D. J. Clarke Department of Mathematics, Wollongong University College, Wollongong, N.S.W., Australia

Transcription

1 Resonant and nonresonant motion in a spindle-shaped basin with an entrance N. D. Thomas Department of Mathematics and Quantitative Methods, Mitchell College of Advanced Education, Bathurst, N.S.W., Australia D. J. Clarke Department of Mathematics, Wollongong University College, Wollongong, N.S.W., Australia Abstract Resonant and nonresonant motions in a spindle-shaped (i.e. formed by the intersection of two confocal parabolae) basin are examined when the assumption is made that they are caused by the formation of a standing wave due to incident wave action through the entrance. The spindle basin is transformed into a square region with two entrances and the modified boundary value problem is solved approximately by the Galerkin method. The trial functions are chosen to satisfy the boundary conditions across the entrances in the square region. The amplification of the incident waves is calculated for the internal points of the spindle basin at various frequencies and those frequencies causing large amplifications are found. The results are applied to Port Kembla Outer Harbor. Wave motion within a bay or harbor can be caused by external waves incident at the entrance. Two basic types of motion are the resonant, when the frequency of the incident wave matches one of the free modes of oscillation of the body of water, and the forced oscillation generated through the medium of a standing wave pattern, or clapotis, at the entrance. The wave motion within a harbor can be of much larger amplitude than that of the disturbance outside and is determined by the geometry of the basin. Design criteria for a harbor can best be obtained by first considering an idcalized port with simple geometry to approximate the real situation. For resonant motion we assumed that the incident wave pattern forms a standing wave system with the reflected wave in such a way as to create an antinode at the entrance. In these circumstances the bay or harbor oscillates as if it were undergoing free oscillations. The analysis is thus simplified and merely requires the frequencies and modes of the free oscillations. McNown (1953) explained the nonrcsonant motion by the formation of an external standing wave pattern such that the antinode is outside the entrance. The horizontal velocity at the entrance is thereby as- LIMNOLOGY AND OCEANOGRAPHY sumed to be periodic and no longer zero as in the resonant motion. McNown illustrated this theory on a circular basin, and later, McNown and Dane1 (1952) treated a rectangular basin. Subsequent papers (Le Mehaute 1955; Ippen and Raichlen 1962; Ippen and Goda 1963) have been confined to a basin of rectangular plan. IIidaka ( I931) studied the free oscillations of water in a closed basin whose perimeter is defined by two intersecting parabolae; he termed the basin a spindle shape. The region inside the spindle was transformed to the region inside a square and a variables separable solution assumed. Clarke and Thomas (1972) found the first 15 modes of oscillation for the spindle region by following the transformation of Hidaka and then solving the resulting bound ary value problem with the Galerkin method : an additional 12 modes were thereby added to Hidaka s first 3 modes. IIidaka had omitted these solutions through an assumption of symmetry about the main diagonal of the square. The problem in the spindle basin is here extended to nonresonant motions by the introduction of an entrance into the basin. The boundary conditions used by Clarke and Thomas arc altered to take account of 199 MARCH 1974, V. 19(2)

