Long Waves in Ocean and Coastal Waters

Water waves occurring in the ocean have a wide spectrum of wavelength and period, ranging from capillary waves of 1 cm or shorter wavelength to long waves with wavelength being large compared to ocean depth, anywhere from tens to thousands of kilometers. Of the various long-wavegenic sources, distant body forces can act as the continuous ponderomotive force for the tides. Hurricanes and storms in the sea can develop a sea state, with the waves being worked on by winds and eventually cascading down to swells after a long distance of travel away from their birthplace. Large tsunamis can be ascribed to a rapidly occurring tectonic displacement of the ocean floor (usually near the coast of the Pacific Ocean) over a large horizontal dimension (of hundreds to over a thousand square kilometers) during strong earthquakes, causing vertical displacements to ocean floor of tens of meters. Other generation mechanisms include underwater subsidence or land avalanche in the ocean and submarine volcanic eruption. Gigantic rockfalls and long-period seismic waves can also produce gravity waves in lakes, reservoirs, and rivers. Generation, propagation, and evolution of such long waves in the ocean and their effects in coastal waters and harbors is a subject of increasing importance in civil, coastal, and environmental engineering and science. Of the various long wave phenomena, tsunami appears to stand out in possessing a broad variation of wave characteristics and scaling parameters on the one hand, and, on the other, in having the capacity of inflicting a disastrous effect on the target area. In taking tsunamis as a representative case for the study of long waves in the ocean, it can be said that large tsunamis are generated with a great source of potential energy (as high as 10^15-10^16J ), though the detailed source motion of a specific tsunami is generally difficult to determine. The large size of source region implies that the "new born" waves would be initially long and the energy contained in the large wave-number part (k, nondimensionalized with respect to the local ocean depth, h) would be unimportant. Soon after leaving the source region, the low wave-number components of the source spectrum are further dispersed effectively by the factor sech kh into the even lower wave-number parts. Tsunamis thus evolve into a train of long waves, with wavelength continually increasing from about 50 km to as high as 250 km, but with a quite small amplitude, typically of 1/2 m or smaller, as they travel across the Pacific Ocean at a speed of 650 km/h-760 km/h. There is experimental evidence indicating that tsunamis continually, though slowly, evolve due to dispersion while propagating in the open ocean; this property has been observed by Van Dorn (16) from the data taken at Wake Island of the March 9, 1957 Aleutian tsunami. One of our primary interests is, of course, the evolution of tsumanis in coastal waters and their terminal effects. Large tsunamis can have their wave height amplified many fold in climbing up the continental slope and propagating into shallower water, producing devastating waves (up to 20 m or higher on record) upon arriving at a beach. The terminal amplification can be crucially affected by three-dimensional configurations of the coastal environment enroute to beach. These factors dictate the transmission, reflection, rate of growth, and trapping of tsunamis in their terminal stage. After the first hit on target, a tsunami is partly reflected to travel once over across the Pacific Ocean, with some degree of attenuation -- a process which is still unclear, but is generally known to be small. Based on observations, Munk (13) suggests the figure of the "decay time" (intensity reducing to 1/e) being about 112 day, and the "reverberation time" (intensity falling off to 10^-6) about a week, while the reflection frequency (across the Pacific) is around 1.7/day. To fix idea, the pertinent physical characteristics and their scaling parameters of a tsunami through its life span of evolution can be described qualitatively in Table I. From the aforementioned estimate we note that the dispersion parameter, h/[lambda], and the amplitude parameter, a/h, are both small in general. However, their competitive roles as rated by the Ursell number Ur, can increase from some small values in the deep ocean, typically of order 10^-2 for large tsunamis, by a factor of 10^3 upon arriving in near-shore waters. This indicates that the effects of nonlinearity (amplitude dispersion) are practically nonexistent in the deep ocean, but gradually become more important and can no longer be neglected when the Ursell number increases to order unity or greater during the terminal stage in which the coastal effects manifest. The small values of the dimensionless wave number, kh = 2[pi]h/[lamda] being in the range of 0.6-0.03 during travel in open ocean, suggests that a slight dispersive effect is still present and this can lead to an accumulated effect in predicting the phase position over very large distances of travel. The overall evolution of tsunamis, as only crudely characterized in Table 1, depends in fact on many factors such as the features of source motion, nonlinear and dispersive effects on propagation in one and two dimensions, the three-dimensional configuration of the coastal region, the direction of incidence, converging or diverging passage of the waves, local reflection and adsorption, density stratification in water, etc. While these aspects of physical behavior are akin to tsunamis, they are also relevant to the consideration of other long wave phenomena. With an intent to provide a sound basis for general applications to long wave phenomena in nature, this paper presents (in the section on three-dimensional long-wave models) a basic long-wave equation which is of the Boussinesq class with special reference to tsunami propagation in two horizontal dimensions through water having spatial and temporal variations in depth. Under certain particular conditions (such as the propagation in one space dimension, or primarily one space dimensional of long waves in water of constant depth) this equation reduces to the Korteweg-de Vries equation or the nonlinear Schrodinger equation. In these special cases we have seen the impressive developments in recent studies of the "soliton-bearing" nonlinear partial differential equations by means of such methods as the variational modulation, the inverse scattering analysis, and modern differential geometry (12,14,17). While extensions of these methods to more general cases will require further major developments, the present analysis and survey will concentrate on the three-dimensional (with propagation in two horizontal dimensions) effects under various conditions by examining the validity of different wave models (based on neglecting the effects of nonlinearity, dispersion, or reflection) in different circumstances. From the example of self focusing of weakly-nonlinear waves (given in the section on converging cylindrical long waves), the effects of nonlinearity, dispersion, and reflection will be seen all to play such a major role that the present basic equation cannot be further modified without suffering from a significant loss of accuracy

