur application, les rend utiles pour la réso-lution des problèmes particuliers de construction navale.
Introduction.
The problem of waves of finite height on the surface of deep water was origi-nally solved by G. G. Stokes [1849]. The solution was in the form of an infinite series, with increasing powers of the amplitude, a, and of the cosines of increasing multiples of the angle k (x - ct), where x is the distance in the direction of the wave propagation, c is the wave celerity, and t is the time. The wave number k = 2rr/A, where ) is the wave length. In the original article three terms of the series were formulated, and in a later article, G. G. Stokes
[1880], the series was extended to five terms, i.e. to the term containing the wave amplitude in 5-th power. J. R. Wilton [1914] extended the series to 10 terms, and computed a very steep wave profile of )./2a = 7.3 to 8-th order. The present author computed the wave profile
of a moderately steep wave of 2/2a = 11.1, using Wilton's series of 8-th order. These wave profiles, computed by series of such a high order, will be assumed to be essentially exact, and will be used as standards of comparison for a simpler solution, to be presently discussed. Lord Rayleigh [1876, page 268] began his discussion by presenting simple, one-term expressions for the velocity potential and stream function, and stating that these expressions satisfy the conditions of continuity and freedom from rotation. However, he did not further investigate the properties of these expressions, but immediately proceeded to develop them
into series. H. Lamb [1945, page 417] followed Lord Rayleigh in stating Eq. (3), and immediately proceeding with the expansion of it in series. It appears that simple original expressions of Lord Rayleigh were likewise neglected by all subsequent writers, and the series solution is usually the only one mentioned, whenever the theory of the waves of finite height is discussed. Third order series of 1876 was extended by Lord Rayleigh [1920] later to fifth order.
Ko rvin-Kroukovsky, Close Approximation to the Exact Theory of Water Waves 107 It is the objective of the present work to investigate the properties of the original Lord Rayleigh's expressions, and to demonstrate that they are not only convenient in use, but that they represent a close approximation to the exact theory of progressive waves on the surface of deep water. In presenting this investigation, the basic expressions for the velocity potential, stream function, and the wave profile will be stated first, and a simple method of computing the wave profile will be described. The resultant wave profiles will be compared with those obtained from the high order series. This will be followed by presenting a set of derived equations, with the demonstration of their compliance with the continuity, freedom
from rotation, and kinematic surface conditions. Subsequently the deviations of the free surface pressure, from the postulated uniform one, will be evaluated and discussed.
Basic equations and the wave profile. Lord Rayleigh [1876, page 268] reduced the problem to the one of steady motion by attributing to the water the velocity - c, equal
and opposite to the celerity of waves. His equations will be here reproduced, with a slightly modified notation, as they apply to progressive waves.
The velocity potential is
cp=KacekYsiri0. (1)
The stream function is
= Kac eky cos 0. (2)
And the corresponding wave profile is
y = - Ka e'
cos 0. (3)In the foregoing, O is written for brevity for k (x - ct). The ordinate, y, denotes the wave surface displacement from a reference line, positive in upward direction. The coefficient K is introduced by the present author in order to make Yrnax- 1/min =2a = H. By fitting empiri-cal curves to results of detailed computations, the following algorism was obtained for a
priori estimate of K:
K = 0.785 (2/H - 6.5)°°. (4)
At first sight, the Equation (3) appears to be awkward, since it contains the dependent
variable, y, on both sides of the equation. Indeed, it appears that explicit mathematical
solution of ky = f (kx) has been possible only by expansion in series, requiring lengthy and laborious computations to determine the wave profile. In practice, however, a simple solu-tion is obtained by transposing Eq. (3), and introducing the auxiliary variable z
z=ye-CY=_Kacosø.
(5)The values of y, corresponding to assigned values of Ka cos O are then found in the Tables 1, 2, or 3, which list z = /(y). Thus the profile of the wave of any steepness can be obtained with but little more labor than in case of linearized cosinusoidal profile.