2 2 Thomas and Clarke is decreased (the paradox) but, because of the effects of friction, it decreases quickly when the opening tends to zero. We will show that, for Port Kembla Outer Harbor, the Miles and Munk paradox is present but that it is modified for larger openings because these interfere with nodal lint structure (Kravtchenko and McNown 19%; Apte and Marcou 1955). Fig. 1. Port Kembla Outer Harbor. incident wave action through the entrance. Solutions arc sought for the basin amplifications at different frequencies of incident waves. Port Kembla Outer IIarbor (Fig. I), which is about 8 km south of Sydney, Australia, has been shown to approximate closely to a spindle shape ( Clarke 1971). The harbor is considered as an example of the nonresonant theory for the spindle basin; it exhibits severe horizontal ranging during times of storm activity (Lucas 1962 ). The effect of varying the width of the entrance of the harbor is also considered. Some controversy has surrounded the question of the likely consequences when the width of the entrance to a harbor is altered. Miles and Munk ( 1961) referred to a paradox in relation to harbor resonance : they found that a narrowing of the harbor mouth diminishes protection from sciching, but that it also would result in a decrease in the frequency at which the maximum wave amplification is attained. Lc Mehaute reported (Lc Mehaute and Wilson 1962) that Bicsel and he found experimental evidence of this phenomenon. The conclusion reached was that the amplitude at resonance begins to increase when the opening Boundary conditions in the transformed region The boundary value problem is determined for the open spindle, i.e. the spindle basin with an entrance. The region is then transformed into a square and a secon d boundary value problem is obtained. The Galerkin method as used by Clarke and Thomas ( 1972) is extended to include the altered boundary conditions. The pcrimetcr of the spindle, as defined by Hidaka ( 1931)) is given by equation 1 (Table 1). The open spindle contains an entrance of width 2b (Fig. 2). Let the velocity potential in the open spindle be given by equation 2. At the center of the entrance where x = a/2, y =, let the velocity be given by equation 3. Here 7c is the wave number, h is the uniform depth of the water, u is the frequency of the incident wave, and V is an arbitrary velocity of reference occurring at the surface for the entire width of the entrance. The wave height, [, outside the entrance of the spindle basin is given by equation 4, where 6 is the amplitude of the clapoti:;. The value of 6 is twice the amplitude of the incident progressive wave. The wave height at the entrance, co, is then given by equation 5, where x has the value x1 - E at the entrance ( Fig. 3). Inside the spirtdle basin let C;= [*(x, y) sin (T t, hence, at the entrance, co = PO * sin (T t. Therefore the maximum value of wabe height at the entrance, &*, is given by equation 6. The horizontal acceleration in the x di-

3 Resonant and nonresonant motion 21 Table 1. Equations concerned with wave motion in a spindle-shaped basin. 1, 2, 3, Equation of spindle shape: y2 = a(a - 2x), x, Y 2 = a(a + 2x), xc Velocity potential in open spindle: 4 = F(x, y) cos u t cash k (2 + h)/cosh k h Velocity at center of the entrance: g= v. cos u t co& k (2 + h)/cosh k h 4, Wave height outside the entrance: c = 6 eos Ii (x + E) sin u t Wave height at the entrance: ci = 6 cos k x1 sin u t Maximum value of wave height at the z; * = 6 cos k x entrance: I 7. a The horizontal acceleration in the x direction outside the basin:. ar g 6 k sin it (x + E) sin u t u= -gax:= Horizontal velocity at the entrance: = g 6 k sin It x1 cos u t/o 1JO Maximum value of horizontal velocity at af entrance: = V. = g 6 k sin k xl/ ax where P = af cos u t TG Amplitude g *2 + of the clapotis: voo 2 = (32 i-l gk Fig. 2. Spindle basin with an entrance of width 2b. horizontal velocity at the entrance is zero, i.c. V. =, co* = 6. The open spindle is mapped into the interior of a square with two opposite entrances of width 2b/a (Kg. 4) by equation 11 (Table 2). The boundary conditions for the square are given by no flow through the boundary, i.e. equation 12. The equation of motion for long waves of frequency (T inside the open spindle becomes equation rection outside the basin is given by cquation 7. Integrating over time with appropriate initial conditions and then taking the horizontal velocity at entrance, i.e. where x=xl- E, gives equation 8. From equations 2, 3, and 8 the maximum value of the horizontal velocity at the entrance is given by equation 9. Eliminating x1 from equations 6 and 9 gives equation 1 entrance which corresponds to the equation given Fig. 3. Clapotis or standing wave at the enbv, McNown (\ 1953). I At resonance the trance to a basin.

