Effects of depth discontinuity on harbor oscillations

CoastaI Engineermg, 10 (1986) 395-404 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

395

Effects of depth d i s c o n t i n u i t y on harbor oscillations PHILIP L.-F. LIU Joseph H. DeFrees Hydrauhc Laboratory, School of Civil and Environmental Engineering, CorneU University, Ithaca, N Y 14853, U.S.A

(ReceivedAugust 28, 1985; revised and accepted February 28, 1986)

ABSTRACT Liu, P.L.-F., 1986.Effects of depth discontinuity on harbor oscillations. Coastal Eng., 10: 395-404. The effectsof water depth discontinuity near the harbor mouth on harbor oscillationsare examined. Linear long-waveequations are used as the basis of the present study. For simplicity, only the normal incident wavesare considered.Assumingthat the harbor mouth is small in comparison with wavelength, the method of matched asymptotic expansion is employedto obtain the ocean impedance and the harbor responses. It is found that the incident waves can be trapped over the depth discontinuity which causes largeoscillationsnear the harbor mouth. The radiation damping also decreasesbecause of the appearance of the depth discontinuity, which leads to large amplifications at the lowest mode.

1. INTRODUCTION H a r b o r oscillations due to t he t s u n a m i attack have been extensively studied in the framework of inviscid linear long-wave theory. A comprehensive review on harbor resonance has been given by Miles (1974) and most recently by Raichlen (1979). Various numerical m e t hods have also been developed to deal with harbors of a r bi t r ar y shape and water dept h (e.g. Lee, 1971; H w ang and Tuck, 1970; Olsen and Hwang, 1971; Berkhoff, 1972; Chen and Mei, 1974). Examining the tsunami responses of Hilo Harbor, Hawaii, Raichlen et al. (1983) d e m o n s t r a t e d clearly t he importance of the pl anform and t he offshore b a t h y m etry on the amplification and a t t e n u a t i o n characteristics of the harbor. Momoi (1976) and Liu (1983) investigated analytically t he interactions of harbor oscillations and wave propagation on a cont i nent al shelf. T h e continental shelf was modeled by a strip of steps parallel to the shoreline. Liu (1983) demonstrated numerically t h a t the harbor responses are significantly increased due to the appearance of the c o n t i n e n t a l shelf. In this paper, using the m e t h o d of asymptotic expansion, we investigate the problem where the cont i nent al

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396

YI~.~~ hz

i

. . . . . . . . . . . . . . . . . . . . . .

":- X

Fig. 1. Sketch of geometry and coordinates.

shelf is modeled by a semi-circular step. Closed-form solutions are obtained for the responses inside the harbor of rectangular shape. It is shown that the increase of the amplification factor is primarily due to two factors. First, waves may be trapped over the submerged step such that wave amplitudes near the harbor mouth become large. Secondly, the radiation damping becomes zero near the lowest resonant frequency. The theoretical results obtained herein suggest that in designing a harbor subject to long-wave attack, the bathymetry outside the harbor must be taken into consideration. The effects of continental shelf on the harbor oscillations are important and can not be ignored for long waves. 2. FORMULATION

2.1 Shallow-water approximation We consider a harbor and ocean system as shown in Fig. 1. The water depth inside the harbor is the same as that outside of the harbor within a semi-circular region (r ~ hi ) along the semi-circle r = L. For simplicity, we assume that the harbor is a long, narrow rectangular basin with a dimension l X 2a. The width of the harbor, 2a, is considered to be small in comparison with the wavelength. The linear shallow-water equations are employed herein to describe the velocity ~(x,y,t), ~(x,y,t) and the free-surface displacement ~(x,y,t). We shall, however, investigate only incident waves and responses which are periodic in time, with radian frequency co. The solutions can be expressed as: (~,0,¢) = R e { (u,v,~) e -'~t}

(2.1)

The linear shallow-water equations can be written in the following forms: u=

v=

o) Ox

0C (o Oy

(2.2a)

(2.2b)

397

o_Ox

-7

(2.a)

The incident waves are normal to the shoreline and can be expressed as: ~m¢= e-ik2x

( 2.4 )

at infinity, where the incident wave amplitude is taken as unit, and k2 denotes the wave re, tuber associated with the water depth h2, i.e.:

k2 = w/(gh2) 1/2

(2.5)

