Integral Equation Coupled Mode Approach Applied to Internal Wave Problems

Abstract

A two-way coupled mode approach based on an integral equation formalism is applied to sound propagation through internal wave fields defined at the 1999 Shallow Water Acoustics Modeling Workshop. Solutions of the coupled equations are obtained using a powerful approach originally introduced in nuclear theory and also used to solve simple nonseparable problems in underwater acoustics. The basic integral equations are slightly modified to permit a Lanczos expansion to form a solution. The solution of the original set of integral equations is then easily recovered from the solution of the modified equations. Two important aspects of the integral equation method are revealed. First, the Lanczos expansion converges faster than a Born expansion of the original integral equations. Second, even when the Born expansion diverges due to strong mode coupling, the Lanczos expansion converges. It is shown that the internal wave problems examined are essentially one-way propagation problems because one observes good agreement between the coupled mode solutions and those provided by an energy-conserving parabolic equation algorithm. In the Workshop examples, at both 25 and 250 Hz, significantly greater coupling between modes occurs in the linear internal wave field case than the nonlinear soliton case.

Modeling Fluctuations in Depth-Integrated Acoustic Intensity Induced by Internal Waves Along a 2-D Track

“…For this work, the range acoustic model (RAM) [28]- [30] was selected for its computational efficiency. RAM was first benchmarked against the integral equation coupled mode (IECM) model [31], [32] to verify that it was accurately computing the acoustic field in the presence of nonlinear IWs. After this verification process, RAM was used to compute the acoustic field every 15 s during the on-time of the acoustic source over a period of 19 days of the SW06 experiment.…”

Seabed rough surface scattering described by 2-way coupled mode approach

“…1 are solved for R n ∞ (r) and R n 0 (r) with an approach first developed by Udagawa and Kim for problems in nuclear structure [4][5] and then for applications in underwater acoustics. [6][7] …”

Broadband modeling of downslope propagation in a penetrable wedge

“…3 The coupling operator is denoted C mn ͑r͒. This is a differential operator and can be conveniently decomposed as…”

“…͑7͒ was solved using a method developed by Udagawa and Kim that transforms this equation into an auxiliary equation that can be conveniently solved with the Lanczos method. 31 In this approach, an expansion with unknown coefficients of orthonormal vector-valued functions is used to solve for the vector of mode amplitude functions ͑see Knobles et al 3 for details͒. The orthonormal vectors are determined by an iterative Gram-Schmidt algorithm.…”

“…Like finite difference schemes, two-way coupled mode approaches introduce no approximations to the PDE apart from those necessary to discretize the problem ͑which may include neglecting a continuum in the form of a branch line integral͒. Coupled mode schemes have been used to study wave propagation in range-dependent environments, not only in underwater acoustics, [1][2][3][4][5] but in seismology 6 and electromagnetics 7-9 as well.…”