A Class of Low-Frequency Modes in Laterally Homogeneous Fluid-Solid Media

Abstract

By a fundamental wave mode in a laterally homogeneous fluid-solid medium, we mean a propagating mode that continues to exist as a propagating mode down to arbitrarily low frequencies. In underwater acoustics with a fluid medium having a pressure-release surface, there is no fundamental mode since each mode has a lower cutoff frequency. In plate acoustics there are two fundamental Lamb modes, a symmetric one (the quasi-longitudinal wave) and an antisymmetric one (the bending wave). There is also a fundamental mode for a fluid plate with rigid boundaries. In seismology, interface waves of Rayleigh-, Scholte-, and Stoneley-type are well known for certain media with one or two homogeneous half-spaces.The purpose of this paper is to give a unified treatment of a class of modes that is related to the fundamental modes. Specifically, we consider low-frequency P-SV modes whose complex phase velocities do not approach zero as fast as the frequency when the frequency is decreased towards zero. Utilizing the analyticity of the dispersion function, a complete characterization is given of these modes and the linearly visco-elastic fluid-solid media in which they occur. All the mentioned particular waves appear in a general setting, and we give asymptotic low-frequency expressions for the modal slownesses and mode forms. As the frequency approaches zero, the phase velocities of these waves will either tend to a nonzero constant or approach zero like the square root of the frequency. In addition, slow modes appear whose phase velocities approach zero like three other powers of the frequency: 1/3, 3/5, and 2/3. (The power 1/5 is possible as well, but only for certain leaky modes.) Concerning the power 1/3, we make a correction to a previous study by Ferrazzini and Aki. The powers 3/5 and 2/3 appear for certain bending-type waves that are slower than the classical bending wave. A less-known type of interface wave also emerges from the analysis. In the nonleaky case, we give precise conditions for its existence and uniqueness. The results can be directly extended to interface conditions with slip. In an extreme case with vanishing specific normal stiffness, we get yet another kind of interface wave.

Mode Structure for Fluid–solid Media as Derived by Low-Frequency Asymptotics

“…The structure for the class of low-frequency modes such that lim inf S " k( )"(R was determined in references [16,17]. It was shown that lim S k( )"q always exists for such a mode.…”

“…Fluid}solid media with an arbitrary number of #uid and solid regions, and with a general variation with depth for the velocity and density parameters, were considered by Ivansson [16,17]. By an asymptotic study of the dispersion function, all normal modes for which the horizontal wavenumber tends to zero with the frequency could be determined explicitly.…”