Copyright 1974. All riqhts reserved

HARBOR SEICHING ): 80!

John W Miles University of California, La Jolla, California

INTRODUCTION

Historical

Free-surface oscillations in enclosed basins, known as seiches or seiching in lakes and harbors or as sloshing in coffee cups and storage tanks, have been observed since very early times. Forel (1895) quotes from "Ies Chroniques de Christophe Schulthaiss," which gives a vivid description of seiching in the Lake of Constance

in 1549, and Darwin (1898) refers to seiching in the Lake of Geneva in 1600 with a peak-to-peak amplitude of over one meter; observations in cups and pots doubtless antedate recorded history. Scientific study dates from the papers of Merian (1828) and Poisson (1828) and, especially, from Forel's observations in the Lake of Geneva beginning in 1869 (Forel 1895, Darwin 1898, Defant 1961).1 Limnological studies have been continuous since Forel, but the past two decades have seen more concen­trated study in connection with natural disasters, such as tsunami amplification in harbors and bays (Wiegel 1964) and surging in nearly closed seas. [Van Dantzig & Lauwerier (1960) begin a series of papers on the North Sea with the statement that "On February 1st 1953 the South-Western part of the Netherlands was stricken by a flood disaster unsurpassed in the memory of this country."] There also has been extensive study of fuel sloshing [Barr (1972) has assembled a bibliography with 656 entries], mostly rather specialized and of limited scientific interest; how­ever, the elegant work of the Russian school merits special mention (see Moiseev 1964, Moiseev & Petrov 1966, Moiseev & Rumyantsev 1968, and references therein). Closely related to fuel sloshing is the problem of earthquake-induced loads on dams and storage tanks. The following review is concerned primarily with the theory of seiching in harbors, which may (by definition) be distinguished from lakes and other closed basins by the existence of an opening into an exterior body of water.

Earlier work on seiching in closed basins is covered in some detail by Chrystal (1905), who gives 136 references (including 22 by Forel) covering a 150-year period that begins with the 1755 observations of seiches induced in Scottish lochs by the

I Liquid oscillations in a U tube may quite properly be included under the rubric of seiching, in which case priority for scientific study belongs to Newton (1686), who showed that the natural period of such oscillations is equal to tb,at of a simple pendulum with a

length equal to half the length of the fluid in the U tube.

17

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18 MILES

Lisbon earthquake, by Lamb (1932, Sections 190-93, 207-12, 257-61), by Defant (1961), who gives a thorough coverage of work up to 1950, and by Wilson (1972). Wehauseil & Laitone (l960) give a rela�ively brief survey, but with extensive references to Russian work, Olson (1962) gives a bibliography of almost 400 entries, including extensive coverage of Japanese work.2 Seiching in harbors is

reviewed by Wiegel (1964), Raichlen (1966a), and Wilson (1972). Some of the high­lights of this earlier work are covered in Section 2 below.

The principal causes of seiching in lakes are storm surges or other atmospheric inputs and, rarely, earthquake ground motion. The same causes are operative for harbors, but excitation through the mouth (even though due ultimately to atmos­pheric inputs) generally dominates direct excitation within a harbor; moreover, earthquakes are relatively more important in that tsunamis are seismic in origin. Storm surges and tsunamis have recently been reviewed by Bretschneider (1967) and Van Dam (1965), respectively, and will not be considered in detail here.

Model studies are discussed in the reviews of Wiegel (1964) and Raichlen (1966a) and the references therein. The fluid-mechanical aspects of such studies have not changed significantly in the interim (see, e.g., Lee 1971), although Shaw & Parvulescu (1971) have suggested that acoustical models be used for the analog study of harbors of constant depth. Reference may be made to recent issues of ASCE Journal of the Waterways, Harbors, and Coastal Engineering DiviSion,

La Houille Blanche, and other civil-engineering journals for papers on model testing and harbor design.

Assumptions

The basic assumptions in the theoretical study of seiching are (i) a perfect, incompressible fluid, (ii) small displacements, (iii) plane level surfaces in an inertial reference frame, and (iv) a perturbation pressure proportional to the vertical dis­placement of the free surface, ((x, t) at x = (x, y) and time t. These assumptions imply the linear, long-wave equations developed in Sections 189 and 193 of Lamb (1932). The additional assumption (v) of simple harmonic motion,

((x, t) = Rl {((x) ejwt} and similarly for the velocity, nix, t), reduces these equations to

jwu = -gV(

and

(1.2)

(1.3)

2 Japanese work is underrepresented in the present review, partially because much of it is in Japanese and/or in difficult-to-obtain sources. Olson's bibliography contains 143 Japanese entries, 37 by Hidaka, whose dedication to the study of seiching rivals that of Fore!. A complete list of Hidaka's papers is given by Yoshida (1964).

3 The time dependence exp (jOlt) is conventional in electric-circuit analysis. Physicists currently use exp( - iM), but most of the literature cited here, including Lamb (1932) and Defant (1961), uses exp (iwt) or its equivalent. The use of j, rather than i, for (_1)1/2 is a mnemonic against confusion.

