Date of Award

Fall 1980

Project Type


Program or Major

Engineering (Theoretical and Applied Mechanics)

Degree Name

Doctor of Philosophy


Sound propagation in shallow water is modelled using the Green's function formalism and normal mode techniques. The water layer is bounded above by a pressure-released condition and below by a halfspace of viscoelastic solid media. Nine different marine sediments are investigated by incorporating their published measured parameters into the bottom boundary condition utilizing the definition of specific acoustic impedance. A depth-dependent sound speed is treated in the ideal liquid layer.

In order to determine the most convenient mathematical representation for this shallow water model an investigation of the normal mode spectra of sound waves in viscous fluids and viscoelastic solids is first undertaken. Due to the non-self-adjoint property of the differential operator describing viscous media, complex-valued propagation constants (eigenvalues) were encountered and had to be dealt with in an appropriate manner.

A tensor Green's function is necessary to completely define vector fields. The velocity vector describes the fluid medium and the displacement vector the solid medium. It is found that both fields are composed of transverse and longitudinal polarizations due to shear and compressional effects, respectively. The (omega)- and k-poles of the inverse of the governing differential equation in the Fourier wavenumber-frequency domain prescribe the character of the sound waves in the wavenumber-time and space-frequency domains, respectively. It is shown that in each case the real- and imaginary- components of these complexed-value poles contribute independently to the propagation and attenuation of the acoustic energy, respectively. The longitudinal polarization of the viscous fluid and both polarizations of the viscoelastic solid are the transport modes of sound waves, while on the other hand the transverse polarization of the viscous fluid diffuses acoustic energy in both space and time.

The normal mode representations are constructed by a two-dimensional transform of a one-dimensional characteristic Green's function. The transforms are determined by the integral expressions of the Dirac delta function for the particular coordinate direction of the appropriate coordinate system. The delta function may be written as a summation of normal modes and hence is a convenient method for the treatment of boundary-value problems.

Using the appropriate normal mode representation numerically generated plots of transmission loss are produced from the shallow water model developed. The input parameters to the model are the conditions of an actual data-collection site in the Baltic Sea (1974). The computer simulation agrees qualitatively with the measured results at the test site and demonstrates the fundamental importance of the viscoelastic boundary on the attenuation of sound waves in shallow water.