Purdue University



Purdue University

EAS 557

Introduction to Seismology

Robert L. Nowack

Lecture 10

Equation of Motion in an Unbounded Medium: Plane Waves

The equation of motion for a linear, isotropic, and elastic solid can be written as

[pic]

or in vector form,

[pic]

where [pic] and [pic] are the Lamé constants, [pic] is the particle displacement, and [pic] is the force term.

We can use the formulation of Helmholtz to decompose [pic] (and [pic]) into scalar and vector potentials. Let [pic], where [pic] and [pic], where [pic], then

[pic]

which are simple wave equations. We will first look at the simple wave equation without the source term. Consider,

[pic]

In order to solve this equation, we will use the technique of separation of variables.

Assume a solution in the form

[pic]

then,

[pic]

This results in,

[pic]

where [pic] is an arbitrary constant. We have chosen the sign in anticipation of the form we want. Since the left term is only a function of [pic] and the right term is only a function of t, they both must be equal to a constant and can be solved separately.

1) [pic]. A solution of this is [pic], with [pic] and [pic] where [pic] is radial frequency in rad/sec.

2a) For the 1-D case, [pic]. A solution of this is of the form [pic] where [pic] is the spatial wavenumber in radians per km.

The combined solution is of the form [pic]. General solutions can be written as [pic]. Our sign convention uses a combined form [pic] with a plus sign for the kx term and a minus sign for the [pic] term.

2b) For the 3-D case,

[pic]

[pic]

[pic]

then,

[pic] and [pic]

where

[pic]

The wavenumber vector can then be written [pic] where

[pic]

Thus, we have a combined solution of the form [pic] which is called a plane wave solution. General solutions can be written as

[pic]

where [pic] is the weighting function for the [pic] term and k3 is defined above. [pic] must be chosen to satisfy the boundary conditions and initial conditions.

We will look at the solution to the wave equation in a slightly different way for the 1-D case. Let

[pic]

We will postulate a solution of the form [pic] (D’Alembert’s solution in 1-D) where [pic] and [pic] are arbitrary functions of the combined variable [pic]. [pic] is a function that has a fixed shaped and for increasing t moves in the +x direction and [pic] is a function that has a fixed shape and moves in the –x direction for increasing t.

[pic]

We verify that this is the solution by letting [pic] where [pic]. Then by the chain rule

[pic]

and

[pic]

Also,

[pic]

and

[pic]

Then, we see that [pic] is a solution of the simple wave equation for any functional shape that moves in an undistorted fashion to the right at a special [pic]. Thus,

[pic]

Doing the same for [pic] and adding, then

[pic]

Thus, the above form satisfies the wave equation.

Thus, the simple wave equation in 1-D has two solutions which propagate undeformed in opposite directions with increasing t with a velocity [pic]. This is one of the fundamental properties of waves. The solution can be written as disturbances that propagate at well-defined velocities.

Let the right propagating solution be

[pic]

which is the right propagating cosine wave, or in complex notation

[pic]

which is the same as the previous solution derived using separation of variables and is sinusoidal over all x and t, where [pic].

Let

k = wavenumber = [pic] in rad/km

with [pic] = wavelength. Also,

[pic] = angular frequency = [pic] in radian/sec

with T = period. The wavelength [pic] and the period T are shown below.

At a fixed time, say t = 0,

as a function of x then

At a fixed x, say x = 0,

as a function of t then

The wave equation also requires that k and [pic] be related through the wave speed [pic] as

[pic]

Alternatively, this can be written as

[pic]

[pic]

From the figure above, we can choose any point A of constant phase [pic] and this propagates with velocity [pic]. The following properties can be inferred:

1) A sinusoidal wave solution is completely nonlocalized in space (as well as time). In this sense it is in steady state. Its propagating nature can be isolated by following a given peak or trough.

2) Arbitrary solutions can be written as a superposition of sines and cosines or (complex exponentials). Thus,

[pic]

[pic]

which can be thought of as a Fourier synthesis of sinusoidal wave solutions to construct more general solutions.

A plane wave solution in three dimensions can be written

[pic]

where A is the amplitude and [pic] is the phase. The wavenumber vector is [pic] and [pic]. Also, [pic], [pic], [pic] where

[pic]

For example, for some fixed time in 2-D then

[pic]

where in the figure [pic], [pic], [pic] and [pic]. Then [pic] are the apparent wavelengths in the x = x1, and z = x3 directions, i equals the angle from vertical, [pic], [pic], and [pic], and [pic]. Also,

[pic]

The wavenumber vector can be written

[pic]

Let [pic] be a unit vector in the plane of constant phase (wavefronts) [pic], then

[pic], Thus, the wave vector [pic] is perpendicular to the wavefront.

Since velocities are related to wavelengths by [pic] (since [pic] is related to particular peaks and troughs, it is called the “phase velocity”), the apparent phase velocity in the x1 and x3 directions can be defined as

[pic]

[pic]

The phase velocity vector is then

[pic]

For example, for a plane wave propagating vertically, then i is zero and

[pic]

Thus, [pic] ! This is similar to a water wave going directly toward a beach and having a wave crest hit all along the beach at the same time. The paradox of having infinite apparent velocities is resolved by the fact that information travels at the signal velocity.

In an isotropic, nondispersive medium, the signal velocity can be written in terms of the group velocity [pic] where [pic] and [pic]. Thus, for a vertically propagating wave [pic]. We will discuss phase and group velocities further when we talk about dispersive waves.

Let’s return to elastic plane waves and write the elastic solution in terms of scalar and vector potentials as

[pic] with [pic]

For a P-wave, let [pic] for [pic] propagating in the positive x1 direction with velocity [pic]. Thus [pic] and [pic]. Then,

[pic] where [pic]

Thus, the particle motion for the P-wave is in the x1 direction parallel [pic] and is in the direction of propagation of the waves.

For an S-wave, let [pic] for [pic] propagating in the positive x1 direction with velocity [pic]. Then,

[pic]

Thus, the particle motion for the S-wave is in a plane perpendicular to [pic] and is in the plane of the wavefront.

For P waves,

[pic]

The uP particle motion is in the direction of the [pic] vector

For S waves (for simplicity, assuming A2 = 0)

[pic]

The uS particle motion is perpendicular to [pic] in the plane of the wavefront

Summary

Plane P waves – These produce longitudinal displacement in the direction of propagation, have associated dilatation, and propagate at the “P” velocity [pic].

Plane S waves – These produce transverse motion perpendicular to the direction of propagation, have associated rotation and shear strain, and propagate at the S velocity [pic].

[pic]

P-wave particle ground motion parallel [pic]

[pic]

S-wave particle ground motion perpendicular to direction of propagation [pic]

Finally, the flux rate of energy density of either plane P-waves or S-waves per unit time across a unit area normal to the direction of propagation is proportional to

[pic]

Thus, the energy flux is proportional to the square of the wave amplitude and also is proportional to [pic] where I is called the seismic wave impedance where V is equal to [pic] for P-waves and [pic] for S-waves.

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