Very simple formulae are derived for the difference in mean level between the two sides of a submerged breakwater when waves are incident on it at an arbitrary angle. The formulae apply also to waves undergoing refraction due to changes in depth and to waves in open channel transitions

Longuet-Higgins, M. S.

English

Yale University

      P.O. BOX 208118 | NEW HAVEN CT 06520-8118 USA | PEABODY.YALE. EDU   JOURNAL OF MARINE RESEARCH The Journal of Marine Research, one of the oldest journals in American marine science, published important peer-reviewed original research on a broad array of topics in physical, biological, and chemical oceanography vital to the academic oceanographic community in the long and rich tradition of the Sears Foundation for Marine Research at Yale University.  An archive of all issues from 1937 to 2021 (Volume 1–79) are available through EliScholar,  a digital platform for scholarly publishing provided by Yale University Library at  https://elischolar.library.yale.edu/.  Requests for permission to clear rights for use of this content should be directed to the authors, their estates, or other representatives. The Journal of Marine Research has no contact information beyond the affiliations listed in the published articles. We ask that you provide attribution to the Journal of Marine Research.  Yale University provides access to these materials for educational and research purposes only. Copyright or other proprietary rights to content contained in this document may be held by individuals or entities other than, or in addition to, Yale University. You are solely responsible for determining the ownership of the copyright, and for obtaining permission for your intended use. Yale University makes no warranty that your distribution, reproduction, or other use of these materials will not infringe the rights of third parties.  This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. https://creativecommons.org/licenses/by-nc-sa/4.0/   On the Wave-induced Difference in M ean Sea Level Between the Two Sides of a Submerged Breakwater' M . S. L onguet-Higgins National Institute of Oceanography, England and Scripps Institution of Oceanography, L a :Jolla, California ABSTRACT Very simple formulae are deri ved for the di fference in mean l evel between the two sides of a submerged breakwater when waves are incident on it at an arbitrary angle. T he formulae apply also to waves undergoing refr acti on d ue to changes in depth and to waves in open channel transiti ons. When sea waves approach a submerged breakwater or an offshore sand bar, the mean level of the water on the far side of the bar or breakwater is com-monly observed to be higher than on the side from which the waves are in-cident. The purpose of this note is to show that the difference in mean water level can be calculated very simply in certain circumstances, once the height of the incident waves and the coeffi cient of refl ecti on are both known. The situati on is as shown in F ig. I. A submerged " breakwater" separates two unifo rm regions in which the undisturbed depths are h, and h2 , say. W aves of amplitude a, are propagated from the left and are incident (not necessaril y normall y) on the " breakwater. " There is a transmitted wave of amplitude az and a refl ected wave of amplitude a:. If the steepness of the waves is suffic iently small everywhere, then the coeffic ients of transmission and refl ecti on, namely T = a2/a, and R = a{/a, , are nearly independent of a, . The coeffi cients R and T may be determined by experiment or, in some ideal cases, by the linear theory of water waves. (For some examples, see the REFERENCES.) In the neighborhood of the breakwater t. Accepted fo r publication and submitted to press 17 February 1967. Contri bution from the Scri pps I nstitu tion of Oceanography. 