Normal Mode Analysis of Three-Dimensional Propagation Over a Small-Slope Cosine Shaped Hill

Abstract

Three-dimensional propagation over an infinitely long cosine shaped hill is examined using an approximate normal mode/parabolic equation hybrid model that includes mode coupling in the out-going direction. The slope of the hill is relatively shallow, but it is significant enough to produce both mode-coupling and horizontal refraction effects. In the first part of the paper, the modeling approach is described, and the solution is compared to results obtained with a finite element method to evaluate the accuracy of the solution in light of assumptions made in formulating the model. Then the calculated transmission loss is interpreted in terms of a modal decomposition of the field, and the solution from the hybrid model is compared to adiabatic and N × 2D solutions to assess the relative importance of horizontal refraction and mode-coupling effects. An analysis using a horizontal ray trace is presented to explain differences in the modal interference pattern observed between the 3D and N × 2D solutions. The detailed discussion provides a thorough explanation of the observed 3D propagation effects and demonstrates the usefulness of the approximate normal mode/parabolic equation hybrid model as a tool to understand measured transmission loss in complex environments.

A coupled mode model for omnidirectional three-dimensional underwater sound propagation

“…As the water depth decreases, a mode which may have been trapped in the water can become strongly bottom interacting and no longer integrable as it transitions into the leaky mode domain. In this paper, the issue is addressed by introducing a small gradient in the bottom attenuation and sound speed, 3,19 which will force the modes to decay exponentially after some amplitude growth in the bottom. However, even this approach has its limitations, as the gradient must remain small (suggested 3 less than about 0.18 dB/wavelength and 5.5 m/s/wavelength) to reduce error in the acoustic field in the water.…”

“…Furthermore, in order to handle the transition from deeper water to shallow regions, a small attenuation and sound speed gradient is inserted into the bottom following the method of Westwood and Koch. 3 This forces leaky modes to eventually decay and become integrable, 3,19 as without the gradient the amplitude of the leaky vertical modes grows exponentially. By including some of the leaky modes, the leading order effects of coupling into lossy modes at the slopes are included.…”

Modeling and Forecasting Ocean Acoustic Conditions

“…It must be emphasized that with accurate knowledge of the sound speed in any 3D ocean volume, 2D modeling can be done with a choice of established codes. Fewer implementations are available for 3D, among them parabolic equation (PE) one-way codes (Lin et al 2013a;Heaney and Campbell 2016), coupled normal mode codes (Shmelev et al 2014;Ballard et al 2015), adiabatic (uncoupled) normal mode codes, and ray-based models such as Gaussian beams.…”

Underwater Sound Propagation Modeling in a Complex Shallow Water Environment

“…The application of normal mode theory in underwater acoustics goes back to the 1940s (Pekeris, 1948). Since the early 1990s several 3D underwater propagation models based on the normal mode theory have been proposed (e.g., Porter, 1992;Luo and Schmidt, 2009;Ballard et al, 2015;DeCourcy and Duda, 2020). In normal mode models, the 2D horizontal refraction equation can be handled by several different techniques, including rays (e.g., Weinberg and Burridge, 1974), Gaussian beams (Porter, 1992), and also PEs (e.g., Petrov et al, 2020).…”

“…Three-dimensional (3D) effects can profoundly influence underwater sound propagation and hence soundscape at different scales in the ocean (e.g., Duda et al, 2011;Ballard et al, 2015;Heaney and Campbell, 2016;Reilly et al, 2016;Oliveira and Lin, 2019;Reeder and Lin, 2019). In the particular case of coastal seas, a range of physical oceanographic and geological features can cause horizontal reflection, refraction, and diffraction of sound.…”