4 22 Thomas and Clarke Table 2. Equations involved with the transformation of the spindle to the square region Mapping x+iy = $P + iqp Boundary conditions for the square: ap 9 af JF E a4 =, p = +1, 141 b b/a = avop, p=+l, 141 < b/a =, q = +1 Equation of motion for long waves inside the open s indle: a2f a2f u : -+-+*zo F ax2 332 gh Equation of motion for long waves inside transformed region: -+-+ a2f a2f X (p2 + q2) F = ap2 392 where A = k2a2 = a2a2/gh 13 under the mapping, where F is the timereduced velocity potential. Equations 12 and 14 describe a boundary value problem which is now solved by the Galcrkin method ( Kantorovich and Krylov 1964). Briefly, to solve a boundary value problem in one independent variable of the form L(a) = with boundary conditions at x and x1, a sequence of trial functions, (pi, satisfying the boundary conditions and also differentiability and completeness properties, is developed so that approximate solutions to a are found as a - a, = 2 ai +i. i=l The ai are arbitrary constants which are found from the orthogonality conditions given by the form of equation 15 (Table 3). Clarke and Thomas (1972) showed that a set of trial functions for determining symmetric oscillations in the closed spindle basin was of the form: 1, r)-l- 2p2, q4-2q2, 2p6-3p, (p -2~ ) (q -299, 2q -3q4, 3p8-4p, ( 2p6-3p ) (q -2q2), (P4-2P2) (29-3q4), 3q8-4q, etc. The functions will be called +1, +2,..., and the linear s-=---hi L 2b a p Fig. 4. Square region transformed from the spindle basin with entrance as shown in Fig. 2. combination xn ai +i will i--l be called rfl. They satisfy the boundary conditions in the closed region given by equation 16. Appropriate trial functions satisfying the boundary conditions 12 are defined as equation 17. The orthogonality relations ate given by equation 18, where L is the differential operator of 14. Selected values of X, corresponding to particular frequencies of incident waves, are substituted into these orthogonality relations and the equations solved for scaled ai, i.e. solved to give coefficients ai, where avoai = ai, assuming a value for the ratio of the entrance width to the length of the major axis of.2. From the cxc values, a scaled value of time-reduced velocity potential, Z,,,, is calculated for any positioil (x, y) in the open spindle basin, i.e. Wave height for resonant and nonresonant motions From the free. surface condition + + g g =, x ==, the time reduced wave height is given by

5 Resonant and nonresonant motion 23 Table 3. Equations involved in the Galerkin method of solution of a differential equation Form of orthogonality condition: Xl L(a,> 4. dx =, j = 1,2,---,?I J 3 xo 16. Boundary conditions in square region: af -=o,p=+ 1 ap af = as, q = Form of trial functions: F=G n = Fn,/q/3 b/a 1 Fn+ a Vop2/2, /q/ -C bla i 18. Orthogonality relations:.i J 1 L(G,)c)~ dp dq =, j = 1,2,--,n -1 -b/a b/a 1 i.e. I L(Fn)$i dq + L(F n + a V, ~~/2)@~ dq+ LC, I Ojdq ldp=o, j =1,2,...,~ -1 J -1 J J -b/a b/a The quantity L(F,+ a Vop2/2) can be expressed as L(F,) + a Vo+ k2 a2(p2 + q2)a V. P2/2, so that the orthogonality 1 relation becomes: (-A + XB)s = -(II + Xl,)a V 1 1 where A(i,j)= - I (@$,, + (Qlqq~dp dq, J J 1 b/a 1 B (i, j> - 11 (P2 + ~j dp &a 11 (j)= 4-j dp dq J J J J i J (P2 + q2) P2 ~j dp dq and z(j) = aj 12(j) = b/a 3 Thus the value of, y *, the time-reduced determined. Similarly, the wave height at wave height at the middle of the entrance any position in the open spindle is given of the open spindle basin, is given by equa- by equation 21. For the special case of a tion 19 (Table 4). After eliminating ~VO node at the entrance, co* =, SO from equafrom equation 1 and 19, a further equa- tion 1 the amplitude of the clapotis betion (2) can be given from which the comes value go* can be calculated, as all quantities on the right are given or have been 6 =Vo o/g k,