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HARBOR SEICHING 19

where V is the two-dimensional gradient operator. We omit the modifiers complex amplitude of and free-surface and refer to , simply as the displacement. If h is constant, as is often assumed in harbor studies but is not generally assumed herein, (1.3) reduces to the two-dimensional Helmholtz equation. The condition of zero mass transport across a boundary B requires the normal component of hV,

to vanish on B. Alternative boundary conditions are that, vanish at an opening into a very deep body of water or that' behave as an outgoing wave at infinity.

The assumption (i) implies the neglect of compression (sound) waves and, more importantly, of viscous and capillary effects, including surface contamination. Capillary and other surface forces may be safely neglected in full-scale studies, but the situation is otherwise in the laboratory, and capillary hysteresis may be the major source of dissipation in model studies (see Miles 1967 for a review of laminar viscous anl capillary effects).

The assumption (ii) of small disturbances is generally adequate for the calculation of motion within the harbor, but it precludes a quantitative description of wave build-up in shallow water (which may be important for tsunami studies) and of the more important dissipative processes, which (for natural scales) usually are associated with turbulent flow in the boundary layers and the harbor mouth. Takano (1959), Gaillard (1960), Chabert d'Hieres (1960), Miles & Ball (1963), and Verhagen & van Wijngaarden (1965) have considered nonlinear effects in seiching, but their results are of limited interest for harbors.

The assumption (iii) of plane level surfaces excludes the effects of the Earth's rotation except as they may be included as first-order corrections. Rotational effects can be significant only for periods that are of the order of twelve hours or more, and even then they may be small for a sufficiently oblong planform [Platzman & Rao (1964) find that rotation increases the longest period of oscillation for Lake Erie, roughly 14 h, by only 1.3%].

The approximation (iv), which is equivalent to the assumption of hydrostatic equilibrium in the vertical if the fluid is homogeneous, requires h <{ g/w2, where h is a representative depth, and implies an upper bound on the number of modes that may be accurately described; however, only the first few modes are important in most analyses. This approximation is not essential for a homogeneous fluid of constant depth, for which, satisfies the two-dimensional Helmholtz equation and wavenumber and frequency are related by a transcendental dispersion relation in which h appears as a parameter (Lamb 1932, Section 228); however, dispersion complicates the analysis of broad-spectrum excitation.

The approximation (iv) also excludes internal gravity waves, which are associated with stratification and have the frequency spectrum (0, N). The cutoff (Viiisiilii) frequency, N, may be as high as 10- 2 rad/sec (periods as small as 10 min) in a summer thermocline, which is comparable with, or significantly larger than, the lowest resonant frequency of many harbors; however, this resonant frequency is usually associated with the Helmholtz mode, which does not couple effectively with internal waves. On the other hand, internal waves may be strongly excited in certain lakes (Defant 1961, Thorpe 1971, Thorpe, Hall & Crofts 1972, Wilson 1972).

The assumption (v) of simple harmonic motion is, in principle, no more restrictive

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20 MILES

than the assumption (ii) of small disturbances, since linearity permits the extension to arbitrary time dependence through Fourier superposition. This is essential for most applications, in which the input is posed as a power spectrum without phase information.

The assumption (v) of uniform depth may be adequate for some harbors, but it is rarely adequate for natural basins, especially if several modes must be included. It is generally inadequate for the exterior ocean.

2 PRELIMINARY SURVEY

Closed Basins

The longest period of oscillation in a shallow rectangular basin of length I and uniform depth h is given by

T = 2n/w = 21/e, e = (gh)1/2 (2.1a,b)

which is known as M erian's fonnula.4 It is often used to obtain rough approxima­tions to the longest period in oblong lakes, using the length of the centerline for I and the mean depth for h, but it is unreliable if the plan form of the basin is significantly nonrectangular and/or the depth is significantly nonuniform. Favorable examples may be cited but are outweighed by unfavorable ones; thus, using Forel's estimates of 1= 70 km and h = 150 m for the Lake of Geneva yields T = 61 min, which is 18% below the observed value of 74 min. [Darwin (1898) obtains 70 min for this same example and adds that "the agreement is suspiciously exact"; in fact, he appears to have made a numerical error. Raichlen (1966a) also considers this example, using h = 160 m (525 ft) to obtain T = 59 min, against which he cites (without reference) an observed value of 83.5 min; however, this last value appears to be in error.]

Merian's formula also may be invoked for higher modes by replacing I by I/n, where n is the number of transverse nodal lines, and for transverse modes by replacing I by the width of the basin. In fact, the higher modes for a basin of variable depth and/or width are not harmonically related, and the resulting estimates are even less reliable than those for the longest period.

DuBoys (1891; cited by Chrystal 1905) suggested that Merian's formula could be improved by invoking Green's approximation for wave propagation in a channel of slowly varying dimensions to obtain

T = 2 I {gh(x)} - 1/2 dx (2.2)

As Defant (1961) points out, however, Green's approximation (which applies primarily to propagation over many wavelengths of gradually varying depth) requires

4 The result (2.lb) for the speed of a shallow-water wave is generally attributed to Lagrange (1781; cited by Lamb 1932, Section 170). It appears that nearly half a century elapsed before Lagrange's result was first applied to standing waves in a closed basin by Merian (1828).

