Introduction to Seismology - d'Alembert's Solution

Introduction to Seismology 1 Introduction At previous lectures, we derived the equation of motion (λ + 2µ)�(� · u(x, t)) − µ� × (� × u(x, t)) = ρ ∂ 2 u(x, t) ∂t2 (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 1 ∂ 2 φ(x, t) (2) �2 φ(x, t) = 2 α ∂t2 with the velocity α = [(λ + 2µ)/ρ]1/2 . Similarly, the vector potential satisfies �2 Ψ(x, t) = 1 ∂ 2 Ψ(x, t) β2 ∂t2 (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) 2 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 kx x + ky y + kz z − ωt = k · x − ωt for the following complex form wave functions: ˆ i(k·x−ωt) Ψ = B × ke φ = Aei(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 d2 Y 1 d2 Z 1 d2 T 1 d2 X + + − 2 =0 2 2 2 Y dy Z dz c T dt2 X dx (7) To satisfy this equation (7), each term has to be constant kx 2 , ky 2 , kz 2 and ω 2 /c2 , respec­ tively. Each term makes four ODEs: d2 X + kx 2 X dx2 d2 Y + ky 2 Y dy 2 d2 Z + kz 2 Z dz 2 d2 T + ω2T dt2 = 0 −→ e±ikx X = 0 −→ e±iky Y = 0 −→ e±ikz Z = 0 −→ e±iωT 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. � ei(k·x−ωt) (8) φ(k, x) = k,ω To make each term in equation (7) constant provides physically important phenomenon, what is called dispersion. kx 2 + ky 2 + kz 2 − ω2 =0 c2 (9) 3 The dispersion relation means that waves with different frequencies travel with different velocities. Furthermore, if we know three of four parameters, one of them can be fixed from the relation (9). If we consider one­dimensional solution form as φ = ei(kxx −ωt) , then dφ/dx = ikx φ, d2 φ/dx2 = −kx 2 φ. From equation (2), φ¨ = α2 �2 φ = −α2 kx 2 φ φ¨ = ∂2φ (10) = −ω 2 φ (11) ω 2 kx 2 = ( ) α (12) ∂t2 Thus, we can get the dispersion relation, We can also get the Helmholtz equation. �2 φ + k 2 φ = 0 (13) We will get back to the dispersion relation later. 4 Fourier Transform Fourier transform lets us understand the relation between space–time (x, t) domain and wave number–frequency (k, ω) domain. The followings are Fourier transform and inverse Fourier transform in term of time–frequency and space­wave number, respectively. � π � ∞ 1 iωt Φ(x, ω) = φ(x, t)e dt ←→ φ(x, t) = Φ(x, ω)e−iωt dω (14) 2π −π −∞ � � 1 φ(x, t)eik·r d3 r ←→ φ(x, t) = Φ(k, t)e−ik·x dkx dky dkz (15) Φ(k, t) = 3 (2π) V k We combine the two relation, i.e. double Fourier transform � π �� ∞ 1 φ(x, t) = Φ(kx , ky , ω, z)ei(k·x−ωt) dkx dky dω (2π)3 −π −∞ (16) This is the solution that satisfies the wave equation (2). It is also shown that this solution is similar to the solution (8) obtained by separation of variables. The integrand, Φ(kx , ky , ω, z), can be considered as amplitude or weight. There are two points to consider the equation (16) in detail . First, we do not need to put z component of wave number in the integration because of the dispersion relation (9). As mentioned before, one parameter can be fixed by dispersion relation if we know three of four parameters, i.e. kz = f (kx , ky , ω). Second, it is difficult in practice to integrate from −∞ to ∞ in terms of wave number and from 4 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 k0 ± dk and frequency ω0 ± dω. � ω0 +dω � � k0 +dk 1 φ(x, t) = Φ(kx , ky , ω, z)ei(k·x−ωt) dkx dky dω (17) (2π)3 ω0 −dω k0 −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). 5 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. ds dt c =c = ≡ cp cx dx dx ds dt c =c = ≡ cη cos(i) = cz dz dz sin(i) = (18) (19) 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≡ 1 sin(i) = cx c η≡ 1 cos(i) = c cz (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), s= � 1 p2 + η 2 = c p2 + η 2 = 1 c2 (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 = � c2 . In addition, we will examine critical � phenomenon in reflection and refraction with the relation (21), η = 1/c2 − p2 . In terms of wave number, each component of wave number can be represented by horizontal and vertical slowness. kx = ω = ωp cx kz = ω = ωη cz (22) 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. 6 6 Nomenclature of body waves in Earth’s interior At this stage, it is useful to introduce the jargon used to describe the different types of body wave propagation in Earth’s interior. There are a few simple basic rules, but there are also some inconsistencies. Capital letters are used to denote body wave propagation (transmission) through a medium1 . On the other hand, lower case letters are used to indicate either reflections or upward prop­ agation of body waves before they are reflected at Earth’s surface. Note that this is always used in combination of a transmitted wave. The followings are summary of nomenclature of body wave. P a P­wave in the mantle S an S­wave in the mantle K a P­wave through the outer core I a P­wave through the inner core J an S­wave through the inner core c a reflection from the mantle – outer core boundary i a reflection from the outer core – inner core boundary p a up­going P­wave from the earthquake focus to the surface of the earth s a up­going S­wave from the earthquake focus to the surface of the earth LR a Rayleigh wave LQ a Love wave The [Figure 3] and [Figure 4]2 describe the ray paths in terms of various phases. Note that the phases are in combination of a wave described above. 1 The phase J has no definitive observations of this seismic phase, although recent research has produced compelling evidence for its existence 2 Adapted from B.L.N. Kennett (2001), The Seismic Wavefield Volume 1: Introduction and Theoretical Development, pp. 78 – 80, Cambridge University Press 7 7000 P Outer Core 5000 Pc ScS Inner Core 6000 Mantle 4000 S ScP P and S 3000 PP and SS P PcP and ScS 2000 SKS SS PKP and SKS 1000 PKP PP 0 PKIKP 1000 2000 PKiKP SKP 3000 4000 PKiKP PKIKP 5000 6000 7000 Figure 3: The main seismic phase in the Earth. 8 30 0 10 20 7000 0 10 20 40 Inner Core 30 6000 40 5000 50 Outer Core 4000 60 S 60 P 3000 70 70 2000 80 Mantle 50 80 1000 SKS 90 PKP 0 90 1000 100 2000 SKiKS 110 100 PKiKP SKIKS 120 3000 110 PKIKP 120 4000 130 130 5000 140 140 6000 150 160 170 150 7000 180 180 170 160 Figure 4: Ray paths for major P and S phases for the AKI35 model of seismic wavespeeds.