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
or in vector form,
where 
and 
are the Lamé constants, 
is the particle
displacement, and 
is the force term.
We can use the formulation of Helmholtz
to decompose (and
) into scalar and vector potentials. Let 
, where 
and 
, where 
, then
which are simple wave equations. We will first look at the simple wave equation without the source term. Consider,
In order to solve this equation, we will use the technique of separation of variables.
Assume a solution in the form
then,
This results in,
where 
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 
and the right term is
only a function of t, they both must
be equal to a constant and can be solved separately.
1) 
. A solution of this is 
, with 
and 
where 
is radial frequency in
rad/sec.
2a) For the 1-D case, 
. A solution of this
is of the form 
where 
is
the spatial wavenumber in radians per km.
The combined
solution is of the form 
. General solutions can
be written as 
. Our sign convention
uses a combined form 
with a plus sign for
the kx term
and a minus sign for the 
term.
2b) For the 3-D case,
then,
and 
where
The wavenumber vector can then be written 
where
Thus, we have a combined solution of the
form 
which is called a
plane wave solution. General solutions
can be written as
where 
is the weighting
function for the 
term and k3 is defined above. 
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
We will postulate a solution of the
form 
(D’Alembert’s
solution in 1-D) where 
and 
are arbitrary
functions of the combined variable 
. is
a function that has a fixed shaped and for increasing t moves in the +x
direction and is a function that has
a fixed shape and moves in the –x
direction for increasing t.
We verify that this is the solution by letting 
where
. Then by the chain
rule
and
Also,
and
Then, we see that is a solution of the
simple wave equation for any functional shape that moves in an undistorted
fashion to the right at a special 
. Thus,
Doing the same for and adding, then
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 . 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
which is the right propagating cosine wave, or in complex notation
which is the same as the previous
solution derived using separation of variables and is sinusoidal over all x and t, where 
.
Let
k = wavenumber = 
in rad/km
with = wavelength. Also,
= angular frequency = 
in radian/sec
with T = period. The wavelength 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 be related through the
wave speed as
Alternatively, this can be written as
From the figure above, we can
choose any point A of constant phase 
and this propagates
with velocity . 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,
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
where A is the amplitude and 
is the phase. The wavenumber vector is 
and 
. Also, 
, 
, 
where
For example, for some fixed time in 2-D then
where in the figure 
, 
, 
and 
. Then 
are the apparent
wavelengths in the x = x1, and z = x3
directions, i
equals the angle from vertical, 
, 
, and 
, and 
. Also,
The wavenumber vector can be written
Let 
be a unit vector in
the plane of constant phase (wavefronts) 
, then
, Thus, the wave
vector 
is perpendicular to the
wavefront.
Since velocities are related to wavelengths by 
(since 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
The phase velocity vector is then
For example, for a plane wave propagating vertically, then i is zero and
Thus, 
! 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 
where 
and 
. Thus, for a vertically
propagating wave 
. 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
with
For a P-wave, let 
for 
propagating in the positive
x1 direction with velocity . Thus 
and 
. Then,
where 
Thus, the particle motion for the P-wave is in the x1 direction parallel and is in the
direction of propagation of the waves.
For an S-wave, let 
for 
propagating in the positive
x1 direction with velocity 
. Then,
Thus, the particle motion for the S-wave is in a plane
perpendicular to and is in the plane of
the wavefront.
For P waves,
The uP particle
motion is in the direction of the vector
For S waves (for simplicity, assuming A2 = 0)
The uS particle
motion is perpendicular to 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 
.
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 
.
P-wave particle ground
motion parallel
S-wave particle ground
motion perpendicular to direction of propagation
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
Thus, the energy flux is proportional to the square of the
wave amplitude and also is proportional to 
where I is called the seismic wave impedance where V is equal to for P-waves and 
for S-waves.