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HARBOR SEICHING 21

the change in h(x) over I to be small. But if this change is small, say h = h 1 (1 +fj), fj � 1, it is readily shown that the periods predicted by (2.1), using the average value of h, and by (2.2) are identical to first order in fj. If, on the other hand, f) is not small, (2.2) may be less accurate than (2.1). Consider the following examples of constant-width basins: (a) h varies linearly from 0 to ho; (b) h varies parabolically and symmetrically from 0 at the ends to ho at the center. Exact solutions of (1.3) are available for both of these profiles, and a comparison of the approxima­tions provided by (2.1) and (2.2) with the exact results yields (co is the wavespeed based on ho)

(1.3) (a) Co Til = 3.279 (b) Co Til = 2.221

(2.1) 2.828 (-14%) 2.449 ( + 10%)

(2.2) 4 (+22%) n (+41%)

from which it appears that (2.2) may be less reliable than (2.1). In brief, duBoys' formula cannot be recommended on either theoretical or empirical grounds (Green's approximation might yield better approximations for the higher modes, for which the change in depth over one wavelength may be small).

Exact solutions may be obtained for any basin geometry for which (1.3) admits solutions in terms of known functions through separation of variables. Obvious examples are rectangular, circular, and elliptical basins of uniform depth; less obvious are such examples as a butterfly-shaped basin of uniform depth bounded by segments of ellipses and hyperbolae (Groves 1968).

The circular basin of uniform depth, which may be regarded as the prototype for variable width, was considered originally by Poisson (1828); however, he did not obtain explicit results for the periods. Rayleigh (1876), who appears to have been unaware of Poisson's paper, gave a more complete analysis and obtained (among other results) T = (2n/1.84)(a/c) = O.85(4a/c) for the longest period of oscillation in a shallow basin of radius a and uniform depth (Merian's formula yields a 17% error for this example). Rayleigh also measured the periods of the first five modes in a tank approximately three meters in diameter by "dipping one or more buckets synchronously with the beat of a metronome" and counting the oscillations for some five minutes after the withdrawal of the buckets. The maximum discrepancy between theory and observation was roughly 1%, and Rayleigh concluded that "The agreement ... is as close as could be expected." Indeed, Case & Parkinson (1957), working with electronic instrumentation, found discrepancies as large as 9% for the period of the dominant mode in a carefully polished, three-inch cylinder and attributed them to "surface tension effects associated with wetting of the wall" (see Miles 1967 for further discussion); whatever their cause, they demonstrate the difficulties of working at too small a scale.

One of the few examples of an exact solution for which both width and depth are variable is provided by an elliptic basin with a paraboloidal bottom, for which Goldsbrough (1930) obtained

� b = n(a, b)/(gh)1/2 (2.3)

for the longest periods of longitudinal and transverse modes (a and b are the

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22 MILES

semimajor and semiminor axes of the undisturbed surface, and Ii is the mean depth). Ball (1965) showed that (2.3) holds for unrestricted amplitudes and that the cor­responding periods for a basin rotating about a vertical axis with angular speed Q are given by

(2.4) for unrestricted amplitudes. This remarkable result for finite-amplitude rotational splitting is identical in fonn with the period equation for a rotating two-degree­of-freedom system (Lamb 1932, Section 205a) or with the Rayleigh approximation for the dominant longitudinal and transverse modes in any rotating basin with two horizontal axes of symmetry and small (QT � 1) rotational coupling (Lamb 1932, Section 212a).

Analytical results for other than the special geometries indicated above may be obtained by regarding more complicated basins as connected systems of simple basins with plane-wave coupling canals. An especially interesting example is provided by two small basins of surface area AI and A2 connected by a short, narrow channel of length I and cross section S. The hypothesis that the potential energy appears entirely in the basins and the kinetic energy entirely in the canal leads to the result (Neumann 1943)

w2 = (gS/I)(Ajl + All) (2.5)

where w is the angular frequency of oscillation. This system is, as Defant points out, a variant of the Helmholtz resonator in acoustics (see below) or, alternatively, of the U tube (see footnote 1). Neumann (1944) also considered other configura­tions and introduced impedance methods for their analysis.

The effect of an opening on the lowest frequency of a rectangular bay may be calculated by analogy with an open-ended, two-dimensional organ pipe, using Rayleigh's (1904) end correction (Honda, Terada & lsitani 1908).5 An approximation for any bay may be obtained by assuming zero vertical displacement at the mouth, but this is accurate only if the exterior body of water is much deeper than the bay.

Numerical methods prior to the advent of high-speed computers have been based on quasi-one-dimensional models developed by Chrystal, Defant, Ertel, Hidaka, Honda, and Proudman and often described as channel approximations; see Defant (1961) for a survey and Platzman & Rao (1964) for an exemplary application to Lake Erie. High-speed computers have been applied to the fully two-dimensional equations, (1.2) and (1.3), during the past decade; see Raichlen (1965), Hwang & Tuck (1970), Lee (1971), Olsen & Hwang (1971), Lee & Raichlen (1972), Platzman (1972), and Wilson (1972). Platzman includes the Earth's rotation and allows for either zero mass transport or zero free-surface displacement along any segment of the boundary. He gives applications to a square basin of uniform depth

5 Neumann (1948) argues that the end correction should be determined by analogy with

a three-dimensional organ pipe and supports his claim by comparison with experiment. His

results may have some empirical validity, but they do not appear (at least to me) to have

any rational basis.