6 24 Thomas and Clarke Table 4. Equations for time-reduced wave height. 19. At the middle of the entrance: Table 5. Wave heights in meters at four positions inside the open spindle for x values neas resonance assuming an external wave of amplitude 1 m incident at the entrance. < *= OaV, g Z a/2, Position A= 13.3 A= 45 A= 49*,6 Center of entrance 2-1* At the middle of the entrance: % co*= 6x2 a/2, '(l+' z2a/2,y 21. At any position in the open spindle: Top extremity l-86 8*37 9*53 Center of spindle '498 1* Vertex opposite entrance ao94 c*= 6% z x,y /Cl + x zza,2, 1% and hence [* = 6 A* Z,,, is the time-reduced wave height at any position in the open spindle when there is a node at the entrance. Near resonance, the computed value, z a/2,, approaches infinity and so co* approaches 6. The orthogonality relations (equation 18) have a singularity at resonance and thus a resonant value of x would cause computer overflow. Resonant motions have been calculated previously by Clarke and Thomas ( 1972). Numerical results for the open spindle Clarke and Thomas calculated seven approximate values of A, using 1 trial functions, for the resonant symmetric modes. These were , , 44.28, , , 13.51, and For the nonresonant motions sample values of X of 1, 4, 9,..., 1 were used initially to examine nodal structure. Significant changes in wave heights, nodal structure, or both occurred only in the neighborhood of the resonant values. Hence a number of calculations were carried out for x values close to the resonant values. Several of these are illustrated in Fig. 5, which also contains some of the results for A values close to the resonant antisymmetric modes, e.g. 4.48, 26.71, and 75. In the latter examples no significant change in wave height or nodal patterns occurred, which is to be expected. [McNown and Dane1 ( 1952) stated that a nodal line cannot terminate at the entrance.] Table 5 gives values of wave height at points within the spindle for A values where the incident wave is supposed to have an amplitude of 1 m, In each example the wave height at the center of the entrance is nearly 2 m, which is the amplitude of the clapotis, as the x values are near resonant values. Noticeably large amplif i- cations of wave height have occurred at the A values of 45 and Some other A values gave large amplifications inside the spindle: for a A value of 98.5 the wave height at the vertex opposite the entrance was 9.94 m. In this example the wave height at the entrance :is very small and the nodal pattern rcscmbles one with a node approximately at the entrance. Similar nodal patterns were found for x values of 2.97 and Fig. 5. Nodal patterns produced by incident waves in terms of particular x = u2a2/gh.

7 Resonant and nonresonant motion 25 X = X = 46

8 26 Thomas and Clarke (i> x = 13.3 (ii> (iii) ) X = 49.6 (ii> (iii) h = 98.5 (ii> (iii>

