Harbor scattering problem

HARBOR RESONANCES IN COASTAL ENGINEERING

In this chapter we consider the application of the half-plane Helmholtz problem described in Chapter III to the computation of harbor resonances in coastal engineering. We consider the problem of computing resonances for the Helmholtz equation in a two-dimensional compactly perturbed half-plane with an impedance boundary condition. One of its main applications corresponds to coastal engineering, acting as a simple model to determine the resonant states of a maritime harbor. In this model the sea is modeled as an infinite half-plane, which is locally perturbed by the presence of the harbor, and the coast is represented by means of an impedance boundary condition. Some references on the harbor oscillations that are responsible for these resonances are Mei (1983), Mei et al. (2005), Herbich (1999), and Panchang & Demirbilek (2001). Resonances are closely related to the phenomena of seiching (in lakes and harbors) and sloshing (in coffee cups and storage tanks), which correspond to standing waves in enclosed or partially enclosed bodies of water. These phenomena have been observed already since very early times. Scientific studies date from Merian (1828) and Poisson (1828–1829), and especially from the observations in the Lake of Geneva by Forel (1895), which began in 1869. A thorough and historical review of the seiching phenomenon in harbors and further references can be found in Miles (1974). Oscillations in harbors, though, were first studied for circular and rectangular closed basins by Lamb (1916). More practical approaches for the same kind of basins, but now connected to the open sea through a narrow mouth, were then implemented respectively by McNown (1952) and Kravtchenko & McNown (1955). But it was the paper of Miles & Munk (1961), the first to treat harbor oscillations by a scattering theory, which really arose the research interest on the subject. Their work, together with the contributions of Le Mehaut ´ e (1961), Ippen & Goda (1963), Raichlen & ´ Ippen (1965), and Raichlen (1966), made the description of harbor oscillations to become fairly close to the experimentally observed one. Theories to deal with arbitrary harbor configurations were available after Hwang & Tuck (1970) and Lee (1969, 1971), who worked with boundary integral equation methods to calculate the oscillation in harbors of constant depth with arbitrary shape. Mei & Chen (1975) developed a hybrid-boundary-element technique to also study harbors of arbitrary geometry. Harbor resonances using the finite element method are likewise computed in Walker & Brebbia (1978). A comprehensive list of references can be found in Yu & Chwang (1994). The mild-slope equation, which describes the combined effects of refraction and diffraction of linear water waves, was first suggested by Eckart (1952) and later rederived by Berkhoff (1972a,b, 1976), Smith & Sprinks (1975), and others, and is now well-accepted as the method for estimating coastal wave conditions. It corresponds to an approximate model developed in the framework of the linear water-wave theory (vid. Section A.10), which assumes waves of small amplitude and a mild slope on the bottom of the sea, i.e., a slowly varying bathymetry. The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as cliffs, beaches, seawalls, and breakwaters. As a result, it describes the variations in wave amplitude, or equivalently wave height. From the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. These quantities, wave amplitude and flow-velocity amplitude, may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects, sediment transport and resulting geomorphology changes of the sea bed and coastline, mean flow fields and mass transfer of dissolved and floating materials. Most often, the mild-slope equation is solved by computers using methods from numerical analysis. The mild-slope equation is a usually expressed in an elliptic form, and it turns into the Helmholtz equation for uniform water depths. Different kinds of mild-slope equations have been derived (Liu & Shi 2008). A detailed survey of the literature on the mild-slope and its related equations is provided by Hsu, Lin, Wen & Ou (2006). Some examinations on the validity of the theory are performed by Booij (1983) and Ehrenmark & Williams (2001). A resonance of a different type is given by the so-called Helmholtz mode when the oscillatory motion inside the harbor is much slower than each of the normal modes (Burrows 1985). It corresponds to the resonant mode with the longest period, where the water appears to move up and down unison throughout the harbor, which seems to have been first studied by Miles & Munk (1961) and which appears to be particularly significant for harbors responding to the energy of a tsunami. We remark that from the mathematical point of view, resonances correspond to poles of the scattering and radiation potentials when they are extended to the complex frequency domain (cf. Poisson & Joly 1991). Harbor resonance should be avoided or minimized in harbor planning and operation to reduce adverse effects such as hazardous navigation and mooring of vessels, deterioration of structures, and sediment deposition or erosion within the harbor. Along rigid, impermeable vertical walls a Neumann boundary condition is used, since there is no flow normal to the surface. However, in general an impedance boundary condition is used along coastlines or permeable structures, to account for a partial reflection of the flow on the boundary (Demirbilek & Panchang 1998). A study of harbor resonances using an approximated Dirichlet-to-Neumann operator and a model based on the Helmholtz equation with an impedance boundary condition on the coast was done by Quaas (2003). In the current chapter this problem is extended to be solved with integral equation techniques, by profiting from the knowledge of the Green’s function developed in Chapter III. This chapter is structured in 4 sections, including this introduction. The harbor scattering problem is presented in Section 6.2. Section 6.3 describes the computation of resonances for the harbor scattering problem by using integral equation techniques and the boundary element method. Finally, in Section 6.4 a benchmark problem based on a rectangular harbor is presented and solved numericall.

