TODAY’S LECTURE 1. Review of ground roll/Love waves 2. Evanescent wave and sensitivity kernel 3. ω – k space 4. Phase velocity vs. group velocity REVIEW OF GROUND ROLL/LOVE WAVES The dispersion relationship for ground roll is given by tan( ω H 1 − p2 ) = c12 ρ2 1 − p2 c12 ρ1 p 2 − (1) 1 c 22 Second higher mode First higher mode Fundamental mode 1/c2 Transmitted waves into half space (leaky 1/c1 Direct wave P Locked mode 1/c2 < p <1/c1 mode) p < 1/c2 In the leaky mode (precritical), the waves can be transmitted into the half space. In the locked mode (postcritical), all of the energy stays in the upper layer. A finite number of modes have discrete phase velocities. The fundamental mode represents a direct wave and there can be higher modes depending on the wave frequency. It is because the left side of equation (1) includes the wave frequency. In the low frequency case, the spacing of the curves increases (the number of higher mode decrease and the fundamental mode can be exist only). On the other hand, the spacing decreases in the high frequency case (the number of higher mode increases. In the extreme case, it can be continuous.) 1 EVANESCENT WAVE AND SENSITIVITY KERNEL Z=0 Evanescent wave : → exp{ −ωη z} exp{i ( kx − ωt )} C1 / \ S1 S1 The first term represents exponential decay with respect to z Z=H C2 < C1 for the postcritical case. Evanescent wave In the layer (0H), the displacement is given by \ u n ( x, z, t ) = 2 S 1 cos{ω 1 1 − pn2 H )}exp{−ω pn2 − 2 ( z − H )}exp{i(k n x − ωt )} 2 c1 c2 (3) As in the layer, we have a wave propagating in the x direction in the half space with wavenumber k n . The displacement decays exponentially with depth as the first exponential term. In the low frequency case, the decay is slow and the amplitude is large at c=c2. In the high frequency case, the decay is fast. If the frequency is infinite, there is no sensitivity. Moreover, as the depth goes to infinity, the displacement or sensitivity is going to zero. The following figure describes the pattern of the wave in the layer and in the half space. We can see that higher modes sample deeper and faster. 2 Fundamental mode First higher mode Second higher mode Frequency increases ω-k SPACE ω – k space is the frequency – wavenumber domain which is obtained by taking Fourier transform from time-space domain. In the ω – k space, we can analyze the previous study of surface wave dispersion relation in equation (1). A constant phase velocity can be represented by a straight line, c = ω / k . The fundamental mode and higher modes can also be represented. ω c2=ω/k1 c fix c1=ω/k2 kfix 3rd higher mode 2nd higher mode 1st higher mode ω2 ω1 ω0 ωfix Fundamental mode k2 k1 k k0 Higher frequency makes the mode approach c1 asymptotically (i.e. no sensitivity). 3 Conversely, smaller frequency makes the mode approach c2 asymptotically, which means low frequency waves are more sensitive to half space and evanescent waves at low frequency sample deep in the half space. We can consider three cases: fixed frequency, fixed wavenumber, and fixed wave speed. (1) At fixed frequency, waves propagate along different directions consistent with each discrete wavenumber. Discrete wavenumber represents discrete phase speed, c n = ω / k n . The smaller wavenumber has the higher phase velocity. The lowest velocity corresponds to fundamental mode and high velocity comes from higher modes. (2) At fixed wavenumber, waves propagate along a fixed direction but with a discrete number of wave frequencies. The higher frequency has the higher phase velocity. The lowest velocity comes from fundamental mode and higher modes give higher velocities. (3) At fixed wave speed, there are infinite numbers of modes. For a given distance, waves of low frequency of the fundamental mode can arrive at the same time as higher modes at a higher wave frequency. Note that the phase velocity is described by a straight line. PHASE VELOCITY VS. GROUP VELOCITY Consider two waves with particular wavenumber k1 and k2 and with particular wave frequency ω1 and ω2. The two waves can be represented by cosine term. u ( x, t ) = cos(k1 x − ω1t ) + cos(k 2 x − ω 2 t ) = 2 cos(kx − ωt ) cos(δkx − δωt ) δω = ω − ω1 = ω − ω 2 , ω1 < ω < ω 2 (4) δk = k − k1 = k − k 2 , k1 < k < k 2 The first cosine term travels with phase velocity, c = ω / k and the second travels with group velocity, u = δω / δk . The following figure describes the ‘beating’ effect. , The solid line has phase velocity and the dotted line has group velocity. 4 We can understand the phase and group velocity on the time-space plane and frequency-wavenumber plane, respectively. Group velocity Phase velocity In time-space plane, the tangent dashed line represents the phase velocity. T dT 1 k = p= = dx c ω The group velocity is (5) defined by the following. U = X X δω = T δk (6) The phase velocity is faster than the group velocity. Phase velocity c2 ω cn c1 In ω-k space, the group velocity is the tangent dashed line which is clear from equation (6). As frequency goes to infinity, group velocity is equal to phase velocity. Group velocity lim U = lim ω→∞ ω→∞ δω ω = δk k (7) In this case, group velocity will be equal to c1 for high frequency waves. k 5