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HARBOR SEICHING 23

(for which analytical results are available), Lake Erie (for which the channel approximation provides good results), Lake Superior (for which the channel approximation does not provide good results), and the Gulf of Mexico (which is complicated by openings through the Straits of Florida and the Yucatan Channel). He also indicates alternative numerical procedures. Reference also should be made to a series of eight papers on the North Sea by Lauwerier and his colleagues (beginning with van Dantzig & Lauwerier 1960, and ending with Lauwerier & Damste 1963) and to the analog-computer (electric-network) methods of Ishiguro (1959) and Joy (1966).

Harbors Many pre-1960 papers on seiching in harbors and other partially enclosed bodies of water assume either no mass transport across, or zero displacement in, the opening(s) and therefore may be grouped with work on closed basins. In fact, the resonant modes of a harbor with a sufficiently narrow opening closely resemble those of the corresponding closed basin, but with two important exceptions: (i) the resonant frequency (for maximum flow through the mouth) of the nth mode in the harbor lies slightly above the corresponding resonant frequency for the closed basin; (ii) the degenerate mode of zero frequency and uniform displacement in the closed basin is transformed to the Helmholtz mode, for which the displacement is approximately uniform within the harbor and the lateral motion is concentrated near the mouth. This mode is of primary importance for tsunami response. [The adjectives pumping (Lee 1971) and co-oscillating (Platzman 1972) also have been used to describe the Helmholtz mode. Helmholtz resonance in harbors appears to have been studied originally by Miles & Munk (1961), but the analogy between the dominant seiching mode in connected basins and the motion in a two­dimensional Helmholtz resonator was recognized by Neumann (1943). Professor Platzman (personal communication) has pointed out that seiching is not really an appropriate description of the motion in this mode.]

McNown (1952) carried out model experiments using a circular harbor with a 2210 opening and found that the first several modes closely resembled those determined by Rayleigh (1876) in both period and shape. He did not report any evidence of the Helmholtz mode, presumably because his incident-wave spectrum did not extend to sufficiently low frequencies (the Helmholtz-resonant frequency for McNown's harbor would have been roughly one-quarter of the lowest resonant frequency for the corresponding closed basin) . Lee (1971) carried out similar experiments and did observe the Helmholtz mode at the theoretically predicted frequency.

Biesel & LeMehaute (1955, 1956) and LeMehaute (1961) used a plane-wave (essentially one-dimensional) mathematical model of a harbor to study the effects of discontinuities in breadth and depth, obstacles, sloping beaches, and radiation into the exterior sea qua semiinfinite canal. Their methods have much in common with those of Neumann (1944) and with the equivalent-circuit methods described in Section 4 below but are not directly applicable to the quantitative analysis of actual harbors.

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24 MILES

Harbor Paradox Miles & Munk (1961) considered a long rectangular harbor coupled to a semi­infinite sea through an aperture, allowing for both inertial loading and radiation damping by the external sea. They concluded that, in the absence of friction, narrowing the mouth of the harbor would increase the mean-square response of the harbor to a broad-band input ("the harbor paradox"\ 6 but their analytical results support this conclusion only for the Helmholtz mode, which exhibits a series­resonant response like that of a single-degree-of-freedom oscillator. Higher modes exhibit responses like that of an oscillator with closely spaced series- and parallel­resonant frequencies (see below), and the mean-square response to a broad-band input then is independent of the width of the opening (Miles 1971). Viscous damping is increased by narrowing the mouth, however, and the response of the higher modes is thereby reduced. Wilson (1962), in discussing the harbor paradox, cited the example of Duncan Basin in Table Bay Harbor (Cape Town), where narrowing the mouth significantly reduced the seiching amplitudes within the harbor. LeMehaute (1962) argued that not only friction, but also a reduced "probability" of resonance, would negate the paradox. The latter effect presumably refers to the fact that the time required for an oscillator to achieve its peak resonant amplitude is inversely proportional to the damping; however, at least in the case of tsunami excitation, this is not likely to be an operative limitation. The decisive question is whether, and to what degree, the reduction of radiation damping of the Helmholtz mode is compensated by an increase in frictional damping; this question, which has important implications for tsunami protection, has not been definitely answered, but it appears that the balance could go either way and is likely to depend on the amplitude of oscillation (see below). Nevertheless, it should be emphasized (especially by one of its perpetrators) that the harbor paradox is significant only for the Helmholtz mode, and therefore typically for tsunami excitation, and that narrowing the mouth of a harbor almost certainly decreases its mean-square response to swell.

The analysis of actual harbors usually can be carried out only with the aid of numerical solutions of the equations of motion (Lee 1971, Olsen & Hwang 1971)

and/or model studies, hut analytical formulations may be helpful either in the resolution of the problem into subproblems (see below) or in providing solutions for certain simple planforms. It should be recognized, however, that basins with regular planforms and bottom profiles yield more regularly spaced normal modes, albeit with some near degeneracies, than do irregularly shaped basins and that only the first few modes in the spectra of the latter may be resolvable. This suggests that the response of a harbor in its higher modes should be calculated by the methods of architectural acoustics (Munk 1963; Morrow 1963, 1966). Moreover, it may I suffice to determine only the resonant frequencies and modal shapes for the lower

6 Carrier, Shaw & Miyata (1971a,b; see also Miles 1971) pointed out that lengthening the channel connecting the harbor to the external sea would have the same effect as narrowing the mouth.

