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1 Introduction
At previous lectures, we derived the equation of motion
(λ + 2 μ)�(� · u(x, t)) − μ� × (� × u(x, t)) = ρ ∂
(^2) u(x, t) ∂t^2 (1) This equation of motion can be expressed by a scalar potential, φ(x, t), and a vector potential, Ψ(x, t) which are Helmholtz potentials, based on the relations u(x, t) = �φ(x, t)+�×Ψ(x, t) and � · Ψ(x, t) = 0. The two potentials give us two wave equations. The scalar potential satisfies �^2 φ(x, t) = 1 ∂
(^2) φ(x, t) α^2 ∂t^2 (2) with the velocity α = [(λ + 2 μ)/ρ]^1 /^2. Similarly, the vector potential satisfies
�^2 Ψ(x, t) = 1 ∂
(^2) Ψ(x, t) β^2 ∂t^2 (3)
with velocity β = (μ/ρ)^1 /^2. These two equations can be solved by the following three ways: d’Alembert’s solution, separation of variables and Fourier Transform. Although they are different methods to get solution, they give the same solution of equation of motion. This lecture explains not only the three ways to get plane wave solutions, but also the concepts of dispersion relation and slowness. In addition, this lecture introduces the nomenclature of body waves in Earth’s interior.
2 d’Alembert’s Solution
We begin with the onedimensional function,
f (±x � ct). (4)
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The phase of this function is x ct, so if we only consider constant phase, we can get phase velocity, c = x/t. For convenience, if we consider slightly different form, x/c t, multiplied by angular frequency, ω,and using the relation of k = ω/c, we can get more convenient form, (kx ωt). In threedimensional case, the phase becomes kxx + ky y + kz z − ωt = k · x − ωt for the following complex form wave functions:
φ = Aei(k·x−ωt)^ Ψ = B × ˆkei(k·x−ωt)^ (5)
Equation (5) represents the solutions of wave equation (2) and (3).
3 Separation of variables
Separation of variables is a method of solving ordinary and partial differential equations. For a partial differential equation, we can use separation of variables to make a substitution of the form φ(x, y, z, t) = X(x)Y (y)Z(z)T (t) (6)
to equation (2), and divide it by X Y ZT , then we have
1 d^2 X 1 d^2 Y 1 d^2 Z 1 d^2 T (^) (7) X dx^2 +^ Y dy^2 +^ Z dz^2 −^ c^2 T dt^2 =^0
To satisfy this equation (7), each term has to be constant kx 2 , ky 2 , kz 2 and ω^2 /c^2 , respec tively. Each term makes four ODEs: d^2 X (^) + kx 2 X = 0 −→ e±ikxX dx^2 d^2 Y dy^2 +^ ky^
(^2) Y = 0 −→ e±iky Y d^2 Z (^) + kz 2 Z = 0 −→ e±ikz Z dz^2 d^2 T (^) + ω (^2) T = 0 −→ e±iωT dt^2 From the equation (6), these equations produce desirable solution form, φ = ei(k·x−ωt). Now we can get the full solution to make use of superposition of plane waves.
φ(k, x) = ei(k·x−ωt)^ (8) k,ω
To make each term in equation (7) constant provides physically important phenomenon, what is called dispersion.
2 ω^2 = 0 (9) kx 2 + ky 2 + kz − (^) c 2
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Figure 1: This figure describe the ray and wavefront. The arrow is used for a ray and dash–line for wavefront. The wave number k indicates direction of the ray. The angle i is take–off angle or angle of incidence. ds, dx and dz are distance along the ray, horizon and vertical, respectively. We can get the slowness vector from this figure.
−π to π in frequency. However, it can be used to produce synthetic seismogram by limit: integration over directions k 0 ± dk and frequency ω 0 ± dω.
φ(x, t) = 1 �^ ω^0 +dω^ � �^ k^0 +dk ( 2 π)^3 Φ(kx,^ ky^ ,^ ω,^ z)e
i(k·x−ωt)dkxdky dω (17) ω 0 −dω k 0 −dk The full solution can be obtained by the superposition of plane wave like equation (8), and we find the displacement by u(x, t) = �φ(x, t).
Slowness
Now we define the slowness vector, so we can easily understand what the dispersion relation means. We define the ray speed, c = ds/dt, horizontal wave speed, cx = dx/dt, and vertical wave speed, cz = dz/dt. From [Figure 1], we can relate the angle of incidence with horizontal and vertical wave speed as below.
sin(i) = ds^ = c dt^ = c ≡ cp (18) dx dx cx cos(i) = ds^ = c dt^ = c ≡ cη (19) dz dz cz
12.510 Introduction to Seismology 5
Figure 2: This figure describe the ray in terms of wave number. The arrow is used for a ray and dash–line for wavefront. The wave number k which magnitude is ω/c, indicates direction of the ray. Here kx = ωp and kz = ωη .
Here p is horizontal slowness and η is vertical slowness.
p ≡ (^) c^1 x^ = sinc( i)^ η ≡ (^) c^1 z^ = cosc(i)^ (20)
The slowness vector is composed of horizontal and vertical slowness, s = (p, η). Let us examine some properties of slowness vector. From equation (20), � (^1 ) s = p^2 + η^2 = (^) c p^2 + η^2 = (^) c 2 (21)
However, the addition of squares of horizontal wave speed and vertical wave speed does not equal to squares of wave speed, cx 2 + cz 2 =� c^2. In addition, we will examine critical phenomenon in reflection and refraction with the relation (21), η = 1 /c^2 − p^2. In terms of wave number, each component of wave number can be represented by horizontal and vertical slowness.
kx = ω^ = ωp kz = ω= ωη (22) cx cz Thus, wave number is related to slowness vector.
k = (kx, ky) = (ωp, ωη) = ω(p, η) = ωs (23) As shown in [Figure 2], if we know kx which is related to horizontal slowness and ω/c, then kz is fixed. This situation is based on dispersion relation.
12.510 Introduction to Seismology 7
SKP
S P
SS
SKS
ScP
ScS PcP
PKP PP
PKIKP
PKiKP
Inner Core Outer Core Mantle
PKiKP PKIKP
P and S PP and SS PcP and ScS PKP and SKS
Figure by MIT OCW. Figure 3: The main seismic phase in the Earth.
12.510 Introduction to Seismology 8
Inner Core Outer Core Mantle
1000
1000 2000 3000 4000 5000 6000 7000
0
2000
3000
4000
5000
6000
7000
(^170 180 ) (^160160)
(^150150)
140 140
130
120 120
100 100
90 90
80 80
70 70
60 60
50 50
40 40
30 30
20 10 10 20
0 0
130
SKS
PKP
SKIKS PKIKP
SKiKS PKiKP
S P
110 110
Figure by MIT OCW. Figure 4: Ray paths for major P and S phases for the AKI35 model of seismic wavespeeds.