A porous prolate-spheroidal model for ciliated micro-organisms

A fluid-mechanical model is developed for representing the mechanism of propulsion of a finite ciliated micro-organism having a prolate-spheroidal shape. The basic concept is the representation of the micro-organism by a prolate-spheroidal control surface upon which certain boundary conditions on the tangential and normal fluid velocities are prescribed. Expressions are obtained for the velocity of propulsion, the rate of energy dissipation in the fluid exterior to the cilia layer, and the stream function of the motion. The effect of the shape of the organism upon its locomotion is explored. Experimental streak photographs of the flow around both freely swimming and inert sedimenting Paramecia are presented and good agreement with the theoretical prediction of the streamlines is found

Oblique Long Waves on Beach and Induced Longshore Current

This study considers the 3D runup of long waves on a uniform beach of constant or variable downward slope that is connected to an open ocean of uniform depth. An inviscid linear long-wave theory is applied to obtain the fundamental solution for a uniform train of sinusoidal waves obliquely incident upon a uniform beach of variable downward slope without wave breaking. For waves at nearly grazing incidence, runup is significant only for the waves in a set of eigenmodes being trapped within the beach at resonance with the exterior ocean waves. Fourier synthesis is employed to analyze a solitary wave and a train of cnoidal waves obliquely incident upon a sloping beach, with the nonlinear and dispersive effects neglected at this stage. Comparison is made between the present theory and the ray theory to ascertain a criterion of validity. The wave-induced longshore current is evaluated by finding the Stokes drift of the fluid particles carried by the momentum of the waves obliquely incident upon a sloping beach. Currents of significant velocities are produced by waves at incidence angles about 45 [degrees] and by grazing waves trapped on the beach. Also explored are the effects of the variable downward slope and curvature of a uniform beach on 3D runup and reflection of long waves

Hydromechanics of low-Reynolds-number flow. Part 4. Translation of spheroids

The problem of a uniform transverse flow past a prolate spheroid of arbitrary aspect ratio at low Reynolds numbers has been analysed by the method of matched asymptotic expansions. The solution is found to depend on two Reynolds numbers, one based on the semi-minor axis b, R[sub]b = Ub/v, and the other on the semi-major axis a, R[sub]a = Ua/v (U being the free-stream velocity at infinity, which is perpendicular to the major axis of the spheroid, and v the kinematic viscosity of the fluid). A drag formula is obtained for small values of R[sub]b and arbitrary values of R[sub]a. When R[sub]a is also small, the present drag formula reduces to the Oberbeck (1876) result for Stokes flow past a spheroid, and it gives the Oseen (1910) drag for an infinitely long cylinder when R[sub]a tends to infinity. This result thus provides a clear physical picture and explanation of the 'Stokes paradox' known in viscous flow theory

Hydromechanics of low-Reynolds-number flow. Part 5. Motion of a slender torus

In order to elucidate the general Stokes flow characteristics present for slender bodies of finite centre-line curvature the singularity method for Stokes flow has been employed to construct solutions to the flow past a slender torus. The symmetry of the geometry and absence of ends has made a highly accurate analysis possible. The no-slip boundary condition on the body surface is satisfied up to an error term of O(E^2 ln E), where E is the slenderness parameter (ratio of cross-sectional radius to centre-line radius). This degree of accuracy makes it possible to determine the force per unit length experienced by the torus up to a term of O(E^2). A comparison is made between the force coefficients of the slender torus to those of a straight slender body to illustrate the large differences that may occur as a result of the finite centre-line curvature

Effects of channel cross-sectional geometry on long wave generation and propagation

