In this paper, the acoustic scattering by an obstacle across a wide frequency range of sound waves is investigated on the basis of the Helmholtz integral formulation. To overcome the nonuniqueness difficulties, the methods proposed by Burton [NPL Report NAC 30 (Jan 1973)] and by Burton and Miller [Proc. R. Soc. London, Ser. A 323, 201–210 (1971)] are adopted for the Dirichlet and Neumann problems, respectively. The aim of this paper is twofold. The first is to bring together completely regular formulations of the Helmholtz integral equation and its normal derivative. The second is to extend these formulations to treat the higher-frequency problems. The weakly singular integrals are regularized by subtracting out one term and adding it back. Depending on the problem concerned, the additional integral can finally be expressed in an explicit form or results in solving a surface source distribution of the equipotential body. The hypersingular kernels are regularized by the method of using some properties of the associated Laplace equation, originally proposed by Chien et al. [J. Acoust. Soc. Am. 88, 918–937 (1990)]. The completely regularized integral equations are amenable to computation by direct use of the standard quadrature methods. To study the acoustic scattering due to higher-frequency waves, Filon’s quadrature method [Proc. R. Soc. Edinburgh 49, 38–47 (1928)] is extended to treat the rapidly oscillatory integrands. Numerical examples consist of acoustic scattering from a hard or soft sphere of radius a across a wide spectrum of wave numbers ka=π–20π. Comparisons of the numerical results with the exact solutions demonstrate the validity and efficiency of the implementation.

SUMMARYThe paper presents the non-singular forms of Green's formula and its normal derivative of exterior problems for three-dimensional Laplace's equation. The main advantage of these modi"ed formulations is that they are amenable to solution by directly using quadrature formulas. Thus, the conventional boundary element approximation, which locally regularizes the singularities in each element, is not required. The weak singularities are treated by both the Gauss #ux theorem and the property of the associated equipotential body. The hypersingularities are treated by further using the boundary formula for the associated interior problems. The e$cacy of the modi"ed formulations is examined by a sphere, in an in"nite domain, subject to Neumann and Dirichlet conditions, respectively.The modi"ed integral formulations are further applied to a practical problem, i.e. surface-wave}body interactions. Using the conventional boundary integral equation formulation is known to break down at certain discrete frequencies for such a problem. Removing the &irregular' frequencies is performed by linearly combining the standard integral equation with its normal derivative. Computations are presented of the added-mass and damping coe$cients and wave exciting forces on a #oating hemisphere. Comparing the numerical results with that by other approaches demonstrates the e!ectiveness of the method.

This work formulates the singularity-free integral equations to study 2-D acoustic scattering problems. To avert the nonuniqueness difficulties, Burton’s and Burton and Miller’s methods are employed to solve the Dirichlet and Neumann problems, respectively. The surface Helmholtz integral equations and their normal derivative equations in bounded form are derived. The weakly singular integrals are desingularized by subtracting a term from the integrand and adding it back with an exact value. Depending on the relevant problem, the additional integral can finally be either expressed in an explicit form or transformed to form a surface source distribution of the related equipotential body. The hypersingular kernel is desingularized further using some properties of an interior Laplace problem. The new formulations are advantageous in that they can be computed by directly using standard quadrature formulas. Also discussed is the Γ-contour, a unique feature of 2-D problems. Instead of dividing the boundary surface into several small elements, a parametric representation of a 2-D boundary curve is further proposed to facilitate a global numerical implementation. Calculations consist of acoustic scattering by a hard and a soft circular cylinder, respectively. Comparing the numerical results with the exact solutions demonstrates the proposed method’s effectiveness.

The free-surface flow generated by an impulsively accelerating, surface-piercing, vertical plate has been studied experimentally in an open channel of constant depth. The flat vertical plate is fixed on a towing carriage that is set off by suddenly dropping a weight bucket through a connecting steel cable in a pulley system. The free-surface profile in front of the plate and the pressure distribution on the plate surface are measured for three different accelerations of the plate. A capacitance-type wave gauge is used to measure the variations of the water surface, while a variable reluctance pressure transducer is used to measure the pressure on the plate surface. The acceleration of the plate is obtained by means of an accelerometer. All response voltage outputs are recorded on an IBM PC-XT personal computer with a data-acquisition electrical board. Experimental measurements are compared with the numerical, viscous-flow results of Yang and Chwang (IIHR Report No. 332; Iowa Institute of Hydraulic Research, The University of Iowa, 1989) and the analytical, inviscid-flow solution of Chwang [Phys. Fluids 26, 383 (1983)]. The agreement of the free-surface profile and the pressure distribution between the numerical results and the present experimental measurements is fairly good. However, the inviscid-flow solution overpredicts the wave amplitude and the pressure distribution on the plate. In the physical experiments, the water surface is observed to rise in front of the vertical plate where the potential-flow theory becomes singular.

SUMMARYThis paper presents the non-singular forms, in a global sense, of two-dimensional Green's boundary formula and its normal derivative. The main advantage of the modiÿed formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element-free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier-Legendre series, together with transforming the integration interval [a; b] to [−1; 1]; the series coe cients are thus to be determined. The hypersingular integral, interpreted in the Hadamard ÿnite-part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands deÿned explicitly when a source point coincides with a ÿeld point. The e ectiveness of the modiÿed formulations is examined by an elliptic cylinder subject to prescribed boundary conditions.The regularization is further applied to acoustic scattering problems. The well-known Burton-Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non-uniqueness problem. A general non-singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made.