Geophysics/Tectonics GLY 325

Elastic Waves, as waves in general, can be described spatially...

…or temporally.

Elastic Waves Lame’s Constant ( ) -- interrelates all four elastic constants and is very useful in mathematical computations, though it doesn’t have a good intuitive meaning. It’s important for you to know the terms and what they represent (when appropriate) because we will be using them in labs.

The Wave Equation We’ll look at the scalar wave equation to mathematically express how elastic strain (dilatation,  ) propagates through a material:   2  = (  + 2  ) 2   t 2 where  xx  yy  zz and 2 is the Lapacian of , or    x     y     z   

Elastic Waves When solving the wave equation (which describes how energy propagates through an elastic material), there are two solutions that solve the equation, V p and V s. These solutions relate to our elastic constants by the following equations:

Elastic Waves It turns out that V p and V s are probably familiar to you from your introductory earthquake knowledge, since they are the velocities of P-waves and S-waves, respectively. So, now you know why there are P- and S-waves--because they are two solutions that both solve the wave equation for elastic media. P S

The Wave Equation The wave equation can be rewritten as   2  = 2     t 2 where   = (  + 2  )/ , or alternatively as   2  = 2     t 2 where   =  /  And you’ll recognize the physical realization of these equations as  = P-wave and  = S-wave velocity.  

The Wave Equation Since the elastic constants are always positive,  is always greater than , and  /  = [  /( +2  )] 1/2 = [(0.5-  )/(1-  )] 1/2 So, as Poisson’s ratio, , decreases from 0.5 to 0,  /  increases from 0 to it’s maximum value 1/√2; thus, S-wave velocity must range from 0 to 70% of the P-wave velocity of any material.

The Wave Equation These first types of solutions–P-waves and S-waves–are called body waves. Body waves propagate directly through material (i.e. its “body”). I. Body Waves a. P-Waves 1. Primary wave (fastest; arrive first) 2. Typically smallest in amplitude 3. Vibrates parallel to the direction of wave propagation. b. S-Waves 1. Secondary waves (moderate speed; arrives second) 2. Typically moderate amplitude 2. Vibrates perpendicular to the direction of wave propagation.

The Wave Equation The other types of solutions are called surface waves. Surface waves travel only under specific conditions at an interface, and their amplitude exponentially decreases away from the interface. II. Surface waves (slowest) 1. Arrives last 2. Typically largest amplitude 2. Vibrates in vertical, reverse elliptical motion (Rayleigh) or shear elliptictal motion (Love)

The Wave Equation The three types of surface waves are: 1) Rayleigh Waves– form at a free-surface boundary. Air closely approximates a vacuum (when compared to a solid), and thus satisfies the free-surface boundary condition. Rayleigh waves are also called “ground roll.” 2) Love Waves– form in a thin layer when the layer is bound below by a seminfinite solid layer and above by a free surface. 3) Stonely Waves– form at the boundary between a solid layer and a liquid layer or between two solid layers under specific conditions.

The Wave Equation For a “typical” homogeneous earth material, in which Poisson’s ratio  = 0.25 (also called a Poisson solid), the following relationship should be remembered between P-wave, S-wave, and Rayleigh wave velocities: V P : V S : V R = 1 : 0.57 : 0.52 In other  words, V S is about 60% of V P, and V R is about 90% of V S. But remember, this only is a guide...

The Wave Equation Modeled The wave equation explains how displacements elastically propagate through material. In models, colors represent the displacement of discrete elements (below: yellow–positive, purple–negative) away from their equilibrium position.

The Wave Equation Modeled As displacements propagate away from the initial source of displacement (i.e., the source), a spherical wavefront is observed. Seismologists define raypaths showing the direction of propagation away from the source. Raypaths are always perpendicular to the wavefront, analogous to flowpaths in hydrology.

The Wave Equation Modeled Boundary Conditions: We’ve seen how body-wave displacements propagate through a homogeneous material, but what happens at boundaries? At boundaries (defined as a place where material elastic properties change ), body waves refract (following Snell’s Law) and reflect. * Without going into details, the potential ENERGY expressed in the propagating displacements is partitioned at every interface into REFRACTED (or transmitted) and REFLECTED energy as stated by the complex Zoeppritz Equations.

The Wave Equation Modeled Boundary Conditions–REFRACTION: Snell’s Law states that an incident raypath will refract at an interface to a degree related to the difference in velocities:

The Wave Equation Modeled Boundary Conditions–REFRACTION: Note that by definition, if the propagation velocity increases across an interface, the ray will refract toward the interface. In the example below, the diagram is drawn such that v 2 > v 1.

The Wave Equation Modeled Boundary Conditions–REFLECTION: At an interface, body wave displacements also reflect. Simply, waves reflect at an interface with an angle equal to the incidence angle, regardless of the propagation velocities of the layers: i 1 = i 2 i1i1 i2i2

The Wave Equation Modeled So, at any interface, some energy is reflected (at the angle of incidence) and some is refracted (according to Snell’s Law). Let’s look at a simple model and just watch what happens to the P-wave energy...

The Wave Equation Modeled FYI, if we used the same model, but only looked at the surface waves, not surprisingly we would just see them move out from the source at a constant velocity.

The Wave Equation Modeled

Each of the things labeled is called a phase. Phases, in layman's terms, represent a part of the original source energy that has done something.

The Wave Equation Modeled One thing we didn’t check out is what happens when the variables are set up just right so that i 2 = 90°. That means the energy will travel right along the interface. It turns out that this phenomenon generates the interesting head wave phase. An important term to know is critical angle. The critical angle (i c )is the incidence angle at which the energy refracts directly along the interface.

The Wave Equation Modeled Head waves are generated by energy refracting along an interface, and along the way “leaking” some energy back toward the surface at the critical angle.

The Wave Equation Modeled As displacements propagate away from the initial source of displacement (i.e., the source), a spherical wavefront is observed. Seismologists define raypaths showing the direction of propagation away from the source. Raypaths are always perpendicular to the wavefront, analogous to flowpaths in hydrology.