148 Longuet-Higgins: Difference in M ean Sea Level 149 Figure 1, itself the waves are not generally sinusoidal; nevertheless, the motion every-where flu ctuates harmonically with time, say with period 2n/a . The wave-length, 2n /k1, of the waves on the near side of the breakwater is related to the frequency a and to the local depth by the usual relationship a2 = gk, tanh k, h1, and similarly for the waves on the far side. These results can be deri ved from the well-known small-amplitude theory of water waves. However, on the two sides of the barrier there will be a differ-ence, LI ( , in mean surface level that is of second order in the wave amplitude. W e shall see that LI C, though of second order, can be determined directly from the fir st approximation in the foll owing way. L et x and y be horizontal coordinates and z be measured vertically upward from the still-water level. L et u , v , and w denote the corresponding compo-nents of velocity and p the pressure. L et (! and g denote the density and the acceleration of gravity, both assumed constant. The free surface is denoted by z = l;(x , y , t) . N eglecting viscous forces, we then have two simple rela-ti onships (cf L onguet-Hi ggins and Stewart 1964). First, consider the flu x of verti cal momentum into a vertical column of water of unit cross section contained between z = o and z = l; . The flu x upward through the base of the column equals (p + ew' ) evaluated at z = o . The flux through the upper surface of the column is zero. The flux of vertical momentum through the sides of the column, which is euw per unit area, is of third order when integrated over the height l; of the column. Hence this can be neglected. The total flu x of vertical momentum into the column is therefore (p + ew' )z=o . This is opposed by gravity, which produces a downward force, egl;. But since the moti on is peri odic, the vertical momentum within the column re-mains, on average, unchanged. Thus, on taking mean values we have (A) 150 'Journal of M arine Research where a bar denotes the average with respect to time. This is our fir st equation. Second, if the motion everywhere is assumed irr otational (which excludes wave breaking, for example), then we have the Bernoulli integral I a~ p+ - e(u2+v2+w2)+ (!gZ+(! - = O. 2 at Here denotes the velocity potential, which includes an arbitrary function of the time, t . If we take time-averages in this equation and set z o , we have - J ( 2 2 2 ) C Pz=o + - (! U + V + W z=o + = 0 , 2 (B) C being at most a constant. From the two equations (A) and (B) we may eliminate the pressure to obtain the basic relationship - J ( 2 2 2) C g( = - - e u +v-wz=o+. 2 (C) From this relationship it is very easy to determine the difference in mean surface level, C, at two different points (x,,y,, o) and (x2,J2, o), say. Clearly the constant C is immaterial, so we have (- - ) I [( 2 2 2) JI (!g C,-(2 = - - e U +v -w Z=O 2. 2 Thus, in the present problem, LI C is given by - I LI C = - [(u2 + v2 - w2)z=o]~ . 2g This is the simple relationship promised earlier. (D) N ow, in a wave of amplitude a traveling in some direction that makes an angle 0 with the x-axis, the components of orbital velocity are given by aa cos 0 u = . coshk(z- h) cos(kx' - at+s) smh kh aasin 0 v = . coshk(z-h)cos(kx'-at+s) smh kh w = . aa sinhk(z- h) sin(kx'-at +s) smh kh ' Longuet-Higgins: D ifference in M ean Sea Level where x 1 = x cos 0 + y sin 0 and c denotes a constant phase. On squaring the velocities and taking averages with respect to time, we find so - I a2 a2 cos2 0 u2 = - ---- cosh2 k ( z - k) 2 sinh2 kh - I a2a2sin2 0 v2 = - . cosh 2 k ( z - h) 2 smh2 kh - I a2 a2 w 2=2 sinh2kh sinh2k (z - h), 1 a2 a2 u2 + v2 - w 2 = - ---2 sinh2 kh . Using the relationship that a2 = g k tanh k h locally, we then have I ----- a2 k - (u2+v2-w2) = ----2g 2 sinh 2kh If two systems of waves are present (as on the seaward side of the breakwater), then in place of a2 we shall have ( ai + a?). There will also be a contribution from the product terms, proportional to ar a~ . However, on averaging with respect to the hori zontal coordinates, x, y , as well as with respect to t, we find that these product terms vanish. From equati ons (D) and (E I), (E 2) we deduce that in the present situation LI - _ a a r + a, a2 l ( 2 12 l ) C - 4g sinh2 k,h, - sinh2 k2h2 ' or alternati vely, A- (ai+a?)k, aik2 LJ C = ~---- - -----2 sinh 2 k, h, 2sinh2k2h (F2) When the depths h, and h2 on the two sides of the breakwater are equal, then we have simply ( 2 12 2) k LI - = a,+ a, - a2 , C 2 sinh 2 kh (G) where k = k, = k2; h = h, = h2. Since a~ .'.'c. ai , the ri ght-hand side is non-negative, showing that the difference in level is then positive in general. The outstanding feature of this result is that in deep water, if both k, h, and k2 h2 are large, LIC = O. Journal of Marine Research In other words, the difference in level is essentially a finite-depth effect. In shallow water, where both k, h, and k, h, are small, equation (F 2) becomes and, if h, h, h, then 2 ' 2 Z ,1 - a, + a, - a, LJC=--- . 