Joint theoretical and experimental studies are carried out to investigate the effects of channel cross-sectional geometry on long wave generation and propagation in uniform shallow water channels. The existing channel Boussinesq and channel KdV equations are extended in the present study to include the effects of channel sidewall slope at the waterline in the first-order section-mean equations. Our theoretical results show that both the channel cross-sectional geometry below the unperturbed water surface (characterized by a shape factor kappa) and the channel sidewall slope at the waterline (represented by a slope factor gamma) affect the wavelength (lambda) and time period (Ts) of waves generated under resonant external forcing. A quantitative relationship between lambda, Ts, kappa, and gamma is given by our theory which predicts that, under the condition of equal mean water depth and equal mean wave amplitude, lambda and Ts increase with increasing kappa and gamma. To verify the theoretical results, experiments are conducted in two channels of different geometries, namely a rectangular channel with kappa[equivalent]1, gamma=0 and a trapezoidal channel with kappa=1.27, gamma=0.16, to measure the wavelength of free traveling solitary waves and the time period of wave generation by a towed vertical hydrofoil moving with critical speed. The experimental results are found to be in broad agreement with the theoretical predictions

Evolution of long water waves in variable channels

This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data availabl

A nonlinear theory for a flexible unsteady wing

This paper extends the previous studies by Wu [Wu TY (2001) Adv Appl Mech 38:291–353; Wu TY (2005) Advances in engineering mechanics—reflections and outlooks. World Scientific; Wu TY (2006) Struct Control Health Monit 13:553–560] to present a fully nonlinear theory for the evaluation of the unsteady flow generated by a two-dimensional flexible lifting surface moving in an arbitrary manner through an incompressible and inviscid fluid for modeling bird/insect flight and fish swimming. The original physical concept founded by Theodore von Kármán and William R. Sears [von Kármán T, Sears WR (1938) J Aero Sci 5:379–390] in describing the complete vortex system of a wing and its wake in non-uniform motion for their linear theory is adapted and extended to a fully nonlinear consideration. The new theory employs a joint Eulerian and Lagrangian description of the wing motion to establish a fully nonlinear theory for a flexible wing moving with arbitrary variations in wing shape and trajectory, and obtain a fully nonlinear integral equation for the wake vorticity in generalizing Herbert Wagner’s [Wagner H (1925) ZAMM 5:17–35] linear version for an efficient determination of exact solutions in general

Propagation of solitary waves through signicantly curved shallow water channels

Propagation of solitary waves in curved shallow water channels of constant depth and width is investigated by carrying out numerical simulations based on the generalized weakly nonlinear and weakly dispersive Boussinesq model. The objective is to investigate the effects of channel width and bending sharpness on the transmission and reflection of long waves propagating through significantly curved channels. Our numerical results show that, when travelling through narrow channel bends including both smooth and sharp-cornered 90°-bends, a solitary wave is transmitted almost completely with little reflection and scattering. For wide channel bends, we find that, if the bend is rounded and smooth, a solitary wave is still fully transmitted with little backward reflection, but the transmitted wave will no longer preserve the shape of the original solitary wave but will disintegrate into several smaller waves. For solitary waves travelling through wide sharp-cornered 90°-bends, wave reflection is seen to be very significant, and the wider the channel bend, the stronger the reflected wave amplitude. Our numerical results for waves in sharp-cornered 90°-bends revealed a similarity relationship which indicates that the ratios of the transmitted and reflected wave amplitude, excess mass and energy to the original wave amplitude, mass and energy all depend on one single dimensionless parameter, namely the ratio of the channel width b to the effective wavelength [lambda][sub]e. Quantitative results for predicting wave transmission and reflection based on b/[lambda][sub]e are presented

Interview with Theodore Y. Wu

An interview in three sessions, February-March 2002, with Theodore Y. Wu, professor of engineering science, emeritus, in the Division of Engineering and Applied Science. Dr. Wu was born in China and received his BSc from Chiao-Tung University (1946), his MS from Iowa State University (1948), and his PhD from Caltech (1952). In this interview, he recalls his boyhood and tribulations during Japan's invasion of China in World War II, his emigration and matriculation at Iowa State in 1948, and his arrival at Caltech a year later. Recollections of H. S. Tsien, R. A. Millikan, Theodore von Kármán, Julian Cole. Works with Paco Lagerstrom's aeronautics group developing asymptotic perturbation method pioneered by Ludwig Prandtl. Joins faculty as a research fellow in 1952. Interest in hydrodynamics. Origins of the department of engineering science in the mid-1950s by Tsien, Milton Plesset, and Charles De Prima. Interest in bioengineering, beginning in 1960; studies bird flight and fish locomotion. Discusses influence of G. I. Taylor and James Lighthill, and recalls his own work on flagellar and ciliary motion of microorganisms. Caltech's 1974 pioneering symposium on Swimming and Flying in Nature; new field of biofluiddynamics. Recollections of Y. C. (Burt) Fung. Recalls his sabbatical, 1964-65, at University of Hamburg with Georg Weinblum. Joins Advisory Committee for Reactor Safeguards. Recollections of Caltech presidents Lee DuBridge and Marvin L. Goldberger. Visit to China in 1979. Discusses his work, since 1996 retirement, on modeling of water waves; solitons and tsunamis. Concludes with comments on good relations between Chinese and Chinese American scientists and the flood of Chinese students to US for graduate work in late 1970s, after reestablishment of diplomatic relations