The boundary conditions are that the normal velocity component vanishes along the solid walls of the harbor and along the coastline. The continuity of the surface elevation and normal flux should be applied along the harbor mouth and the discontinuity of water depth. Finally, the radiation condition requires that radiated waves must be outgoing waves at infinity. For later use the incident wave displacement (eqn. 2.4) can be written in terms of the polar coordinates (r,0): oo

~m¢= ~ ~ ( - - i ) m J m

(k2r) costa0

(2.6)

rn=O

where Era=0, m = 0 ; e~=2, m = 1,2 ..... is Euler's constant and J~ is the Bessel function of the first kind of m-th order.

2.2 Solution inside the harbor Since the harbor is long and narrow, 0 (kla)<
u(x) = ( - i g k i T / e o ) sin k, ( x + l )

}x0

(2.7)

where

T = A [cos ki l - i Z sin k, l] -1

(2.8)

k, = o) l (gh, ) 1/2

(2.9)

A is the wave amplitude of the incident and reflected waves at the entrance of the harbor and Z is the ocean impedance. The real part of the ocean impedance corresponds to the radiation damping while the imaginary part corresponds to the mass reactance.

398 3. SOLUTIONS OUTSIDE THE HARBOR

The free-surface displacement outside the harbor can be divided into two parts: ~=~x+~a,

x>_-0

(3.1)

where ~t represents the free-surface displacement of the incident and reflected wave field in the absence of the harbor and ~R corresponds to the radiated waves from the harbor. Both ~I and ~R depend on the bottom topography outside of the harbor.

3.1 The incident-reflected waves Consider first the wavefield without the presence of the harbor. The incident waves are reflected by the coastline as well as the submerged circular step. The boundary conditions are the no-flux condition along the coastline: 0~I=0, Ox

x=O

(3.2)

and the continuity of the free-surface elevations ~i and the volume flux per unit length, h O~I/Or, across the edge of the step, i.e.: [~I] =0,

r=L

[h<'l ar ]=O,

(3.3a)

r=L

(3.3b)

where [ ] denotes the difference in the quantities calculated from both sides ' of the edge of the step. Due to the symmetry, we can consider the incident-reflected wave problem as the superposition of solutions concerning the scattering of two opposite wave trains by a submerged circular step. The solutions for these scattering problems are well-known and can be readily written as:

{ n=O ~ An(+i)n (.nJn (klr)cos nO,

r
Bn ( + i ) n E.H(nl) (k~r) cos nO

~-+=

(3.4)

n~O

+ ~

~n ( +__i)"

Jn(k2r)

cos

nO,

r>L

n=O

where ~ + represents the wavefield with the incident wave propagating in the

399 positive x-direction and ~- represents the case where the incident waves are moving in the negative x-direction. The Hankel function of the first kind, H (1~, has been used in eqn. (3.4) so that the radiation boundary condition is satisfied. The coefficients A and B, can be found by satisfying boundary conditions in eqn. (3.3). Thus: n

2i ~l[J,(1) H;, (el) -eJ~ (1) H,(EI)]

A,-

(3.5)

J,(1) Jn (el) - J , (el) J~ (1) B"-Jn(l) H~(el)-eJ~, (1) Hn (el) with l=klL,

e=/~,/(h2)1/2,

(3.6)

kl=W/(ghl) 1/2

(3.7)

The prime in eqns. (3.5) and (3.6) denotes the first derivative with respect to the argument and the superscript of the Hankel function of the first kind has been dropped. The solution for the incident-reflected wavefield can be readily written as: =

~ 2A2,,, e2., (i) 2" J2r. (klr) cos 2m0,

=

f rn~O

~2B2m e2m(i) 2~ H2~ (k2r) cos 2m0 + 2 cos kax,

r<~L (3.s)

r>~L

m=O

PAt the middle of the harbor mouth, r= O,the incident-reflected wave amplitude becomes: ~L-*2Ao= A =

4i ~l[J0(1)H~ (el)-eJ~(1)Ho(el)]

(3.9)

