# Integral Equation Method for linear Water Waves

TL;DR: In this article, the Boundary Integral Equation Method (BIEM) is applied to transient water wave problems and the stability limits and frequency distortion of the numerical method are examined and given.

Abstract: The Boundary Integral Equation Method (BIEM) is applied to transient water wave problems. Only two-dimensional linearized waves are considered. As is general practice, free-surface boundary conditions are applied at the equilibrium surface rather than the actual free surface; thus the problems become fixed-boundary problems rather the free-surface problems. For the cases in which fluid domain is unbounded in the horizontal direction, a radition condition is formulated such that waves pass through the computational boundaries without reflection. The stability limits and frequency distortion of the numerical method are examined and given. Numerical results are compared with analytical solutions or experimental data in three examples. Excellent agreement is observed.

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TL;DR: The mild-slope equation is a vertically integrated refraction-diffraction equation, used to predict wave propagation in a region with uneven bottom slopes as mentioned in this paper, which is based on the assumption of a mild bottom slope.

198 citations

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TL;DR: In this article, the source of nonlinear gravity waves in a boundary integral method is reported, and it is demonstrated that virtually any type of two-dimensional wave field can be generated.

137 citations

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TL;DR: The boundary integral equation method (BIEM) was developed as a tool for studying two-dimensional, nonlinear water wave problems, including the phenomena of wave generation, propagation and run-up.

71 citations

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TL;DR: In this article, the interaction of small amplitude water waves with bottom ripples using the Boundary Integral Equation Method (BIEM) was studied using the laboratory data of Davies and Heathershaw.

Abstract: The interaction of small amplitude water waves with a patch of bottom ripples is studied using the Boundary Integral Equation Method (BIEM). Normal and oblique wave incidence is examined for reflection coefficients and the laboratory data of Davies and Heathershaw are compared to the numerical results, with the conclusion that the analytical model of Davies and Heathershaw overestimates the reflection coefficient at resonance. Comparison of BIEM results for oblique incidence is made with an extension of the recent work of Mei showing good agreement, even for large amplitude ripples.

70 citations

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TL;DR: In this paper, a theory for reflected waves was developed, based on an asymptotic analysis for stochastic differential equations when both the horizontal and vertical scales of the bottom variations are comparable to the depth but small compared to a typical wavelength so the shallow water equations cannot be used.

Abstract: The linear water-wave equations for shallow channels with arbitrary rapidly varying bottoms was analysed. A theory for reflected waves was developed, based on an asymptotic analysis for stochastic differential equations when both the horizontal and vertical scales of the bottom variations are comparable to the depth but small compared to a typical wavelength so the shallow water equations cannot be used. The full, linear potential theory was used and the reflection-transmission problem for time-harmonic (monochromatic) and pulse-shaped disturbances was studied. For the monochromatic waves a formula is given for the expected value of the transmission coefficient which depends on depth and on the spectral density of the 0(1) random depth perturbations. For the pulse problem an explicit formula if given for the correlation function of the reflection process. The theory is compared with numerical results produced using the boundary-element method. Several, realisations of the bottom profile are considered and a Gaussian-shaped disturbance is propagated over each topography sampled and the reflected signal for each realisation is recorded. The numerical experiments produced reflected waves whose statistics are in good agreement with the theory.

52 citations