geophysics/tectonics gly 325. elastic waves, as waves in general, can be described spatially

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.€

=k −2μ

3=

νE

(1+ ν )(1− 2ν )

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 t2

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, Vp and Vs . These solutions relate to our elastic constants by the following equations:

Elastic Waves

It turns out that Vp and Vs 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 t2

where = ( + 2)/, or alternatively as

2 = 2 t2

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 amplitude2. 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 last2. Typically largest amplitude2. 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:

VP : VS : VR = 1 : 0.57 : 0.52

In other words, VS is about 60% of VP, and VR is about 90% of VS.

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.


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.


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.


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:


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 v2 > v1.


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:

i1 = i2

i1i2


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...


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.


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.


One thing we didn’t check out is what happens when the variables are set up just right so that i2 = 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 (ic)is the incidence angle at whichthe energy refracts directlyalong the interface.


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.


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.

geophysics/tectonics gly 325. elastic waves, as waves in general, can be described spatially

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