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# Fundamentals of Seismic Wave Propagation

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c2 < c1. 2.1.2.2 The critical angle and total reflection If c2 > c1 and θinc = θcrit = sin−1(c1/c2), then pz for the transmitted ray is zero. This is known as the critical angle. If θinc > θcrit, the z slowness component for the transmitted ray becomes imaginary, (ptrans)z = − ( 1 c22 − p2x )1/2 = − ( 1 c22 − sin 2 θinc c21 )1/2 = − i sgn(ω) ( sin2 θinc c21 − 1 c22 )1/2 . (2.1.9) Note that the sign of the imaginary root is taken positive imaginary, i.e. Im(ωpz) < 0, so the transmitted wave decays in the negative z direction (as it must do physi- cally). Thus eiωptrans·x = eiωpx x eiωpz z = eiωpx x e|ωpz |z → 0, (2.1.10) as z → − ∞. In the wavefront diagram (Figure 2.3), the wavefronts for the transmitted wave are perpendicular to the interface (from the x dependence in expression (2.1.10)), prefl pinc ptrans interface normal preflpinc ptrans (a) (b) Fig. 2.3. Plane waves incident and totally reflected at a plane interface between homogeneous half-spaces. The transmitted wave is evanescent. Part (a) shows the wavefronts and (b) the slowness vectors. The dashed lines are constant amplitude lines for the evanescent transmitted waves. 2.1 Plane waves 11 and lines of constant amplitude are parallel to the interface (from the z dependence in expression (2.1.10)). Such a wave is known as an evanescent wave (we avoid the term inhomogeneous wave as the word inhomogeneous is overloaded in the subject of wave propagation, being used to describe different features in waves, media and differential equations). This is in contrast to a travelling wave, as con- sidered so far, where the constant amplitude and phase surfaces coincide. In the evanescent wave, the propagation is parallel to the interface and the amplitude de- cays exponentially away from the interface. No energy is transmitted away from the interface in the second medium, so we describe the reflected wave as being a total reflection. This contrasts with the situation when θinc < θcrit or c2 < c1, when the transmission propagates energy away from the interface and we describe the reflected wave as a partial reflection. Note that because of the signs used in the exponent of expression (2.1.3), we have been able to identify the positive real or imaginary root for pz with propagation in the positive z direction (travelling or evanescent), and negative with negative propagation. Sometimes in the literature, different signs have been taken, making the identification confusingly mixed. We should also mention that Figure 2.3 has been drawn assuming no phase shift be- tween the incident and reflected or transmitted waves. In fact, when total reflection occurs, there is normally a phase shift (e.g. Section 6.3.1, equation (6.3.11)), and the wavefronts should be shifted accordingly. 2.1.3 Time-domain solutions As we are only considering non-dispersive waves (the velocity, c, is independent of frequency, ω), it is straightforward to write the solution in the time domain. All frequencies can be combined to describe the propagation of an impulsive signal. In a homogeneous medium, a solution is φ = f ( t − p · (x − x0) ) , (2.1.11) which satisfies the wave equation (2.1.1) for any function f . For ∇φ = −p f ′ and ∇2φ = |p|2 f ′′ (2.1.12) ∂φ ∂t = f ′ and ∂ 2φ ∂t2 = f ′′, (2.1.13) so provided |p| = 1/c, equation (2.1.1) is satisfied. (The prime indicates ordinary differentiation of the function with respect to its argument. The arbitrary constant p · x0 in the argument of the function f (t) is introduced as it is convenient to measure phase with respect to some origin, x0.) 12 Basic wave propagation φ φ x x + �x �t t t t + �t �x x (a) (b) Fig. 2.4. An arbitrary propagating pulse in: (a) the time domain; and (b) the spa- tial domain. The pulse is propagating in the x direction. R xS x p Fig. 2.5. A spherical wavefront and three rays in a homogeneous medium. In Figure 2.4, we illustrate a propagating impulse in the temporal and spatial domains. Obviously the shape of the pulse is reversed between the two domains. 2.2 A point source So far we have considered plane waves, necessarily of infinite extent, and there- fore not physically realizable. Let us now consider the wavefronts from a point source which might represent a physical source, e.g. an explosion. We only con- sider the geometry of the wavefronts and postpone a more detailed analysis. Obviously in a homogeneous medium, the wavefronts from a point source are spherical (Figure 2.5). Denoting the position vector of the point source as xS, 2.2 A point source 13 we define the radial vector, etc. r = x − xS (2.2.1) R = |x − xS| = |r| (2.2.2) rˆ = r/R = sgn(r). (2.2.3) Then the slowness vector must be p = rˆ c , (2.2.4) and the solution (2.1.11) reduces to φ = f ( t − r · r cR )/ R = f ( t − R c )/ R. (2.2.5) In Figure 2.5, we illustrate a spherical wavefront in a homogeneous medium. The wavefront is defined by t = T (x) = constant, (2.2.6) where T (x) is the travel-time function. Thus the solution is φ = f ( t − T (x) )/ R, (2.2.7) where for a point source in a homogeneous medium, the travel-time function is T (x) = |x − xS|/c = R/c. (2.2.8) We will describe later, in much greater detail, the definition and methods of solving for the travel-time function. For the moment we note that p = ∇T, (2.2.9) which follows as the slowness vector is perpendicular to wavefronts defined by equation (2.2.6), and the variable (2.2.9) has the magnitude of the slowness (time divided by distance). By analogy with potential fields (cf. electric potential and electric field, gravity potential and acceleration due to gravity), given the travel- time function, T (x), we can construct continuous lines perpendicular to wavefronts in the slowness direction, called ray paths (e.g. Figure 2.5). As the wavefronts are propagating in the direction of the rays, we can consider the energy as propagat- ing along rays. Again later we will describe in much greater detail the properties of rays in more complicated media (inhomogeneous and/or anisotropic media – Chapter 5). For now we introduce them as a useful, intuitive concept. Given the 14 Basic wave propagation transmission reflection direct z zS z1 x (a) (b) Fig. 2.6. Wavefronts at a plane interface with a velocity decrease. Part (a) shows the wavefronts at two times, and part (b) the ray paths. The direct wave is the solid line, the reflection is the dashed line, and the transmission is the dotted line. definition (2.2.9) we can compute travel times as T (x) = ∫ x x0 p · dx, (2.2.10) where usually the integration is performed along a ray, although this is not neces- sary. 2.2.1 A point source with an interface Again we consider two homogeneous half-spaces with a point source in the first medium. The cases of a velocity increase and decrease are significantly different. In Figure 2.6, wavefronts from a point source in a model with a velocity de- crease are illustrated. The first wavefront is at a time before it reaches the inter- face. Only the direct, spherical wavefront (the solid line) exists. At a later time, the wavefront has interacted with the interface. A reflected wavefront (the dashed line), also spherical but centred on the image point of the source, exists. The trans- mitted wavefront (the dotted line) spreads out slower and is not a simple spherical shape. Where the three wavefronts intersect the interface, the wavefronts satisfy Snell’s law (as in Figure 2.2 except for a velocity decrease). In Figure 2.7, wavefronts from a point source in a model with a velocity in- crease are illustrated. The direct and reflected wavefronts are similar to before 2.2 A point source 15 transmission reflection