The second and third columns in Tables 4, 5, and 6 give the comparison of the wave surface displacements, ky, computed by Eq. (3), with the ones obtained by serial solutions of high order for three waves of the steepness A/H of 30.9, 11.1, and 7.3. Third column in the Table 4 contains the data computed by Lord Rayleigh [1917, page 483) by the series of fifth order. These data, computed with high precision, are rounded here to four decimal points. Third column of the Table 5 contains the data computed by the present author, using J. R. Wilton's [1914] series of 8-th order. Table 6 contains the data computed by
J. R. Wilton [1914] by the series of 8-th order. It will be observed that the data computed by Eq. (3) are essentially identical with the ones computed by series of high order, even in case of waves as steep as A/H = 7.3.
It appears that in all theoretical work, mentioned in the introduction, the primary
interest was in periodically varying ky as a function of kx, and little attention was paid to constant factors. The necessary equality of the water raised above the mean water level,
108 Deutsche Hydrographisehe Zeitschrift, Jahrgang 26, 1973, Heft 3
and displaced below it, has not been observed. In order to achieve this equality, it is necessary to place the reference line at a depth Yo below the water surface, so that the final wave
profile ordinates are evaluated as = y - Yo' thus increasing the depression, and reducing the elevation. The constant Yo is defined by
Yo = (1/it) f ky dkx. (6)
By fitting empirical curves to results of detailed computations, the following algorism was obtained for a good approximation to Yo
= 3 (2/H)-85. (7)
Derived equations, and the proof of validity. The following equations are derived from Equations (1), (2), and (3): The horizontal water velocity is given by
u = -
= -
= - Kakc eky cos O = ¡coy. (8)Cy Cr The vertical water velocity is given by
The continuity equation,
is satisfied, since
v=.L
= -
= Kakc eky sin O.Cx Cy
Cu Cv
-+-= O
Cr Cy Cu CvKalc2 C e'Y sin O.
By substitution of u = - Cp/Cx and y = - C(p/Cy in the foregoing equation, theLaplace equation is obtained
a2 C2p Cx2+
Cy
-In the remainder of the analysis, it will be convenient to impose on the wave system the velocity - e. The kinematic surface condition, in the resultant steady state flow, takes the simple form
f dy\
(uc).
(13)The wave slope, dy/dx, is evaluated on basis of Eq. (5), using the auxiliary variable z, and remembering that eky is a function of y
=
(1 - ky)' = eku(1 + Kak eky cos O)', (14)=
((\
= Kak e' (1 + Kak eku cos O) sin 0, (15) dz \dz \dx/(uc) =_c(1+KakekY cosO).
(16)The substitution of Equations (15) and (16) into Eq. (13) results in the equation identical with Eq. (9), thus proving that the kinematic surface condition is satisfied.
Thus Equations (1), (2), and (3) appear to be exact, in the sense that all hydrodynamic conditions are satisfied without resorting to approximations. However, the question arises as to the pressure distribution over the water surface, which will be considered in the next section.
Korvin - Kr ou ko vsky, Close Approximation to the Exact Theory of Water Waves 109 Wave surface pressure and wave celerity. The wave celerity is evaluated by considering the balance of energies in a stream tube adjacent to the free water surface. The usual assump-tion is made that the air pressure is uniform over the water surface, and that the only external force acting on the water is that of gravity. Therefore, changes of kinetic energy must be equal in magnitude to changes of potential energy. This is expressed as
U2 + 2gy = constant, (17)
where U2 = ( c + u)2 + y2 = c 2c2 icy + (Kak)2 c2 e2kY (18) and, after substitution of the foregoing into Eq. (17), and incorporating c2 into the constant,
- c2 icy + (Kak)2 c2 e2kY = constant. (19) For very low waves, i.e. a small ratio y/2, the exponential can be taken as unity, and the second term on L.H. side can be incorporated into the constant. Setting the coefficient of icy equal to zero, leads to the evaluation of the celerity as c2 = g/k.