Harbor scattering problem

 We are interested in computing the resonances of a maritime harbor, as the one depicted in Figure 6.1 The sea is modeled as the compactly perturbed half-plane Ωe ⊂ R 2 +, where R 2 + = {(x1, x2) ∈ R 2 : x2 > 0} and where the perturbation represents the presence of the harbor. We denote its boundary by Γ, which is regular (e.g., of class C 2 ) and decomposed according to Γ = Γp ∪ Γ∞. The perturbed boundary describing the harbor is denoted by Γp, while Γ∞ denotes the remaining unperturbed boundary of R 2 +, which represents the coast and extends towards infinity on both sides. The unit normal n is taken outwardly oriented of Ωe and the land is represented by the complementary domain Ωc = R 2 \ Ωe.

To describe the propagation of time-harmonic linear water waves over a slowly varying bathymetry we consider for the wave amplitude or surface elevation η the mild-slope equation (Herbich 1999) div(ccg∇η) + k 2 ccgη = 0 in Ωe, (6.1) where k is the wave number, where c and cg denote respectively the local phase and group velocities of a plane progressive wave of angular frequency ω, and where the time convention e −iωt is used.

Computation of resonances

 The resonance problem (6.18) is solved in the same manner as the half-plane impedance Helmholtz problem described in Chapter III, by using integral equation techniques and the boundary element method. The required Green’s function G is expressed in (3.93). If we denote the trace of the solution on Γp by µ = u|Γp , then we have from (3.156) that the solution u admits the integral representation

Benchmark problem

Characteristic frequencies of the rectangle 

As benchmark problem we consider the particular case of a rectangular harbor with a small opening. Resonances for a harbor of this kind are expected whenever the frequency of an incident wave is close to a characteristic frequency of the closed rectangle. To obtain the characteristic frequencies and oscillation modes of such a closed rectangle we have to solve first the problem where we denote the domain encompassed by the rectangle as Ωr and its boundary as Γr. The unit normal n is taken outwardly oriented of Ωr. The rectangle is assumed to be of length a and width b. The eigenfrequencies and eigenstates of the rectangle are wellknown and can be determined analytically by using the method of variable separation. For this purpose we separate u(x) = v(x1)w(x2), (6.37) placing the origin at the lower left corner of the rectangle, as shown in Figure 6.2.

Rectangular harbor problem

We consider now the particular case when the domain Ωe ⊂ R 2 + is taken as a rectangular harbor with a small opening d, such as the domain depicted in Figure 6.3. We take for the rectangle a length a = 800, a width b = 400, and a small opening of size d = 20. To simplify the problem, on Γ∞ we consider an impedance boundary condition with a constant impedance Z∞ = 0.02 and on Γp we take a Neumann boundary condition into account. The rectangular harbor problem can be thus stated as where the outgoing radiation condition is stated in (6.16). The boundary curve Γp is discretized into I = 135 segments with a discretization step h = 40.4959, as illustrated in Figure 6.4. The problem is solved computationally with finite boundary elements of type P1 by using subroutines programmed in Fortran 90, b..