1. Geology 5640/6640 Introduction to Seismology 16 Mar 2015 © A.R. Lowry 2015 Last time: Rayleigh Waves Rayleigh waves are interference patterns involving a combination of P & SV waves at a free surface (i.e.  i3 = 0 ), with identical surface velocity c x = 1/p Imposing identical c x = 1/p and stress  i3 = 0 at a free surface, we can solve for amplitudes A, B, of the P-wave and SH-wave that satisfy the constraints, leading to the Rayleigh function : Read for Wed 18 Mar: S&W 86-100 (§2.7–2.8)

2. Geology 5640/6640 Introduction to Seismology 16 Mar 2015 © A.R. Lowry 2015 Last time: Rayleigh Waves Cont’d Rayleigh waves for a uniform Poisson solid have phase velocity c x = 0.92  (and no dispersion!) Displacements are In the Earth though, velocity increases with depth (and depth sampling of a Rayleigh wave increases with wavelength) so waves are dispersive Remember: Take-Home Exam I due Friday 1 pm!

3. Love Waves Consist of SH motion only Need a positive velocity jump or gradient to exist Are inherently dispersive (Note Rayleigh waves are also dispersive on Earth, but they don’t have to be: They are because different wavelengths sample different depths, and velocity increases with depth).

4. For a simple Love wave, let’s consider a (lower-velocity) layer over a (higher-velocity) infinite half-space: Consider a single SH wave which is a simple harmonic ( sin ). As it bounces around in the layer, there are multiple places on the wavepath where the phase is the same. Let’s look for places where the wave might experience constructive interference at the surface, i.e. conditions for which the phase at point A is the same as at point Q.

5. First we need to remind ourselves what phase is! A complex number can be represented in the complex plane as z = x + iy = r e i , where  is the phase angle: So for a typical SH wave represented as displacement: the phase is  =  t – k x x – k z z. To simplify, we can rotate our spatial axes into the propagation direction (or ray) to get: where k  is the modulus of the wavenumbers k x, k z ; and d is distance along the raypath. (real) (imag) x y r 

6. Some useful quantities in this way of representing things are the angle which we rotated the coordinate axes (also angle of incidence!) j : and by the relationship to V = f, Phase is now  =  t – k  d. If the phase difference between point A and point Q is some multiple of 2 , then the waves will constructively interfere! We’ll arbitrarily choose time t = 0 :

7. But now we need to consider a complicating factor: Recall that for an evanescent wave, a phase change will occur at the interface between the layer and the halfspace if the ray is post-critical! The phase change is given by (S&W 2.6.23): So we’ll get constructive interference IF After some trig substitutions this turns out to be:

8. Since k z /k  = cos j and r  = k z /k x, we can rewrite as: If we take the tangent of both sides: Or: We rewrite this in terms of (S&W 2.5.28) and k x =  /c x :

9. The tangent is defined only for real values, so the square roots above have to be real: So  1 < c x <  2. The equation above is transcendental, (no, not in the Buddhist sense), meaning it can’t be solved analytically through normal algebraic means. But we can look at it graphically to get some intuition about what it signifies…

10. First we’ll define a new variable  as: Note that  = 0 when c x =  1 and if c x =  2. With , our equation becomes: The LHS of the equation is zero at  = n  and goes to ±infinity at  = m  for m = 1,3,5 … The RHS decays with a 1/  dependence.

11. In this plot, where the lines intersect are values of  that solve the equation. The number of modes decreases with wave period. The apparent velocity of the waves increases with wave period: thus Love waves are innately dispersive.