By expanding the exponential in series, i + 2Icy +
-..., and incorporating the term
(Ka 1cc)2 into the constant, the following relationships are obtained: The celerity
e2 = (g/lc) [1 (Kak)2] -1 (20)
and the neglected pressure term
Q = (Kalc)2 ¿ (e2k - i - 2ky). (21)
The designation R will be used for the deviation ofQ from the mean value, computed over
the wave length, O < O < 2it, divided by the total potential energy change, g11. The values of R are listed in the fourth column of the Table 5, for the moderately steep wave of 2/H = 11.1. The root-mean-square value of R in this case is 0.074. This magnitude of R can be somewhat reduced by using true wave elevation, , instead of the elevation, y, with respect to the
reference line. After substitution of y = j + Yo in Eq. (19), the expansion of the exponential takes the form
e2k7e2kYo=(1+2k+ ...)(i+2ky0+2k2y+ ...).
If the terms containing icy0 « 1 were neglected, Equations (20) and (21) retain the same form, but with in place of y. Preliminary calculations showed that a further decrease of R
can be obtained by including the terms with Yo and y, and by empirically adjusting the
coefcients of these terms. This action leads to the expressions
c2
=
() [1
(Kak)2(1 + 41cy0 + lOk2Y)] -1 andQ = (Kalc)2 c2 [e2k'7 (2 + 81cYo + 2Ok2Y) ku].
The resultant values of R are listed in last columns of Tables 4, 5, and 6. The coefficients of g/k, and root-mean-square values of R are summarized as follows:
The value of kc2/g = 1.176, for waves of 2/H = 7.3, bears a reasonable relationship to 1.20, found by J. H. Michell [1893, page 437] for waves of extreme height, 1/H= 7.05.
2/H lcc2/g R
30.9 1.01 0. 000 19
11.1 1.071 0.0038
i 10 Deutsche Hydrographische Zeitschrift, Jahrgang 26, 1973, Heft 3
Concluding remarks. The comparison of second and third columns in Tables 4, 5, and 6 shows that, within computational accuracy, the wave proffles generated by Eq. (3) are identical with the ones obtained by high order series: fifth order for A/H = 30.9 waves, and 8-th order for A/H = 11.1 and 7.3 waves. J. R. Wilton [1914], whose data were used for Table 6, did not discuss the accuracy of his figures, but presented ¡cx and ky to two decimal points. The present author carried his computations, for A/H = 11.1 waves, to four decimal points, but
in view of the large number of steps, involved in the use of Stokes-Wilton series, only two decimal points are reasonably certain. Much simpler calculations by Eq. (3) are probably valid to three decimal points. Lord Rayleigh [1917] carried his computations for A/H = 30.9 waves to extreme accuracy: the last adjustment of coefficients in a five-term series produced the change of ky at wave crest from 0.11188 to 0.11191.
The original postulate of the uniform pressure distribution on the water surface is satis-fied by Eq. (19), provided that the terms containing y2, and higher powers, in the expansion of the exponential, are neglected. The quest for a closer approximation led G. G. Sto k e s and Lord R ayleig h to the expansion of Eq. (3) in series, with various adjustments of arbitrary constants. The only evaluation of the magnitude of the pressure deviation, from the postulated uniform one, is found in the work of Lord Rayleigh [1917, page 483], in case of waves of A/H = 30.9. After the last adjustment of the five-term series, the deviation became but one-millionth of the static difference between the crest and trough.
The foregoing discussion of the accuracy shows that the problem was treated as a purely
mathematical one, without regard to physical limitations, inherent in the formulation of
initial postulates. Within the realm of theoretical physics, the limitation on the meaningful level of accuracy is partially determined by the difference in behavior of the real, slightly viscous, fluid and the ideal frictionless one. Furthermore, the postulated uniform pressure distribution on water surface is an idealization, and does not exist in nature. In case of waves propagating in still air, there is a certain drop of the pressure at wave crests, and an increase of the pressure at wave troughs. In case of the wind velocity equal to wave celerity, there is a formation of a shear air flow, which modifies the pressure distribution.
In the realm of oceanography, the meaningful accuracy is severely limited to magni-tudes which can be discerned in experiments or observations. Apart from problems connected with instrumentation, this level is low, since one can hardly find monochromatic waves in nature. Even the best wavemaking installations in towing tanks, used in naval architecture, contain an appreciable amount of harmonics, while even the purest swell in the ocean is described by a frequency spectrum. It appears to the present author that the deviations from the idealized pressure distribution, summarized at the end of the previous section, arefar smaller than the ones which can be discerned in observations or experiments, and that for
all practical purposes Equations (1), (2), and (3) can be accepted as 'exact." The simple
form of these equations permits them to be utilized in problems, which have been habitually treated by assuming a simple sinusoidal waves. Considerable difference in results may be expected in such problems, for instance, as the maximum forces exerted by sea waves on piles, and the forces, and their time rate of change, causing the vibration of ships.