4h (H) (I) Of course these results are subject to the usual limitations of the small-amplitude theory of surface waves, in particular that a k (< 1 and a k (< ( k h ) 3 in each particular region. In addition, the loss of energy by friction or other means (such as breaking) must not be so great as to affect the results. N evertheless, the formulae are so simple and their application so straight-forward that it would seem worthwhile to check their range of validity by experiments in the laboratory. REFERENCES AND BIBLIOGRAPHY BARTHOLOMEUSZ, E . F. 1958. The reflexion of long waves at a step. Proc. Cambridge philos. Soc., 54: 106-118. BIESEL, F ., and B. LE MEHAUTE 1955. Etude theorique de la reflexion de la houle sur certains obstacles. La Houille Blanche, IO: 130-138. BouRODIMOS, E. L., and A. T . lPPEN 1966. Wave reflection and transmis ion in open channel transitions. Hydrodynamics Lab. Rep., Mass. Inst. Techn., No. 98 (August); 201 pp. CooPER, R. I. B., and M. S. LoNGUET-HI GGINS 1950. An experimental study of the pressure variations in standing water waves. Proc. Roy. Soc., (A) 206: 424-435. D EAN, R . G . 1964. Long wave modification by lin ear transiti ons. Proc. Amer . Soc. c,v. Engrs., 90 (WW1): 1-29. D EAN, R . G ., and F. URSELL 1959. Interaction of a fixed, semi-immersed cir cular cylinder with a train of surface waves. Hydrodynamics Lab. Rep., Mass. Inst. T echn., No. 37; unpublished. DEAN, w. R. 1945. On the reflexion of surface waves by a submerged plane barrier. Proc. Cambridge philos. Soc., 4I: 231- 238. 1947. On the reflexion of surface waves by a submerged cir cular cylinder . Proc. Cam-bridge philos. Soc., 44: 483-491. Longuet-Higgins: Difference in Mean Sea Level 153 HASKIND, M. D . 1948. The pressure of waves on a barrier. Inzhen. Sb., 4: 147-160. JEFFREYS, H. 1944. Motion of waves in shallow water. Wave Rep., U. K. Ministry of Supply; un-published. JOHN, F. 1948. Waves in the presence of an inclined barrier. Comm. appl. Math., I: 149-200. JOLAS, P. 1960. Passage de la houle sur un seuil. La Houille Blanche, I5: 148-152. KREISEL, G . 1949. Surface waves. Quart. appl. Math., 7: 21-44. LAMB, SIR HORACE 1932. Hydrodynamics. 6th ed. Cambridge Univ. Press. 738 pp. LEVINE, H. 1957. Scattering of surface waves on an infinitely deep fluid, pp. 712-716. Proc. Sym-posium on Behaviour of Ships in a Waterway. Wageningen, 1957. 1045 pp. LEVINE, H., and E. RoDEMICH 1958. Scattering of surface waves on an ideal fluid. Tech. Rep. Appl. Math. Stat., Stan-ford Univ., No. 78; 64 pp. LONGUET-HIGGINS, M. S., and R. W. STEWART 1964. Radiation stresses in water waves: a physical discussion, with applications. Deep-sea Res., II: 529- 562. MACAGNO, E. 0. 1954. Houle clans un canal presentant un passage en charge. La Houille Blanche, 9: 10-37. NEWMAN, J . N. 1965. Propagation of water waves over an infinite step. J . fluid Mech., 23: 399-415. TAKANO , K. 1959a. Effet de passage d'une houle lineaire plane sur un seuil. C.R. Acad. Sci. Paris, 248: 1768-1771. 1959 b. Effects de second order d'un seuil semi-indefini sur une houle irrotationelle. C.R. Acad. Sci. Paris, 249: 622- 624. 1960. Effets d'un obstacle parallelepipedique sur la propagation de la houle. La Houille Blanche, I5 : 247-267. 1963. Effet du second order d'un obstacle parallelepipedique sur la propagation de la houle clans un canal. Rec. oceanogr. Wks. Jap., 7: 9-17. URSELL, F. 1947a. The effect of a fixed vertical barrier on surface waves in deep water. Proc. Cam-bridge philos. Soc., 43: 374-382. 1947 b. The reflection of waves from a submerged low reef. Admiralty Res. Lab., Ted-dington, Rep. R5/103.41/W (unpublished). 1950a. Surface waves on deep water in the presence of a submerged circular cylinder, I. Proc. Cambridge philos. Soc., 46: 141-152. 1950 b. Surface waves on deep water in the presence of a submerged circularcylinder, II. Proc. Cambridge philos. Soc., 46: 153-1 58. 

On the wave-induced difference in mean sea level between the two sides of a submerged breakwater

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On the wave-induced difference in mean sea level between the two sides of a submerged breakwater

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