Introduction to Seismology Geology 5640/6640 Introduction to Seismology 8 Feb 2017 Last time: The solution to the Wave Equation! • Solution of the wave equation is made easier by expressing displacement in terms of scalar & vector displacement potentials: In that case: These can be solved by plugging in eigenfunctions of the form: and solutions are: Read for Fri 10 Feb: S&W 62-75 (§2.5) © A.R. Lowry 2017

Introduction to Seismology Geology 5640/6640 Introduction to Seismology Last time cont’d: Harmonic wave motion • Harmonic wave motion can be described entirely by specifying a period T (= 1/f = 2/), wavelength  (= 2/k), and a phase  (= x/c = tan-1[a1/a2]).

Polarizations of P- and S-waves: The polarization (or “polarity”) of a wave is a way of describing the direction of initial particle motion of a wave (i.e., the “first motion”). For example, a seismologist might say “The first motion on the North-South component was down.” This implies a reference frame however… First consider a P-wave (scalar displacement potential) with equation: solution: We could write this as: where the first part (1) travels in the +x direction, and the second (2) travels in the –x direction. Displacement is related to the potential  by the gradient:

Here our y, z components disappeared because there’s no y, z dependence for a plane wave traveling in the x-direction: So, u = (–ik, 0, 0), and particle displacement is always in ± the direction of wave propagation, i.e., perpendicular to the wave front. So if first motion is compressive, all points on the wavefront move in the direction of the black arrows (“positive” or “upward” polarity); if dilatational, they all move in the direction of the red arrows. This is also called longitudinal displacement…

Now consider the S-wave (vector displacement potential)… Like before we’ll consider a plane wave propagating only in the x-direction: equation: solution has three components: To write this in terms of displacement we take the curl: Since it’s propagating only in the x-direction, the y- and z-derivatives again vanish, leaving:

So for an S-wave propagating in the x-direction, particle motion is only in the y- and z-directions, and not the x-direction. We further decompose the wavefield using a convention that: SH = horizontally polarized shear wave SV = vertically Both of these are always perpendicular to the propagation direction, with SH in the horizontal plane and SV in the vertical plane orthogonal to SH.

Thus for our example of a shear wave propagating only in the x-direction: Different texts illustrate this differently; one way is to think of a series of blocks:

To visualize SH and SV a little better, consider a point source deep within a constant-velocity medium: The ray paths must be straight lines; the wavefronts are spheres.

Now imagine the same source-receiver pair for a medium in which velocity increases with depth: The definition’s the same, but now the angle between direction of propagation & the horizontal is changing (so angle of SV motion WRT horizontal also changes).

First, a quick look at some data and phases Seismic Data and Analysis: First, a quick look at some data and phases

Ray Naming Conventions P, S - P or S wave segments in the mantle p, s - up-going energy from an earthquake (not shown here) K - P wave segment in the fluid outer core I - P wave segment in the solid inner core J - S wave segment in the solid inner core c - a reflection off the core-mantle boundary i - a reflection off the inner core boundary Example: SKS—S-wave that converts to a P-wave in the outer core, then back to an S-wave when it leaves the outer core and travels up towards the seismometer.

(More like what it really looks like:) A ray is the normal to a propagating wavefront.

More ray paths…

Ray paths in the upper part of the Earth (the “lithosphere”)