Korvin-Kroukovsky, Close Approximation to the Exact Theory of Water Waves 111
Table i
Table 2
Icy 1cyeY Icy
z = kye1'Y y negative
kyekY ky kyeY
kyeY
kykyeY
Icy0.007 0.0070 0.030 0.0309 0.074 0.0797 0.118 0.1328 0.200 0.2443 0.008 0.0081 0.032 0.0330 0.076 0.0820 0.120 0.1353 0.205 0.2517 0.009 0.0091 0.034 0.0352 0.078 0.0843 0.122 0.1378 0.210 0.2591 0.010 0.0101 0.036 0.0373 0.080 0.0867 0.124 0.1404 0.215 0.2666 0.011 0.0111 0.038 0.0395 0.082 0.0890 0.126 0.1429 0.220 0.2741 0.012 0.0121 0.040 0.0416 0.084 0.0914 0.128 0.1455 0.225 0.2818 0.013 0.0132 0.042 0.0438 0.086 0.0937 0.130 0.1480 0.230 0.2895 0.014 0.0142 0.044 0.0460 0.088 0.0961 0.132 0.1506 0.235 0.2972 0.015 0.0152 0.046 0.0482 0.090 0.0985 0.134 0.1532 0.240 0.3051 0.016 0.0163 0.048 0.0504 0.092 0.1009 0.136 0.1558 0.245 0.3130 0.017 0.0173 0.050 0.0526 0.094 0.1033 0.138 0.1584 0.250 0.3210 0.018 0.0183 0.052 0.0548 0.096 0.1057 0.140 0.1610 0.255 0.3291 0.019 0.0194 0.054 0.0570 0.098 0.1081 0.260 0.3372 0.020 0.0204 0.056 0.0590 0.100 0.1105 0.145 0.1676 0.265 0.3481 0.021 0.0214 0.058 0.0615 0.102 0.1130 0.150 0.1743 0.022 0.0225 0.060 0.0637 0.104 0.1154 0.155 0.1810 0.023 0.0235 0.062 0.0660 0.106 0.1179 0.160 0.1878 0.024 0.0246 0.064 0.0682 0.1OS 0.1203 0.165 0.1946 0.025 0.0256 0.066 0.0705 0.110 0.1228 0.170 0.2015 0.026 0.0267 0.068 0.0728 0.112 0.1253 0.175 0.2085 0.027 0.0278 0.070 0.0751 0.114 0.1278 0.180 0.2155 0,028 0.0288 0.072 0.0774 0.116 0.1303 0.185 0.2226 0.029 0.0299 0.074 0.0797 0.118 0.1328 0.190 0.2297 0.030 0.0309 0.195 0.2370 0.200 0.2443 ky kyekY z = kyekY y positive <0.200 Icy kye"Y
ky kyeY Icy kyekY
0.007 0.0070 0.030 0.0291 0.076 0.0704 0.120 0.1064 0.008 0.0079 0.032 0.0310 0.078 0.0721 0.122 0. 1080 0.009 0.0089 0.034 0.0329 0.080 0.0738 0.124 0. 1095 0.010 0.0099 0.036 0.0347 0.082 0.0755 0.126 0.1111 0.011 0.0109 0.038 0.0366 0.084 0.0772 0.128 0.1126 0.012 0.0119 0.040 0.0384 0.086 0.0789 0.130 0.1142 0.013 0.0128 0.042 0.0403 0.088 0.0806 0.132 0.1157 0.0140.015 0.01380.0148 0.044 0.0421 0.090 0.0823 0.134 0.1172 0.046 0.0439 0.092 0.0839 0.136 0. 1187 0.016 0.0157 0.048 0.0457 0.094 0.0856 0.138 0. 1202 0.017 0.0167 0.050 0.0475 0.096 0.0867 0.140 0. 12 17 0.018 0.0177 0.052 0.0494 0.098 0.0888 0.019 0.0186 0.054 0.0512 0.100 0.0905 0.145 0.1254 0.020 0.0196 0.056 0.0530 0.102 0.0921 0.150 0. 1291 0.021 0.0206 0.058 0.0547 0.104 0.0937 0.155 0. 1327 0.022 0.0215 0.060 0.0565 0.106 0.0953 0.160 0. 1363 0.023 0.0225 0.062 0.0583 0.108 0.0969 0.165 0. 1399 0.024 0.0234 0.064 0.0600 0.110 0.0985 0.170 0. 1434 0.025 0.0244 0.066 0.0618 0.112 0.1001 0.175 0. 1469 0.026 0.0253 0.068 0.0635 0.114 0.1017 0.180 0.1504 0.027 0.0263 0.070 0.0653 0.116 0.1033 0.185 0. 1538 0.028 0.0272 0.072 0.0670 0.118 0.1049 0.190 0. 157 1 0.029 0.0282 0.074 0.0687 0.120 0.1064 0.195 0.1605 0.030 0.0291 0.076 0.0704 0.200 0. 1637