This corresponds to the lowest mode of the responses over the semi-circular step, givenin eqn. (3.8). As shown in Fig. 2, the greater the depth discontinuity (smaller e values) the higher and sharper are the resonant peaks. For the limiting case where the water depth is a constant, i.e. e = 1.0, using the Wronskian identity:

Jo(1)H~(1)-J~(1) Ho(1)--2i/l we can show that the amplitude at the harbor mouth takes the value of two. 3.2 The radiated waves Since the harbor width is assumed to be small in comparison with the wavelength, the radiated waves from the harbor can be considered as waves gener-

400

0

0

I

1

I

I

I

I

I

I

I

2

3

4

5

6

7

8

Fig. 2. Amplitudes of incident-reflected waves at the harbor mouth

ated by a point source at the origin ( at the mid-point of the harbor mouth) i.e.: 0~ R

Ox-J(y),

x=0

(3.10)

The radiated waves are, therefore, independent of 0. The solutions satisfying the shallow-water equation (eqn. 2.3 ) and the radiation condition at infinity can be written as: ,R=

{"~BH°(klr)+~CI~°2)(klr)' "D Ho ( k2r),

r
(3.11)

r>L

where B, C and D are constants to be determined. Note that the radiation boundary condition at infinity is satisfied by eqn. (3.11). Substituting eqn. (3.11 ) into eqn. (3.10), we obtain: B+C=I

(3.12)

The continuity of the free-surface displacement ~R and the volume flux per unit length, hO~R/Or,is required across the edge of the step:

[,R]=0, [h~rRl=0,

r=L

(3.13)

Substituting eqn. ( 3.11 ) into eqn. ( 3.13 ), we obtain: i

{B Ho( ~) - C H(o2)(~) } =D Ho( e~)

ie {B H~(~) - Cd H(2)(2:) -~ "

}=D H~( E~)

(3.14)

(3.15)

401

Solving eqns. (3.12), ( 3.14 ) and ( 3.15 ), we find the following solutions: -

2[Jo(~)H~(~)-EJ~(~)Ho(E~)]

B=I

Ho(

)Hi(e

(3.16)

) - eH ( )Ho

C-2[J~(~)H~(e~)- eJ~(~)Ho(~) ]

(3.17)

where D may be found from either eqn. (3.14) or ( 3.15). Once again, the Hankel function and its derivative appearing in eqns. (3.16) and (3.17), are Hankel functions of the first kind of zero order. From eqns. (3.11) and (3.12), the radiated waves near the harbor mouth ( r < L) can be rewritten as:

----~1 /_/(1 o ) (k,r)+iCJo(kl,r),

rR_

r
(3.18)

The second term on the right-hand side of the above equation represents standing waves in the r-direction. This standing wave affects the impedance of the ocean as shown in the following. For a small harbor entrance, the ocean impedance can be approximated as (Miles, 1974):

Z=iklf:a~R (0~v) dy

(3.19)

Substitution of eqn. (3.11) into eqn. (3.19) yields:

Z:Zo +Zp

(3.20a)

where •

3

Zo : kla{ 1 + ~ [ In (7kla) - ~ ] }

(3.20b)

and ln7 = 0.5177216... is Euler's constant. It is remarked here that Zo represents the ocean impedance for the uniform depth h=hl, outside the harbor, while Zp is the contribution due to the appearance of the depth discontinuity. The typical features of radiation damping caused by depth discontinuity are demonstrated in Fig. 3. It is shown that for the case of e=0.5, l/L= 1.20952, and alL = 0.1175, the value of - Re (Zp) oscillate about zero and intersect with Re(Zo) curve. As a result, the total radiation damping, Re (Zo+Zp), becomes zero at several frequencies, i.e. k,L~ 1.25, 2.05, 4.40, and 5.41. Since these frequencies do not necessarily coincide with the resonant frequencies, finite harbor responses are expected.

402

Fig. 3. Ocean impedances for e =0.5,

l/L= 1.20952 and a/L=0.1175.

