SEISMOLOGY: WAVES Phase and Group Velocities of a Wave Packet The velocity of a wave can be defined in many different ways, partly because there are different kinds of waves, and partly because we can focus on different aspects or components of any given wave. The wave function depends on both time, t, and position, x, i.e.: A = A(x,t), where A is the amplitude. At any fixed location on the x axis the function varies sinusoidally with time.
fixed position
1
A 0
-1 0 3 6 t
The angular frequency, ω, of a wave is the number of radians (or cycles) per unit of time at a fixed position. Similarly, at any fixed instant of time, the function varies sinusoidally along the horizontal axis:
fixed time
1
A 0
-1 0 3 6 x
The wave-number, k, is the number of radians (or cycles) per unit distance at a fixed time. A pure traveling wave is a function of ω and k as follows:
A = A0 cos(ωt kx), − where A0 is the maximum amplitude.
A wave packet is formed from the superposition of several such waves, with different A, ω, and k:
A = X An cos(ωnt knx) n − Here’s the result of superposing two such waves with A1 = √A0 and k1 =1.2k0 (or ω1 =1.2ω0): 8 4
A 0 -4 -8 0 8 16 x or t The envelope of the wave packet (dashed line) is also a wave. 8 4
A 0 -4 -8 0 8 16 x or t
The speed at which a given phase propagates may not coincide with the speed of the envelope (see animations at: www.csupomona.edu/ ajm/materials/animations/packets.html)
The group velocity is the velocity with which the overall shape of the waves amplitude, or the envelope of the wave packet, propagates through space. The phase velocity is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity. Question Is the P-wave speed a phase or a group velocity? P-wave and S-wave velocities are phase velocities. Dispersive Waves A wave packet is said to be dispersive if different frequencies travel at different speed. Surface waves are dispersive:
Question Why do the longer periods travel faster than shorter periods? Surface waves are dispersive because waves with longer wavelengths (lower frequency) traveling deeper than short wavelengths (higher frequency). Since in general seismic speed increases with depth, longer wavelengths travel faster than shorter wavelength. Calculated dispersion curves for Love and Rayleigh waves (from Shearer’s text book): Fresnel Zone The Fresnel zone is a circular region surrounding the ray path with a diameter A-A’.
The size of the Fresnel zone can be calculated to help interpreters determine the minimum size feature that can be resolved (see equation 4.126 in Fowler’s book). Subsurface features smaller than the Fresnel zone usually cannot be detected using seismic waves. Consider a direct wave traveling in a straight line from the source (TX) to the receiver (RX). Other waves that travel slightly off the direct path may still arrive to the same receiver if they bump into something. If they are out of phase with the direct wave they will have a phase cancelling effect. Each Fresnel zone is an ellipsoidal shape as shown below. In zone 1 the signal will be 0 to 90 degrees out of phase in zone 2, 90 to 270 degrees in zone 3, 270 to 450 degrees and so on.