Rayleigh-wave Coupling to Atmospheric

THE THEORY of dispersive Rayleigh waves coupled to atmospheric compressional waves is derived for the case of a solid surface layer. Numerical computation of phase and group velocity curves indicates that an additional branch may be introduced to the dispersion curves as a result of air coupling. Amplitudes of waves propagated according to the various branches are briefly discussed. INTRODUCTION RECENT experimental and theoretical work has indicated that coupling of surface waves to the atmosphere is not an uncommon phenomenon. Press, Crary, Oliver and Katz1 identified air-coupled flexural waves on a floating ice sheet. Press and Ewing2 made use of Lamb's elementary theory3 to show that coupling between air waves and surface waves of all types is appreciable when the phase velocity of the surface wave is very close to the speed of sound in air. These authors also developed the exact theory for air-coupled flexural waves in a floating ice sheet and presented numerical computation of dispersion and relative amplitudes. An experimental investigation of air coupling to Rayleigh waves has recently been completed with the cooperation of the Field Research Laboratory of Magnolia Petroleum Company.4 Coupling of the atmosphere with hydrodynamic gravity waves is now being investigated. In this paper, the theory for air-coupled Rayleigh waves originating in an impulsive point source in the air, is outlined for the case of a solid surface layer. Numerical computation of dispersion is made for a particular case, the results being applicable for a source in the air or in the ground. THEORY The existence of air-coupled Rayleigh waves was shown by Bateman5 on the assumpof a homogeneous earth. Lee investigated the propagation of Rayleigh waves in two solid layers. 6 We now consider Rayleigh Waves coupled to atmospheric compressional waves for the case of a three-layered space formed by air in the half space z = 0 to z = oo, an "upper" solid layer (1) between the planes z = 0 and z = Hand a "lower" layer (2) between z = Hand z = oo. We assume a point source in the air at a distanced * The research reported in this paper has been made possible through support and sponsorship by the Geophysical Research Division of the Cambridge Air Force Research Center. Manuscript received for publication March 9, 1951. 1 Frank Press, A. P. Crary, Jack Oliver, and Sam Katz, "Air-coupled Flexural Waves in Floating Ice," Trans. Am. Geophys. Union, 32:166-172 (1951). 2 Frank Press and Maurice Ewing, "Theory of Air-coupled Flexural Waves," Jour. Appl. Physics, 22:892-899 (1951). 3 H. Lamb, "On Waves Due to a Traveling Disturbance, with an Application to Waves in Superposed Fluids," Phil. Mag., 31:387 (1951). 4 Frank Press and Maurice Ewing, "Ground-Roll Coupling to Atmospheric Compressional Waves," Geophysics, 16:416-430 (1951). 5 H. Bateman, "Rayleigh Waves," Proc. Nat. Acad. Sci., 24:315-320 (1938). 6 A. W. Lee, "The Effect of Geological Structure upon Microseismic Disturbance,'' Mon. Not. Roy. Astron. Soc., Geophys. Suppl., 3:83-105 (1932); idem, "Further Investigation of the Effect of Geologic Structure on Microseismic Disturbance,'' ibid., 3:238-252 (1934). [ 135] 136 BULLETIN OF THE f>EISMOLOGICAL SOCIETY OF AMERICA above the plane z = 0. We denote by p0, Pi, p2, ao, a1, a2, f31, (32 the densities and velocities of dilatational and shear waves in the air and the elastic media 1 and 2. Applying the well-known method of Lamb, 7 we represent the horizontal component of displacement, q, and the vertical component, w, by the formulas