2o

A I0

!, t, 0

I

2

3

4

Fig. 4. A m p l i f i c a t i o n factors f o r

5

6

7

l/L= 1.20952, u/L =

O. 1175 and E= 1.0 ( - - - ) o r ~ = 0.5 ( - - ) .

4. NUMERICAL RESULTS AND DISCUSSION

In Fig. 4, the amplification factor of the harbor oscillations, I~ I/A, is calculated from eqns. (2.7) and (2.8), using the following numerical data: l/L= 1.20952, alL= 0.1175, e = 0.5 or 1.0. The same set of data with E--0.5 has been used to produce Figs. 2 and 3. At the lowest resonance mode, the amplification factor for the case e = 0.5 is almost three times that for the constant depth case, e-- 1.0. This is caused by the smallness of the radiation damping near the resonance frequency, klL,.~ 1.2; recall that the radiation damping is zero at k i L ~ 1.25 (Fig. 3). The radiation damping becomes small again near the third mode, k~L ~ 5.55, the amplification factor is significantly increased. It should be pointed out that the amplification factor is normalized by the wave amplitude at the harbor mouth, A. In the case of constant water depth, A equals 2 for all frequencies. However, with the appearance of the depth discontinuity A oscillates as a function of ~ = klL (Fig. 2 ). The maximum value

403 28 39

15 6 0

2 I

0

0

I

I

[

I

I

I

I

I

2

3

4

5

6

7

I

Fig. 5. Amplification factor normalized by the incident wave amplitude for l/L=1.20952, a/L=0.1175 and e=0.5. TABLE 1 The wave amplitude of the lowest resonance mode for I/hl = 1.20952, a/L=O.1175

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

24.15 138.48 78.61 112.95 131.45 59.42 33.71 24.10 19.20 18.01

1.31 1.30 1.29 1.26 1.24 1.20 1.16 1.13 1.10 1.10

of A increases as e decreases. This implies that additional resonance peaks exist if the amplification factor is normalized by the incident wave amplitude. For the case discussed in Fig. 4, additional peaks should appear at klL'~2.2 and 5.4. The latter peak almost coincides with the third resonance mode shown in Fig. 4. To stress this point, a portion of the harbor responses is plotted in Fig. 5. For certain values of depth ratio, (hi/h2) 1/2, the resonance frequency of the lowest mode may become very close to the frequency at which the radiation damping becomes zero. When this situation occurs, the wave amplification factor becomes very large. Table 1 lists the amplitude responses at various values of (hi/h2) 1/2. It should be remarked here that at the resonance peak, the nonlinearity and viscous effects become important and should be considered. ACKNOWLEDGEMENT

The research work reported herein was sponsored by the New York Sea Grant Institute.

404 REFERENCES Berkhoff, J.C.W., 1972. Computation of combined refraction-diffraction. Proc. 13th Coastal Eng. Conf., American Society of Civil Engineers, pp. 471-490. Chen, H.S. and Mei, C.C., 1974. Oscillation and wave forces in a man-made harbor in the open sea. Proc. 11th Syrup. Naval Hydrodyn., pp. 573-594. Hwang, L.S. and Tuck, E.O., 1970. On the oscillation of harbors of arbitrary shape. J. Fluid Mech., 42: 447-464. Lee, J.J., 1971. Wave-induced oscillations in harbors of arbitrary geometry. J. Fluid Mech., 45: 375-394. Liu, P.L.-F., 1983. Effects of the continental shelf on harbor resonance. In: K. Iida and T. Iwasaki (Editors), Tstmemis-- Their Science and Engineering. Terra Scientific Publishing Company, Tokyo, pp. 303-314. Miles, J.W., 1974. Harbor seiching. Annu. Rev. Fluid Mech., 6: 17-35. Momoi, T., 1976. Scattering of long waves at the mouth of estuaries bordering on a continental shelf, Part I and II. J. Phys. Earth, 24:1-25 and 237-250. Olsen, K. and Hwang, L.-S., 1971. Oscillation in a bay of arbitrary shape and variable depth. J. Geophys. Res., 76: 5048-5064. Raichlen, F., 1979. Bay and harbor response to tsunamis. Tsunamis, Proceedings the National Science Foundation Workshop, pp. 188-221. Raichlen, F., Lepelletier, T.G. and Tam, C.K., 1983. The excitation of harbors by tsunamis. In: K. Iida and T. Iwasaki (Editors), Tsunamis - - Their Science and Engineering. Terra Scientific Publishing Company, Tokyo, pp. 359-385.