Rayleigh Wave Spectra and Group Velocity Minima, and the resonance of P waves in layered structures

Abstract

Summary A connection is established between the group velocity of Rayleigh waves, the spectral amplitudes of surface waves generated by a source, and the resonance of vertically travelling P waves. It implies that a minimum in a group velocity curve is reflected in the spectral amplitudes as a maximum. That this is so, appears to have been first noticed by Longuet-Higgins in a study of microseisms. Also when a sharp impedance contrast occurs in a plane-layered model of the crust, the group velocity minimum in the fundamental mode occurs close to a period equal to four times the travel time of P-waves from the surface to the interface. More than one such contrast gives rise in general to more than one minimum. Similar relations hold for the higher modes. In his paper on the generation of niicroseisn1s, Longuet-Higgins (1950) derives expressions for the amplitudes of Rayleigh waves generated by a point source of harmonic waves acting in fluid layer overlying a solid half space. The amplitude spectrum of the fundamental mode has a distinct maximum when the depth of the fluid layer is about 0.27 times the wavelength of a compressional wave in the fluid; that is, when the period is just a little less than four times the time taken by a P-wave to traverse the layer vertically. This effect is attributed to resonance. Longuet-Higgins also notes that the group velocity curve of the surface waves has a minimum very close to the same value of the period. It appears, therefore, that these three things (the maximum in the amplitude spectrum, the minimum in the group velocity, the resonance of vertically travelling P-waves) may be related. This paper examines this possible relationship. 2. The group velocity of Rayleigh waves (a) Fluid layer over a half space Consider, first, the simplest case, that of waves guided along a fluid layer of depth H with plane surfaces, the upper one free, the lower rigid. The waves with plane wavefront and harmonic time dependence have a dispersion relation for the fundamental mode given by