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HARBOR SEICHING 25

modes of the closed harbor and to determine detailed response characteristics only for the Helmholtz mode. A knowledge of the resonant frequencies and the corresponding locations of maximum motion may suffice for mooring and naviga­tional problems (see Wilson 1972) and for the design and placement of artificial dampers for the alleviation of such problems. The response of the harbor in the Helmholtz mode, on the other hand, may be sensitive to the details of both the harbor entry and the exterior domain; moreover, much more detailed study of this mode may be justified by tsunami danger to life and property.

Exterior Spectrum

The exterior domain generally possesses both a continuous spectrum, which is associated with the approximately semiinfinite abyss, and a discrete set of trapped modes associated with bottom topography, especially the continental shelf. There are no trapped modes for a half-space of uniform depth. Trapped modes on the continental sbelf are known as shelf waves and are detected in off-shore spectra (Munk 1962); although trapped by the shelf, they are radiated along the coast and may provide radiation damping, as well as inertial loading, for harbor seiching. Wilson (1971) has suggested that off-shore basins may possess modes that are trapped with respect to radiation in all directions and that could couple with harbors at tsunami frequencies, but the supporting evidence (proximity of calculated and observed tsunami frequencies in San Pedro Bay) is rather limited.

3 THEORETICAL FORMULATION Green's-Function Solution

We consider (Figure 1) a harbor H that is bounded by a wall B, across which there is zero mass flux, plus a mouth M, through which H opens into an external domain

B

Figure 1 Schematic of harbor (H) opening into semiinfinite, exterior domain (E) through mouth (M).

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E; H is characterized by its free-surface area A and bottom profile h(x); E is typically a half-space but could be a second basin of finite extent; M is assumed to be straight (a segment of x = 0) but could be taken to be a smooth continuation of B. Following Miles (1971), hereinafter M71, we pose the solution of (1.3) in the form

((x) = I L �H(x,1J)f(1J)d1J (x in H) (3.1 a)

= (O)(x)-I L �E(x,l'/)f(1J)dl'/ (x in E) (3.1b)

where: I is the total volumetric flow from E to H through M; If(y) is the volumetric flow per unit width, withfnormalized according to

LfdY = 1 (3.2)

'fJH is a Green's function for H ('fJH = jwG/gh in M71, Section 2, wherein h is constant); ((0) is the displacement that would exist in E if there were no flow into

M and includes both an incident wave and such reflected and trapped waves as are required to satisfy the condition of zero mass flux across the external coastline; 'fJ E is a Green's function for E.

Integral Equation Equating the displacements given by (3.1a,b), with x in M, yields an integral equation for the unknown If(y). The analytical solution of this integral equation in closed form appears to be possible only in special parametric limits. It may be solved numerically, typically in conjunction with a numerical determination (not necessarily explicit) of '!J H (Lee 1971), but the following, approximate procedure is likely to be adequate for most purposes by virtue of an associated variational principle of Schwinger's type (see M71).

Averaging the integral equation over M with the weighting function f* (the complex conjugate off) yields

V = ZHI = V(O)-ZEI (3.3) where

v = L If*dy, (3.4a,b)

and

ZE,H = L Lf*(Y)�E'H(y'I'/)f(l'/)d1JdY (3.5)

Choosing an appropriate trial function for f, subject to the constraint (3.2), allows ZE and ZH to be calculated from (3.5) and (3.3) to be solved for I, after which ( may be determined for x in H by either (3.1a) or (3.7) below. This indirect procedure, which formally reduces the coupled boundary-value problem for H + E

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HARBOR SEICHING 27

to the independent determination of the Green's functions '§ Hand '§ E' is typically more efficient than a direct numerical solution, even though it may be necessary to determine '§ Hand '§ E numerically.

Interior Solution

The Green's function '§ H' qua function of x, satisfies (1.3), subject to the boundary condition that its outwardly directed normal derivative on B + M be equal to (jw/gh)(j(Y-I1), where (j is Dirac's delta function and (0,11) is a point in M (Y-11 may be replaced by x - � if M is not a segment of x = 0). This boundary-value problem may be solved by separation of variables in a few special cases, including rectangular, circular, and circular-sector basins of uniform depth (Miles & Munk 1961, Carrier, Shaw & Miyata 1971a, Lee 1971, Miles 1971), but in general it must be solved numerically to obtain a matrix representation of '§ H'

If h is regarded as constant within H, the boundary-value problem may be transformed to an integral equation for '§ H(X, 11) with x on B + M. Lee (1971) carries out this formulation to obtain '§ H in the form M/�, where M is an IV x p matrix, Band M are divided into IV -p and p segments of length �, and the element at (i,j) in his notation corresponds to the element at (x, 11) in the present notation. It

follows that ZH = M*Mp C, where Mp is the p x p truncation of M, C is a column­matrix representation ofJ(y) , subject to the constraint that the sum of its elements be equal to 1/�, and f* is the conjugate transpose of C. [Hwang & Tuck (1970) also give a numerical formulation of the harbor problem on the assumption of constant depth, but their formulation does not distinguish sharply between Hand E and is less efficient than that of Lee, at least for narrow-mouthed harbors. ]

