Phonological has to do with sounds and acoustics which mathematically is generally modeled using a form of Helmholtz equation to describe its associated wave propagation, reflection, and refraction. Therefore, using something equivalent to Helmholtz equation will help in designing the optimum acoustic medium for the sound to propagate through giving the best sound, best transmission, etc..

Thomas Lynn Warren

That's an interesting point. When it comes to designing the optimum acoustic medium for sound propagation, are there any specific applications where this approach is commonly used? Additionally, are there any alternative mathematical models or equations that are also utilized in studying and optimizing acoustic mediums apart from the Helmholtz equation?

For acoustics you use a scalar velocity potential function, then derive a boundary integral representation giving a BIE and solve it using boundary elements. Myself, I did elastodynamic scattering for ultra sonic QNDE (see Achenbach) it works real good and the BIE is in my opinion most rigorous an you get displacements you can use in Hook's law etc. you can get everything from solving the BIE (see my current spotlight paper on this topic). You can also use a truncated multipole expansion in terms of Lagender and spherical Hankel functions then employ the addition theorem to each scatterer; however, these use approximate wave functions are not as accurate as directly solving Helmholtz eq in a BIE form using BEM for integration. See my comparison with a Born approximation, its just not quite as accurate, but less computational overhead.

Thomas Lynn Warren

Thank you for this information on how you tackle problems using two different methods. The first involves using a scalar velocity potential function and a boundary integral representation, leading to a boundary integral equation (BIE) that can be solved using boundary elements. The second approach uses a truncated multipole expansion with Laguerre and Hankel functions, offering reduced computational overhead.

Thank you for providing me with your work, particularly the comparison you made with the Born approximation and the insights gained from that comparison.

I have been doing BIE since 1988 for ultrasonic for QNDE. I did my MS with Achenbach.

Thomas Lynn Warren

That's impressive! Your extensive experience makes your insights in the field of phonological theories highly valuable.