Question

Consider a vertically incoming (incident) P-wave striking a horizontally stratified earth (Fig. 2). The amplitude of the transmitted P-wave at the top boundary of layer 1 is 1 (don’t worry about units for amplitude). What is the amplitude of the P-wave reflected from the boundary between layers 3 and 4? What is the amplitude of the P-wave transmitted into layer 4? What are the amplitudes of the corresponding S-waves? Express amplitudes in relative terms (relative to the amplitude of the transmitted P-wave at the top boundary of layer 1). Assume that there is no absorption (no amplitude attenuation due to the conversion of elastic energy into heat) and no spherical spreading.

Fig. 2 Vertically incident P-wave (θ'p = 00) = 2 km/s; ρ'= 2.50 g/cm layer 1 10 m %" 2 km/s, player50 g/cm' V., 3 km/s; P, 2.65 g/cm 5 m ayer 2 layer 3 20 m V 4km/s; p 2.75 g/cm layer 4

Answer 1

Ans:

We have reflection formula : A₁/A₀ = (ρ₂v₂ – ρ₁v₁)/ (ρ₂v₂ + ρ₁v₁) = (Z₂-Z₁)/(Z₂+Z₁)

Transmission formula : A₂/A₀ = 2 ρ₁v₁ / (ρ₂v₂ + ρ₁v₁)

A0 = incident ray amplitude = 1

A1, A3, A5 = reflected ray amplitude

A2, A4, A6 = transmitted ray amplitude

For layers

A1/A0 = 0.99, So A1 = 0.99

A2/A0 = 1.64 * 10^-4, A2 = 1.64 * 10^-4

A4/A2 = 1, A4 = 1.64 * 10^-4

A6/A4 = 0.7722, A6 = 1.266 * 10^-4

A8/A6 = 0.8390, A8 = 1.06 * 10^-4

The amplitude of the P-wave reflected from the boundary between layers 3 and 4 = 0.99987

The amplitude of the P-wave transmitted into layer 4 = 1.06 * 10^-4

For S wave

Poisson ration for layer 1 and 2 = 0.29

For layer 3 = 0.12

For layer 4 = 0.11
find speed of S wave

v = V_p2 – 2V_s²/ 2 (V_p² – V_s²)

V_s1 = 1087.77 m/s

V_s2 = 1087.77 m/s

V_s3 = 4602.9 m/s

V_s4 = 6288.21 m/s

A2/A0 = 3.0266 * 10-4, A2 = 3.0266 * 10^-4

A4/A2 = 1, A4 = 3.0266 * 10^-4

A6/A4 = 0.3645, A6 = 1.103 * 10^-4

A8/A6 = 0.8272, A8 = 9.12 * 10^-5

The amplitude of the S-wave reflected from the boundary between layers 3 and 4 = 0.99987

The amplitude of the S-wave transmitted into layer 4 = 9.12 * 10^-5