# Using convolution to simulate acoustic dispersion in shallow water

### Background

I'm a marine biologist who's trying to wrap my head around shallow water propagation. I'm interested in how acoustic dispersion (as described by Pekeris' waveguide) alters how sounds affect marine animals. Following this, I'd like to develop a simple method that biologists can use to create their own simulated sounds at a given range, provided they have a high quality recording of the source. The goal would to be use this tool to compare the effects of sounds at different ranges and provide insight into how bottom properties drive the changes in acoustic properties of transient sounds in shallow water.

### My proposal

Lets assume that I can calculate the following required metrics from the simple pekeris dispersion equation given in Bucker, 1963:

• horizontal velocity/arrival time for each frequency in each mode
• the attenuation loss for each mode (based on bottom reflections)

The image below shows the results of the authors calculations applied to the waveform he was examining.

The next step is to convert a source sound into the predicted dispersed waveform. My thoughts were to:

• Apply multiple convolution filters to the source sound.
• Each filter would use a time-reversed template consisting of a sine wave following the frequency change of a given mode. I'm going to call these sine-modes for simplicity (and due to not knowing a better term)
• The amplitude of each sine-mode at a given time point would be related to its $$\frac{\delta \mathit{frequency}}{\delta \mathit{time}}$$, so the asymptotic tails of the modes don't get over represented, and also the attenuation loss specific to each mode.
• Sum the resulting sine-mode filtered source sounds to make the resulting waveform.

The origin of this idea comes from reading about how convolution is used to simulate reverb in recordings.

### The question

On a scale ranging between:

• this is a dumb idea and illustrates a flawed understanding of discreet signal processing and normal mode propagation
• Cool, go for it!

How feasible is using convolution to achieve my goal? And also... am I reinventing the wheel and missing out much easier methods to achieve this.

I'm planning to give this a try later this week... but being self-conscious about my lack of physics/acoustic background... I wanted to get a litmus test on this idea before I go for it.

### Update

...after much reading into shallow water propagation models... I now learned that what I described is very analogous to how actual propagation models work in contexts where people are trying to simulate what a sound is at a receiver.

In normal mode propagation models, it's commonplace to convolve signals with the greens functions solutions (these greens functions essentially define what modes satisfy boundary conditions in a duct, and hence describe acoustically propagating normal modes).

Long story short, what I was suggesting is essentially a amateur attempt to reinvent the wheel. After realizing this, I opted to use publicly available shallow water propagation models and just convolved my sounds with the model solutions.

With respect to answering my question... I would now answer this as:

• I was on the right track with my thought process, but didn't understand that there are already straight forward ways to achieve this in the field of underwater acoustics.
• Cool question! I would be shocked if this hasn't been done before in a sophisticated way, as there are obvious connections between your question, and say, detecting a submarine, and hence there has been funding to look at this. Also, you will get a lot more help if you make your post self contained; i.e. explain Pekeris' waveguide if it's necessary to answer the question. Finally, I would re-write this whole question in terms of the equations of motion for acoustics in a (potentially rough) wave guide. As of right now there's no physics in your discussion. Sep 24, 2018 at 20:13
• Thanks for the feedback! I was conflicted about posting this here vs DSP. I need advice on the signal processing aspect of this question... but the goal of the operation is fundamentally physics based. I restrained from going into detail on the Pekeris waveguide as I didn't want to open up a can of theoretical worms (especially due to the age of the model I'm working with) which may distract from the more practical nature of the question. I guess my problem is... I'm looking for a "applied physics" Q&A, rather than a theoretical. Would u suggest posting this on a different stack-site? Sep 25, 2018 at 11:39