112 Deutsche Hydrographische Zeitschrift, Jahrgang 26, 1973, Heft 3
Table 3
kyekv 0.200
z = y positive >
Table 4
= 30.9 Wave Profile and Pressure 1)eviation from the Mean, R
Root-mean-square of R = 0.00019
ky kyekr ky kyeku ky kyekY ky kyekY
0.200 0.1637 0.325 0.2348 0.450 0.2869 0.575 0.3236 0.205 0.1670 0.330 0.2372 0.455 0.2887 0.580 0.3247 0.210 0.1702 0.335 0.2396 0.460 0.2904 0.585 0.3259 0.215 0.1734 0.340 0.2420 .0.465 0.2921 0.590 0.327 1 0.220 0.1766 0.345 0.2443 0.470 0.2938 0.595 0.3282 0.225 0.1797 0.350 0.2467 0.475 0.2954 0.600 0.3293 0.230 0.1827 0.355 0.2489 0.480 0.2970 0.605 0.3304 0.235 0.1858 0.360 0.2512 0.485 0.2986 0.610 0.3314 0.240 0.1888 0.365 0.2534 0.490 0.3002 0.615 0.3325 0.245 0.1918 0.370 0.2556 0.495 0.3017 0.620 0.3335 0.250 0.1947 0.375 0.2577 0.500 0.3033 0.625 0.3345 0.255 0.1976 0.380 0.2599 0.505 0.3048 0.260 0.2005 0.385 0.2620 0.510 0.3063 0.265 0.2033 0.390 0.2640 0.515 0.3077 0.270 0.2061 0.395 0.2661 0.520 0.3092 0.275 0.2089 0.400 0.2681 0.525 0.3106 0.280 0.2116 0.405 0.2701 0.530 0.3120 0.285 0.2143 0.410 0.2721 0.535 0.3133 0.290 0.2170 0.415 0.2740 0.540 0.3147 0.295 0.2196 0.420 0.2760 0.545 0.3160 0.300 0.2222 0.425 0.2779 0.550 0.3173 0.305 0.2248 0.430 0.2797 0.555 0.3186 0.310 0.2274 0.435 0.2819 0.560 0.3199 0.315 0.2299 0.440 0.2834 0.565 0.3211 0.320 0.2324 0.445 0.2852 0.570 0.3223 0.325 0.2348 0.450 0.2869 0.575 0.3236 kx
degrees by Eq. (3)ky by Lord Rayleighky
R based on Eq. (23) 0 -0.0916 -0.0912 0.000268 11.25 -0.0900
-
0.000255 22.5 -0.0851 -0.0848 0.000205 33.75 -0.0765-
0.000130 45 -0.0663 -0.0662 0.000045 56.25 -0.0525-
-0.000052 67.5 -0.0365 -0.0369 -0.000145 78.75 -0.0190-
-0.000220 90 0 0 -0.000275 101.25 0.0200-
-0.000275 112.5 0.0400 0.0398 -0.000247 123.75 0.0586-
-0.000185 135 0.0767 0.0763 -0.000082 146.25 0.0915-
0.000028 157.5 0.1027 0.1024 0.000130 168.75 0.1100-
0.000200 180 0.1124 0.1119 0.000228Korvin-Kroukovsky, Close Approximation to the Exact Theory of Water Waves 113
Table 5
1/H = 11.1 Wave Profile and Pressure Deviation from the Mean, R
* Computed by the present author using J. R. Wilton's [1914] series of 8-1h order.
