A generalized inner product-based wave scattering from an underwater source in a compressible ocean

Abstract

Motivated by applications to underwater explosions and volcanic eruptions, this paper considers the evolution of an initial pressure disturbance in the ocean, including effects due to the dynamic and static compression of water and the free surface. In order to solve the equations of motion of a linear compressible ocean, a special inner product is introduced, which allows us to apply self-adjoint operator theory. What results is a Hilbert space in which the acoustic-gravity modes are orthogonal in the generalised sense. This allows the time-domain evolution of the free surface and subsurface pressure field resulting from an initial disturbance to be calculated. Our simulations show initial radial propagation of the pressure pulse and subsequent reflection from the water surface and the rigid ocean floor, eventually leading to horizontal propagation away from the source point. The solutions with and without the inclusion of the static compression are compared, and the effect of static compression is shown to be small but not negligible.

Figures (6)

• Figure 1: Schematic diagram of the physical problem showing wave propagation in a compressible ocean (without including static compression) due to an underwater explosion (red star). Pressure waves generated by the explosion (red dashed lines) are reflected off the sea bed and free surface (blue dashed lines) and induce propagating waves on the fluid surface (solid red lines)
• Figure 2: Compression waves caused by an initial Gaussian pressure distribution centred at $(0,-2000)$.
• Figure 3: Compression waves caused by an initial Gaussian pressure distribution centred at $(0,-2000)$, including the effects of static compression. In each panel, the line graph represents the free surface displacement, and the colour plot represents the hydrodynamic pressure distribution below the surface.
• Figure 4: A plot of the pressure difference between Figure \ref{['fig:underwater_gaussian']}(a) and Figure \ref{['fig:underwater_gaussian_static']}(a) as a percentage of that maximum magnitude of pressure at that time ($t=1$), where a positive difference indicates that the pressure solution including static compression is greater than the pressure solution without static compression. The bottom panel provides a cross-section of the pressure difference at $x=0$.
• Figure 5: Compression waves caused by an initial Gaussian pressure distribution centred at $(0,0)$, including the effects of static compression.