9 Resonant and nonresonant motion 27 Table 6. Magnitude of incident wave height at the entrance to Port Kembla Inner Harbor for various periods (set). x * *5 Period *1 186*2 151-s 1* *1 56*3 Mag. O * * * * Application to Port Kembla Outer Harbor The Outer I-Iarbor is about 1.7 km long and about.9 km wide. It is known for excessive range action during stormy periods; Lucas ( 1962) refers to an average loss of 1 working days per year. Clarke ( 1971) approximated the harbor geometry by a spindle with parameter a of 85 m and a depth h of 9.4 m. Using Clarke s figures and taking the width of the entrance to be the fraction.2 of the length of the major axis, we can apply the results of the preceding section directly to the harbor. Table 6 gives the magnification of the incident wave at the top extremity of the spindle, which corresponds to the Inner I-Iarbor entrance, for particular periods of the incident wave. Large magnifications exist at periods of 11, 83, 79, and 56 sec. For the position directly opposite the Outer Harbor entrance large magnifications exist at periods of 83, 79, and 56 sec. The factors are 14.8, 4.1, and 9.9 respectively. Comparison with the hydraulic model studies of Lucas ( 1962) indicates good agreement in that large horizontal ranging was observed at periods of 8 set and exccssivc ranging at periods of 56 to 58 sec. Variation of entrance width Variation of the width fraction of.2 is now considered in relation to the concept of the harbor paradox put forward by Miles and Munk ( 1961). Width fractions of.1 and.3 were substituted for b/a and amplifications com- pared with previous results. In most cases the nodal patterns were only slightly altered and there was no marked change in wave heights. For example at the x value of 13.3 the wave heights at the entrance to the Inner IIarbor for the three widths.1,.2, and.3 were 1.95, 1.86, and IIowever, two of the A values corresponding to large amplification factors showed quite different effects. For the A value 49.6 there was a complete shift of nodal lines as the entrance width varied. The amplifications were 6.275, 9.53, and 4.43 respectively. Similarly for the x value of 98.5 where the amplifications were 2.631, 4.439, and For both x values the maximunl amplification occurred at the width fraction of.2. The nodal structure for the above three A values is illustrated in Fig. 6. Further calculations were carried out to determine the A in the neighborhood of 98.5, for which the maximum amplification at the Inner Harbor entrance occurred with each of the widths.1,.2 and.3 (Table 7). A decrease in the entrance width has resulted in a downward shift of Table 7. Variation of frequency (in terms of the x value ) with entrance width at which maximum amplification is attainecl. Width fraction Max amplification x '5 = *5.3 4* 99*6 t Fig. 6. Variations in nodal patterns at cntrancc width fractions of (i).1, (ii).2, and (iii).3 for particular X values.

10 28 Thomas and Clarke the frequency at which the maximum amplification was reached, which agrees with the prediction of Miles and Munk. References APTE, S. A., AND C. MARCOU. 19%. Seiche in ports. Proc. Conf. Coastal Eng. (5th) Grenoble, p CLARKE, D. J Seiche motions for a basin of rectangular plan and of non-uniform depth. J. Mar. Res. 26 : , AND N. D. THOMAS The twodimensional flow oscillations of a fluid in a spindle-shaped basin; application to Port Kembla Outer Harbour, N.S.W., Australia. Aust. J. Mar. Freshwater Res. 23: l-9. HIDAKA, K The oscillations of water in spindle-shaped and elliptic basins as well as the associated problems. Mem. Imp. Mar. Obs. Kobc 4: IPPEN, A. T., AND Y. GODA Wave induced oscillations in harbors. The solution for a rectangular harbor comlected to the open sea. Hydrodyn. Lab. Dep. Civil Eng., M.I.T. Rep, p. -, AND F. RAICHLEN Wave induced oscillations in harbors. Hydrodyn. Lab. Dep. Civil Eng., M.I.T. Rep p, KANTOROVICH, L. V., AND V. I. KRYLOV Approximate methods of higher analysis. Intersciencc. KRAVTCHENKO, J., AND J. S. MCNOWN Seiche in rectangular ports. Quart. App l. Math. 13: LI;: MEIIAUTE, B Two-dimensional seiche in a basin subjected to incident waves. Pro<:. Conf. Coastal Eng. (5th) Grenoble, p , AND B. WILSON Harbor paradox. Discussion. J. Waterways Harbors Div. Am. Sot. Civil Eng. 88: LUCAS, A. II Port Kembla: Inner Harbour long period wave investigation. Report of model studies. Dep. Public Works, N.S.W., Harbours Rivers Branch, Hydraulics Lab. 7 p. MCNOWN, J. S Sur l entrctien des oscillations des eaux portuaircs sous l action dc I.1 hautcmer. Publ. Sci. Tech., Ministere de l Air, Paris. -, AND I'. DANEL Seiche in harbours. Dock Harbour Auth , p MILES, J., AND W. MUNK Harbor paradox. J. Waterways Harbors Div. Am. Sot,. Civil Eng. 87: Submitted: 2 February 197;3 Accepted: 31 October 197i3