Table 6
2/H = 7.3 Wave Profile and Pressure Deviation from the Mean, R
Lamb, Sir Horace, 1945: Hydrodynamics. New York: Dover Pubi.
Michell, J. H., 1893: The highest waves in water. Philos. Mag., Ser. 5, 3ò, 430.
Rayleigh, (Lord) J. W. S., 1876: On waves. Philos. Mag., Ser. 5, 1, No. 4, 257.
Rayleigh, (Lord) J.W.S., 1917: On periodic
irrotational waves at the surface of deep water. Philos. Mag. 33, 381.
Rayleigh, (Lord) J. W. S., 1920: Periodic waves in deep water advancing without change of type. Scientific Papers 6, 11.
Root-mean-square of R = 0.0140
References
Eingegangen im Februar 1973 Anschrift des Verfassers:
Prof. B. V. Korvin-Kroiìkovsky, East Randolph, Vermont 05041, U.S.A.
8
Stokes, G. G., 1849: On the theory of oscil-lating waves. Trans. Cambridge Philos. Soc. 8,
441.
Stokes, G. G., 1880: Supplement to a paper on the theory of oscillating waves. Mathe-matical and Physical papers 1, 314.
Wilton, J.R., 1914: On deep water waves. Philos. Mag., Ser. 6, 27, No. 158, 385. kx
degrees by Eq. (3)ky by Wilton*ky based on Eq. (21)R based on Eq. (23)R
0 -0.206 -0.0202 - 0.0035 0.0045 20 -0.196
-
-0.0040 0.0042 37.88 -0.167 -0.164 - 0.0050 0.0024 60 -0.113-
-0.0066 - 0.0005 76.84 -0.058 -0.055 - 0.0076 - 0.0030 100 0.048-
- 0.0077 -0.0060 120.56 0.148 0.145 - 0.0051 - 0.0064 137.52 0.235 0.233 -0.0003 -0.0043 157.17 0.318 0.323 0.007 1 0.0002 164.57 0.340 0.345 0.0095 0.0019 172.19 0.355 0.359 0.0113 0.003 1 180 0.360 0.364 0.0119 0.003 1 Root-meari.square of R = 0.0074 0.0038 kxdegrees by Eq. (3)k1j by Wiltonky based on Eq. (23)R
o -0.256 -0.24 0.0157 20 -0.237 0.0133 36.67 -0.214 -0.20 0.0105 60 -0.143 0.0025 74.49 -0.083 -0.09 -0.0036 110 0.063 -0.0148 115.17 0.166 0.13 - 0.0191 130.64 0.286 0.26 -0.0192 148.98 0.439 0.42 -0.0097 157.58 0.507 0.53 -0.0011 167.89 0.562 0.59 0.0083 180 0.605 0.62 0.0172
Sonderdruck aus der Deutschen Hythoi phischen Zeitschrift, Band 26, 1973, Heft 3
Tiefenkarte voir zentralen
i1 des östliche%'Gotlandbeckens
/
(Hierzu e1 1)Von Eckard Ho flan
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,/'
//
\UDC 551.462.32 (084.3); ANE Baille Sea Zuimmentassung. Eine 1970 durchgefüf'te Vermessiing\4es zentralen Teils des östlichen Gotlandbeckens durch W.F.S. ,,Planet" ergibt eine in EinzeIheiteiìehr abweichende Tiefenvertei-lungegenüber der von B. Schulz [1956j veröffentlichten Karte s dem Jahrç 1942. Dio neue Tiofenkarte umfaßt den tiefsten Teil des Beckens zwischen 56°50' N und 5703v N, 19°25' E und ø°4O' E im Maßstab i : 200 000. Eine Weiserkarte zur Tiefonkarte gj Abb. i wieder. Tiefen1inien sind in Schritten von 5 m gezeichnet. Das Gotlandtief hat'eine Tiefe von 240 m / (57° 15,5' N ; 2006' E). Die Ortsbestimmung wurde durch Decca-Navgátion vorgenommen. Der tageslichtabhängige Fehler beträgt bis auf die südöstliche Ecke der/Karte"± 0,5 Sm. Dort erhöht
sich der Fehler auf ± i em. Der feste Fehler der Decca-Navigation vernachlässigbar, da das Ver-messungsgebiet außerhalb nnmittelbarer Küstennähe liegt. D)1 verwendeten Lotproflle sind in der Karte durch Reihung<on Tiefenangaben erkennbar. Di'Lotungen wurden ach der während der Vermessung aufgeri,ommenen mittleren hydrographiscen Schichtung korrigiert
