1.510 Introduction to Seismology Lecture 5 Feb., 005 1 Introduction At previous lectures, we derived the equation of motion (λ + µ) ( u(x, t)) µ ( u(x, t)) = ρ u(x, t) (1) t 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 φ(x, t) = 1 φ(x, t) () α t with the velocity α = [(λ + µ)/ρ] 1/. Similarly, the vector potential satisfies Ψ(x, t) = 1 Ψ(x, t) β t (3) with velocity β = (µ/ρ) 1/. 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. d Alembert s Solution We begin with the one dimensional function, f (±x ct). (4)
Feb., 005 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 three dimensional case, the phase becomes k x x + k y y + k z z ωt = k x ωt for the following complex form wave functions: ˆ φ = Ae i(k x ωt) Ψ = B ke i(k x ωt) (5) Equation (5) represents the solutions of wave equation () 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 (), and divide it by X Y ZT, then we have 1 d X 1 d Y 1 d Z 1 d T X dx + Y dy + Z dz c T dt = 0 (7) To satisfy this equation (7), each term has to be constant k x, k y, k z and ω /c, respectively. Each term makes four ODEs: d X + kx X = 0 e ±ikxx dx d Y dy + k y Y = 0 e ±iky Y d Z + kz Z = 0 e ±ikz Z dz d T + ω T = 0 e ±iωt dt From the equation (6), these equations produce desirable solution form, φ = e i(k x ωt). Now we can get the full solution to make use of superposition of plane waves. φ(k, x) = e i(k x ωt) (8) k,ω To make each term in equation (7) constant provides physically important phenomenon, what is called dispersion. ω k x + k y + k z = 0 (9) c
1.510 Introduction to Seismology 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 one dimensional solution form as φ = e i(kxx ωt), then dφ/dx = ik x φ, d φ/dx = k x φ. From equation (), Thus, we can get the dispersion relation, We can also get the Helmholtz equation. We will get back to the dispersion relation later. φ = α φ = α k x φ (10) φ φ = = ω φ (11) t k x = ( ω ) (1) α φ + k φ = 0 (13) 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 space wave number, respectively. 1 π Φ(x, ω) = φ(x, t)e iωt dt φ(x, t) = Φ(x, ω)e iωt dω (14) π π 1 Φ(k, t) = φ(x, t)e ik r d 3 r φ(x, t) = Φ(k, t)e ik x dk (π) 3 x dk y dk z (15) V k We combine the two relation, i.e. double Fourier transform 1 π φ(x, t) = Φ(k (π) 3 x, k y, ω, z)e i(k x ωt) dk x dk y dω (16) π This is the solution that satisfies the wave equation (). It is also shown that this solution is similar to the solution (8) obtained by separation of variables. The integrand, Φ(k x, k y, 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. k z = f (k x, k y, ω). Second, it is difficult in practice to integrate from to in terms of wave number and from
4 Feb., 005 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. 1 ω0 +dω k 0 +dk φ(x, t) = Φ(k (π) 3 x, k y, ω, z)e i(k x ωt) dk x dk y 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). 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, c x = dx/dt, and vertical wave speed, c z = dz/dt. From [Figure 1], we can relate the angle of incidence with horizontal and vertical wave speed as below. ds dt c sin(i) = = c = cp dx dx c x (18) ds dt c cos(i) = = c = cη dz dz c z (19)
1.510 Introduction to Seismology 5 Figure : 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 k x = ωp and k z = ωη Here p is horizontal slowness and η is vertical slowness.. p 1 = sin(i) η 1 = cos(i) (0) c x c cz c The slowness vector is composed of horizontal and vertical slowness, s = (p, η). Let us examine some properties of slowness vector. From equation (0), 1 1 s = p + η = p + η = (1) c c However, the addition of squares of horizontal wave speed and vertical wave speed does not equal to squares of wave speed, c x + c z = c. In addition, we will examine critical phenomenon in reflection and refraction with the relation (1), η = 1/c p. In terms of wave number, each component of wave number can be represented by horizontal and vertical slowness. Thus, wave number is related to slowness vector. ω ω k x = = ωp k z = = ωη () c x c z k = (k x, k y ) = (ωp, ωη) = ω(p, η) = ωs (3) As shown in [Figure ], if we know k x which is related to horizontal slowness and ω/c, then k z is fixed. This situation is based on dispersion relation.
6 Feb., 005 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 medium 1. On the other hand, lower case letters are used to indicate either reflections or upward propagation 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 S K I J c i p s LR LQ a P wave in the mantle an S wave in the mantle a P wave through the outer core a P wave through the inner core an S wave through the inner core a reflection from the mantle outer core boundary a reflection from the outer core inner core boundary a up going P wave from the earthquake focus to the surface of the earth a up going S wave from the earthquake focus to the surface of the earth a Rayleigh wave a Love wave The [Figure 3] and [Figure 4] describe the ray paths in terms of various phases. Note that the phases are in combination of a wave described above. Notes: Kang Hyeun Ji (Feb., 005) 1 The phase J has no definitive observations of this seismic phase, although recent research has produced compelling evidence for its existence Adapted from B.L.N. Kennett (001), The Seismic Wavefield Volume 1: Introduction and Theoretical Development, pp. 78 80, Cambridge University Press
1.510 Introduction to Seismology 7 7000 ScS 6000 5000 PcP Inner Core Outer Core Mantle SS S ScP SKS 4000 3000 000 1000 0 PKP P PP P and S PP and SS PcP and ScS PKP and SKS PKiKP PKIKP 1000 000 3000 PKiKP SKP 4000 PKIKP 5000 6000 7000 Figure by MIT OCW. Figure 3: The main seismic phase in the Earth.
1.510 Introduction to Seismology 8 50 40 30 0 10 0 7000 6000 5000 0 10 0 30 40 50 Inner Core Outer Core Mantle 60 70 S 4000 3000 000 P 60 70 80 90 SKS 1000 0 PKP 80 90 100 110 SKiKS SKIKS 1000 000 3000 PKIKP PKiKP 100 110 10 4000 10 130 140 150 160 170 180 5000 6000 7000 180 170 160 150 140 130 Figure by MIT OCW. Figure 4: Ray paths for major P and S phases for the AKI35 model of seismic wavespeeds.