The most direct (although not necessarily the most efficient) numerical procedure for nonuniform depth appears to be the conversion of (1.3) to a finite-difference form, together with a discrete representation of B + M, such that either x or y is constant on each segment (Olsen & Hwang 1971); ZH may be approximated by assuming the flow per unit width through M in the form C, normalized as above (such that 1== 1), solving the difference equations for the matrix representation of

((x) for x in M, say 'M, and then calculating ZH == f*'M (I know of no explicit example at this time) ,

The matrix representations of '§ Hand J may be used to calculate ( from (3.1 a) after first determining I from (3.3), but the description of resonant response is facilitated by a normal-mode representation of ( in H. The normal modes, say I/In(x) for W = Wn' are eigensolutions of (1.3), subject to the boundary condition of vanishing normal derivative along B+M. They form a complete set in H, are orthogonal, and may be normalized according to

f 0 (m =f. n)

HlftmlftndA=l (m=n)

(3.6)

The single-sequence ordering is such that 0 < WI < W2 < . . . , with appropriate allowance for degeneracies; the subscript zero is reserved for 1/10 = A - 1/2, for which Wo = O. Both the I/In and the wn may be calculated by any of the methods cited in Section 2 for closed basins. The neglect of dissipation implies that both are

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real, although small dissipatIOn may be incorporated empirically by allowing wn

to be complex (see below) while still regarding I{In as real. The Green's function '#H may be formally expanded in the I{In to obtain

Fn= LNndY (3.7a,b) n

where the summation is over the complete sequence of the I{I n' The series is dominated by the nth mode if w is in the resonant neighborhood of Wn' The trial function f. and hence also F n' is typically (but not necessarily) real.

Exterior Solution

The explicit calculation of '# E is possible only for quite special configurations, including a half-space of uniform depth (Miles & Munk 1961). Numerical representations could be obtained by patching the numerical solution of (1 .3) in a finite region of variable depth to the solution in a semiinfinite abyss of a constant depth, but the shelf modes offer special difficulties. [Olsen & Hwang (1971) develop such a patching technique, but they assume zero mass transport across two lines extending from the coastline at finite lateral distances from H across the region of variable depth to the abyss, thereby implicitly eliminating radiation damping through the shelf modes. The effects of this constraint on their solution are not dear.]

Fortunately, , is not required in E, and the calculation of ZE requires the calculation of � E only for x in M. The end result for a narrow-mouth harbor has the form (Miles 1972)

ZE = (w/gho){IX+U/n)[f3-1n koaJ} (koa � 1) (3.8)

where: ho is the depth in E just outside of M (x i 0 in Figure 1) and may be assumed to differ from the depth just inside of M (x! 0 in Figure 1); ko is the wavenumber based on ho; CI. is a positive-definite, oscillatory function of the product of a representative wavenumber and the shelf width that tends asymptotically to

t (CI. == t for a half-space of uniform depth); f3 depends on the topography in a complicated way, but results for a few examples suggest that it may be approximated by 1.5 if h does not vary rapidly over the shelf (Miles 1972). The form of (3.8) rests 'on reasonably firm foundations, but the determination of CI. and f3 requires further study.

4 EQUIV ALENT-CIRCUIT ANALYSIS Equivalent Circuit?

The equivalent electric circuit implied by (3.3) is shown in Figure 2, in which VIOl is an externally applied voltage, ZE is the radiation impedance (which contains

7 Neumann (1944) and Defant (1961) define impedance as the ratio of perturbation pressure to transverse volumetric flux, thereby introducing an additional factor of pg vis-a-vis the present definition.

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HARBOR SEICHING 29

0 rz:l 0

t t � VIol V

I

Figure 2 Equivalent circuit for harbor and external domain, as inferred from (3.3).

both resistive and reactive components) of the external domain, ZH is the harbor impedance (purely reactive), I is the input current, and V is the voltage across Zw The normal-mode representation (3.7) implies the series decomposition

(4.1)

in which Zn is equivalent to an inductor and capacitance III parallel, with Cn = IFnl-2. Ln = IFnI2Iw;, and parallel-resonant frequency Wn; Zo = l/jwA is equivalent to a simple capacitor. The coefficient of 1/In in (3.7a) is VjF:. where V. is the voltage drop across Zn'

Internal damping (exclusive of that associated with flow through M) may be incorporated in an ad hoc fashion by replacing w� by w� + (jwwnIQn). where Qn is the Q of the nth mode in the closed harbor (it differs from Qn in M71, Section 4, which is denoted by Qn below). This corresponds to placing a resistor, Rn = wn Ln Qn. in parallel with Ln, but it must not be overlooked that Qn and Rn may be amplitude dependent in consequence of nonlinearity (see below). Motion in the Helmholtz mode (for which the preceding formula gives Ro = 00) is concentrated in the neighborhood of M, and the corresponding boundary-layer damping must be represented by a resistor in series with Co. Damping associated with flow through

M may be represented by placing a resistor RM between (in series with) ZE and ZM (see Dissipative Elements).

