Transmission, Reflection, and Guiding of an Exponential Pulse by a Steel Plate in Water

The problem of the action of an underwater explosion—a pulse of the form pmaxe−t/τ—on an elastic plate in water can be divided into two parts. In the first, the plate is considered as an infinite slab with the explosion wave as an incident plane wave transient. In the second, the plate is considered as a wave guide, with the explosion wave setting the initial conditions at the edge of the plate. The first is an inhomogeneous boundary value problem, the second a homogeneous one. The inhomogeneous case has been solved in the literature, for continuous waves, and the results of the theory are in qualitative agreement with the experiments on transients. The homogeneous problem forms the principal part of this paper. The phase velocity curves, as functions of frequency, for the principal normal modes of a plate in water, are closely related to the corresponding curves for a plate in a vacuum. The effect of the water on the principal symmetric mode of the plate is not to change the real part of the phase velocity from its value for a plate in a vacuum, but to add a small imaginary component, or attenuation due to radiation loss into the water. The principal antisymmetric mode is slightly attenuated at high frequencies, and strongly attenuated at low frequencies. The cut-off comes at that frequency where the phase velocity along the plate equals the velocity of sound in water. In addition to these two modes, which correspond closely to the principal modes for a plate in a vacuum, the presence of the water introduces two additional modes, one symmetric and one antisymmetric. At high frequencies, their phase velocities are very close to, but a constant fraction of, the velocity of sound in water. At low frequencies, the phase velocity of the symmetric mode approaches the velocity of sound in water, the phase velocity of the antisymmetric mode approaches zero. Both these modes are unattenuated, i.e., their phase velocities are all real. The problem of the propagation of a transient in a wave guide presents the interesting mathematical problem, as yet unsolved rigorously, of the contour integral of a function of a complex variable defined implicitly. Brief comparison of the theory with experiment shows that the pure real antisymmetric mode accounts for most of the properties of the precursor. This is an oscillation which precedes the explosion wave along the plate, at a velocity slightly greater than that of sound in water, but less than that of rotational or dilational waves in the plate.