Freeman Gilbert

Long waves on the continental shelf: an experiment to separate trapped and leaky modes

Random fluctuations in sea level, ζ, in the frequency range 0·1-60 cycles per hour were measured along the coast near Oceanside, California, where the coastline and bottom contours are fairly straight and parallel for 30 km. The two-dimensional covariancen

Scattering of gravity waves by a circular dock

R(eta, tau) = langle zeta (y,t) zeta (y + eta, t+ tau) rangle

Scattering of Impulsive Elastic Waves by a Rigid Cylinder

was computed for points separated by various distances η along the coast. The Fourier transformn

First Motion Methods in Theoretical Seismology

S(f,n) = int int R(eta, tau)exp [2pi i (n eta + f tau)]d eta d tau

Scattering of Impulsive Elastic Waves by a Smooth Convex Cylinder

gives the contribution towards the ‘energy’

THE DIRECTIVITY PROBLEM FOR A BURIED LINE SOURCE

langle zeta ^2 rangle

Seismic scattering from topographic irregularities

per unit temporal frequency f per unit spacial frequency (long-shore component) n . It is found that most of the energy is confined to a few narrow bands in ( f, n ) space, and these observed bands correspond very closely to the gravest trapped modes (or edge waves) computed for the actual depth profile. The bands are 0·02 cycles per km wide, which equals the theoretical resolution of the 30 km array. Very roughly S(f,n) ≈ S(f, -n) , corresponding to equal partition of energy between waves travelling up and down the coast. Theory predicts ‘Coriolis splitting’ between the lines f ± (n) corresponding to these oppositely travelling waves, but this effect is below the limit of detection. The principal conclusion is that most of the low-frequency wave energy is trapped.

Radiation from a strike-slip fault

The scattering of a gravity wave of wave number k by a circular dock of radius a and draft d – h in water of depth d is calculated through a variational approximation. The total and differential scattering cross-sections, the peripheral displacement, and the lateral force on the dock are presented as functions of ka with d/a and h/d as parameters and compared with the classical results for a circular cylinder ( h = 0). A pronounced resonance is found near ka = 2 for certain values of d/a and h/d.

Free oscillations of the Earth: 1. Toroidal oscillations

The exact solution to the problem of the scattering of compressional elastic waves from a line source by a rigid, infinitely dense cylinder imbedded in an isotropic, homogeneous, perfectly elastic medium is obtained in integral form. The integrals are evaluated asymptotically obtain the motions on the wave fronts. In the illuminated zone the saddle point method of integration yields the geometrical optics approximation to the reflected field. In the shadow zone the diffracted field is obtained by evaluating the integrals by the method of Dougall and Watson. In the case of an incident P wave the observed events in the illuminated zone are (1) direct P, (2) reflected P, and (3) reflected S. In the shadow zone the observed events are (1) diffracted P and (2) diffracted S. Both diffracted wave fronts travel around the cylinder with the velocity of P waves.

Low Frequency Discriminants

Techniques are presented for approximating integral solutions to some problems in theoretical seismology. The approximations obtained are the first terms of asymptotic series in powers of t−t0, where t is the time and t0 is an arrival time. The approximations are obtained by evaluating the integral form of the Laplace transform of the time solution by the saddle point method or a variation of it. To the resulting expression is applied a Tauberian limit theorem from which is obtained the time solution. Two examples are given which illustrate some of the specific techniques for the use of the method.