The equivalent-circuit formulation facilitates the analysis of the harbor response to a prescribed input by resolving the problem into subproblems that may be attacked separately. [Such a resolution does not require the invocation of an equivalent circuit and may be effected by other means (Lee 1971, Lee & Raichlen 1972), but the equivalent-circuit approach offers conceptual advantages and facilitates the introduction of semiempirical descriptions of dissipation. ] Strictly speaking, these subproblems are not independent, and their separation depends on the assumption, rather than the rational deduction, ofJ(y). In practice, the limitations imposed by a reasonable assumption for J(y) are not significant vis-a-vis other uncertainties in the problem provided that wale == ka � 1, where a is the width of

M, the wave speed e is based on a representative depth in H, and k is the wave­number; the known results for harbors of uniform depth suggest that ka < 1 may be sufficient. A detailed analysis is not likely to be justified if ka is large, and the

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methods of architectural acoustics then should be considered (see penultimate paragraph in Section 2).

Additional input terminals must be provided if the harbor has more than one mouth. This occurs, in particular, when a harbor is regarded as a system of two or more connected basins. A generalization of the formulation in Section 3 then yields a set of coupled integral equations (cf Lee & Raichlen 1972), the weighted averages of which yield a set of equivalent-circuit equations analogous to (3.3). It may prove convenient to introduce lossless transformers to provide for the differences among the generalizations of the weighting factor Fn (3.7b) associated with multiple mouths. The circuit also may be generalized to allow for inputs due to a prescribed free-surface pressure (such as a storm surge); however, surh inputs appear to be of limited importance for harbors, for which the dominant excitation is through M.

Harbor Impedance The convergence of the series in (4.1) is quite slow for a narrow-mouthed harbor; accordingly, it is expedient to rewrite Z H in the form

ZH = (jwAtl+jwLW)+jw3I'IF.12w;2(w;-w2r2 (4.2) n

where

LW) = I'lFnI2w;2 = lim r r f*[(jwrl�H+(w2Arl]fd1]dy (4.3) n wlO JM JM

and the prime implies the exclusion of n == O. It typically suffices to retain only the first few (none if W � WI) terms in the series in (3.7a) and (4.2). On the other hand, the series in (4.3), which is not a modal expansion in the conventional sense, con­verges quite slowly and is unsuitable for the numerical calculation of L<;1).

Analytical approximations to L<;1) are known for circular and rectangular harbors of uniform depth (M71, wherein A<;}) == ghL<;}»). Explicit results do not appear to be available for other configurations but could be calculated from (4.3) if '§ H were known. The approximation (Miles 1973)

(4.4)

where r is the radius from the midpoint of M and < > implies the mean value of the bracketed quantity in H, is roughly applicable to any harbor of approximately uniform depth (h should be evaluated just inside M if the depth is variable), regular planform, and moderate aspect ratio (say t to 3).

The parameter L<;1) may be inferred from an experimental measurement of the Helmholtz-resonant frequency, wO' if the other parameters in the system are known. In particular, if identical models are coupled through M and the measured Wo is sufficiently small to justify the neglect of the series in (4.2),

LW) = (W�Atl (4.5)

Note that this value of Wo is larger than that for H + E. See also (4.7) below.

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Entry Channel

HARBOR SEICHING 31

If H and E are connected by a narrow (compared with both the wavelength and the lateral dimensions of H) channel, in which the motion may be regarded as one­dimensional, the equivalent circuit of Figure 2 may be generalized by inserting a four-terminal network between Zll and ZE' The required development is given in M71 for a channel of constant breadth and depth. Channels of gradually varying breadth and depth may be described either by a single network, the elements of which may be deduced from the solution of the quasi-one-dimensional wave equation (Lamb 1932, Section 185), or by a cascaded sequence of networks obtained by resolving the channel into a corresponding sequence of finite-length elements. Either procedure also may be used to obtain 2H for narrow bays of the type that exist along the coast of Japan and are especially susceptible to tsunami excitation.

If the entry channel is short, as well as narrow, the four-terminal network may be reduced to a simple series inductor having the inductance

LM = I [gS(x)] -1 dx (4.6)

where S(x) is the vertical cross section at x. The inductance LM for even a relatively short channel may exceed Lfji.

An interesting example, which provides an experimental procedure for determining LV/), is presented by a narrow channel of width a and depth h (equal to the depth, or at least the mean depth, of H in M) that is closed at one end and connected to H through M. The short-circuit input impedance of the channel, as seen from M, is given by 2M = j(ac)-l tan kl, where k is the wavenumber based on h. The Helmholtz­resonant frequency, Wo (which differs from that of H + E), then is given by the smallest zero of 2M + 2w The assumption that Wo is sufficiently small to justify the neglect of the series in (4.2) yields [cf (4.5)J

L<jj) = (w�A)- 1 - (ghako) - 1 tan kol (4.7) The resonant frequency Wo decreases with increasing 1 up to the point kol = �n (for which the channel resonates like an open-ended organ pipe) and thereafter exhibits an oscillatory behavior; in particular, w� = 1/ ALV/) for kol = n.