Bathymetric cbltrt of the otland Deep in the Centrúl Baltic (Summary). Recent hosoundings in the central part óf the eastern Gotland Basin by ,R.V. "Planet" yield other details ót the depth distribution than stated by B. Sc h u 1 z [1956] i"a chart of 1942. The new bathymèric chart comprises the deepest part of the basin betwén 56°50' N and 57°30' N, 19°25' E and20°40' E with a scale of 1:200000. A large-seal map in i'g. i shows the location both of the surveyed al'ea and two recent Swedish bathymetric charts of'the Central Baltic. Depth contours are drawn"every
5m. TheGot1and Deep has a depth o2"40 m (57°l5,5' N; 20°6' E). For purposes of localizdtion Decca navigation has been used. The' inherent day-light-dependent inaccuracy has,an amo*it of ± 0,5 nra except in the southeastern corner of the chart, where it increases up to-4- i rim. Sinòe the area is far off the coast, the stationary error can be neglected. The profiles are illustrated b rows of depth numbers a1on the corresponding straight courses. The ßóundings have been 9orrected according to the actual mean hydrographie stratification.
Carte des fonds de lapartie centrale du bassin oriental de Gotlaiíd (Résumé). Une campagne J de sondages exécutée en 1970 par le bâtiment hydrographe «Planet», dans la partie centrale du bassin oriental de Got,land montra que, dans ses détails, la disposition des fonds différait notable-ment de celle indiqmie par la carte que publia B. Schulz 11956] au cours de l'année 1942. La nouvelle carte r000uvre, à l'échelle de 1:200000ème, Ia partie la plus profonde du bassin, entre 56°50'N et 57°3,ON', 19025'E et 20°40'E. La figure 1 reprsente la location de la région des sondages bathymétrique. Les lignes dos fonds sont tracées)lë 5 e' 5 mètres. La fosse do Gotland a une profondeur,,4e 240 mètres (57°15,5'N; 2006'E). LoeositionsQnt été déterminées par le système de navìgati9x Decca. L'erreur due à la lumière d'jour atteint 0,5 mille marin jusqu'à l'angle Sud-Est de ,.l'a cartOE. Là, l'erreur monte jusqu'à ± i mille marin\L'erreur systématique du système Deccíest négligeable car la zone des mesures n'est pas au voisinge immédiat de la côte. Les profils dessondes sont reconnaissables sur lacarte en vue de la disposition des fonds. Les sondes ont été 7irigées
d'après la stratification hydrographique moyenne relevée au cours des mesures.
Das
55°40' N, der Breit'von Memel, bis 59°20' N, der Breite vontockholm. Durch einenGotlandbecken wufaí3t den größten Teil der östlichen Osqee. Es erstreckt sich von von Norden nach Süden verlaufenden Rücken, der die Insel Gotlandägt, ist esin ein west-liches und ein ösIihes Beckèx unterteilt. Die größte Tiefe, das Gotlanief, befindet sich im östlichen Teil äétlich von Gotland im Ùbergangsgebiet zwischen den ausgeglichenen Boden-formen ira Süden und den unruhigen im Norden.In theses Gebiet sind seit eingen Jahren größere ozeanographisehe id geologische Forschzngsunternehmen durch die Aiainerstaaten der Ostsee gerichtet gewèn, ohne daß