Dissipative Elements

A calculation of the rate at which energy is dissipated in M on the hypothesis that the entry-separation loss may be represented as a head loss (as in pipe-flow calcula­tions) yields an equivalent resistance RM = KI/gS2, where K is an empirical coefficient, and S is the cross-sectional area of M. Ito's (1970) study of breakwaters at tsunami frequencies suggests K � 0.6, but experimental results for the related problems of oscillating fiat plates (Keulegan & Carpenter 1958) and acoustical orifices imply that K depends on both the Strouhal and Reynolds numbers (although the variation of K with Reynolds number is probably unimportant for natural scales). Similar calculations may be carried out for boundary-layer damping, but, in

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32 MILES

the absence of reliable measurements, it appears inappropriate to pursue this aspect of the problem here.

Resonant Response

The response of the harbor (including the entry channel, if any) in the resonant neighborhood of ill = illn will be dominated by thr nth mode if radiation plus frictional damping of that mode is sufficiently small. Proceeding on this assumption, we define Rln) and Lin) by

R(n)+ jwVn) == ZE+ZH-Zn (W-4 wn) (4.8)

and approximate the equivalent circuit of Figure 2 by that of Figure 3. The frequency dependence of R(n) and L(n) may be neglected in the neighborhood of W = wn > 0, but not in the neighborhood of w = O. The input impedance Z E + Zu exhibits parallel resonance at ill = OJn and series reSonance at ill = wn' where 030 is determined by

w6AL(O) = 1 (4.9a)

and

w; = w;(l + en), en = Ln/L(n) � 1 (n > 0) (4.9b)

The Q of the resonant peak, defined as the ratio of the resonant frequency to the half-power bandwidth, is given by

Q- = [wL(O)/R(O)] _-(4.10a) o (I)-Wo

and

Qn = [(eJQ(n»)+(l/Qn)]-l, Q(n) = wnL(n)/R(n) (4. lOb)

The resonant-peak amplitude of jV,.!V(O)1 is given by <20 for n = 0 and by 8n <2n for n > O. Here Qn is typically larger than Qo for the first few of the normal modes, for which en � Q(n)/Qn and cOn is quite close to w •.

lt is possible to provide an approximately monochromatic excitation in the laboratory, but in nature the excitation is random and must be described by either an amplitude or a power spectrum without phase information. Let S(Ol(w) be the power spectral density of (0) and assume that the variation of S(O) across M may be neglected (ka � 1) so that the power spectral density of VIOl, (3.4b), may be

Figure 3 The approximate equivalent circuit for the resonant neighborhood of W = wn; Rln) and L(n) are defined by (4.8).

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HARBOR SEICHING 33

approximated by S(O). Invoking the preceding approximations for resonant response then yields

S(w, x) = S(O)(w) I 1Y,,/V(O)121V!n(x)/FnI2 (4.11) n

for the power spectral density of ((x), where the sum is over those modes that exhibit sharp resonant peaks. The (temporal) mean-square displacement at x is given by

2 1 foo 1 S(O)(wn) IV! n(xW « (x) = 2rr 0 S(w, x)dw = 4 � R(n)[l + (Q(n)/en Qn)] (4.12)

where the square-bracketed term is unity for n = 0 (BoQo == 00). The spatial mean of (4.12) may be obtained by replacing 11/1.12 by l/A.

5 CONCLUSION

The calculation of the normal modes for a closed basin is straightforward in principle and amenable to known, albeit still developing, numerical techniques. The calculation of the inertial parameter Lfi) for the Helmholtz mode is also straight­forward in principle, but its numerical calculation, especially for variable depth, requires further investigation; however, (4.4) may provide an adequate estimate for many configurations.

The calculation of dissipation, necessarily semiempirical, requires further investigation. The calculation of boundary-layer dissipation is relatively straight­forward, but our knowledge of entry losses is quite limited. Laboratory tests are required (not necessarily with complete harbor models) over a sufficient range of scales to establish parametric trends.

The calculation of the radiation impedance also requires further investigation, especially with respect to the continental shelf. The approximation (3.8) may suffice for many applications, especially if the depth outside the harbor (at distances small compared with the wavelength) is large compared with that inside the harbor.

The input at the harbor mouth, (0), usually may be inferred from tidal-gauge records, but tsunamis may be an important exception. The theoretical calculation of (0) for tsunamis is possible in principle (Van Dorn 1965) but is complicated by bottom topography, especially that of the continental shelf, and much remains to be done.

Model tests continue to be important, but are limited by their necessarily inadequate representation of the exterior domain, by Reynolds-number dissimilarity, and by cost. In balance, it seems likely that numerical calculations are now, or soon will be, at least as reliable as, and much cheaper than, model tests in predicting the main features of seiching in a particular harbor; accordingly, future model tests should perhaps be directed at those features, such as entry loss, that are not directly accessible through theory.

ACKNOWLEDGMENT This work wa� partially supported by the Atmospheric Sciences Section, National Science Foundation, NSF Grant GA-35396X, and by the Office of Naval Research,

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under Contract N00014-69-A-0200-6005. I am grateful to Professors George Platzman and John Wehausen for helpful comments and bibliographical aid.

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