Download The Wave Equation & Velocity

Geology 5640/6640 30 Jan 2015
Introduction to Seismology
Last time: The Equations of Motion (Wave Equation!)
• Any imbalance of stress will be offset by acceleration
(Newton’s 2nd: F  ma). This leads to the dynamic
equations of motion:
Ýi   j ij  fi
uÝ

• We neglect the body force fi (for now) and express in
terms of displacement, by substituting Hooke’s law and
the definition of the strain tensor. This results in the
 equation:
P-wave
 2
2 2



2
t
in which  is the
propagation velocity:


2
2 2

u


  u
2
t
 
 
  2


Read for Mon 2 Feb: S&W 53-62 (§2.4); 458-462
© A.R. Lowry 2015
This is the wave equation for dilatations only (i.e., a P-wave!)
and is more commonly written:
 2
2 2



2
t
where:


  2

represents the propagation velocity! (Note the units:
sqrt(Pa (kg m-3)-1) = sqrt (kg m-1 s-2 kg-1 m3) = sqrt (m2/s2)
or just m/s). 
If we recall moreover that
u u u 
   1  2  3   u
x1 x 2 x 3 
We can write in terms of displacement as:

2
2 2

u


  u
2
t
 
 
We arrived at the P-wave equation using
 2 ui

 2        2 u i
x i
t
by taking the derivative with respect to xi and summing over
i. We could instead take derivatives with respect to xj and
 set of steps arrive at:
by a similar
2
2 2

u


  u
2
t
 
 
the S-wave equation, in which the S-wave propagation
velocity is given by




Note the important implication: For the P-wave we have
dilatation, but no shear; for the S-wave we have shear,

but no dilatation!
  2

• Here  
and  


represent the
propagation velocities for the P and S waves
respectively.


• Changes in elastic properties contribute more to velocity
variation than changes in density
• Velocity is sensitive to rock chemistry, packing structure,
porosity & fluid type, pressure and temperature. The
tricky part is distinguishing which we’re seeing…
Rock properties that affect seismic
Crustal Rocks
velocity include:
• Porosity
• Rock composition
• Pressure
• Temperature
• Fluid saturation
 = Vp,  = Vs are much more
sensitive to  and  than to 
Mantle
Rocks
Seismic velocity depends on a lot of fields, but not all
are independent:
Velocity
Temperature
Composition
Partial Melt
Pressure
Porosity/Fluid
Density
And some fields can be determined to within
small uncertainty (e.g. pressure at given depth)
So now we have our expressions for the wave equation in
terms of displacements:
2
2 2

u


  u
2
t
 
 
2
2 2

u


  u
2
t
 
 
Question is, how do we solve these?
expressing displacements u
 Solution is simplified by 
in terms of displacement potentials. Helmholtz’
decomposition theorem holds that any vector field u
can be expressed in terms of a vector potential  and
a scalar potential  as:
u     
In our application,  is a scalar displacement potential
associated with the P-wave, and  is a vector
displacement potential associated with the S-wave.

It’s first worth noting a pair of useful vector identities:


 
    0    0
Then, if we substitute our potentials into our P-wave
equation:

2
2 2








     
2
t
 
Rearranging:

 
 2

2 2
  2      0
t

2

And hence:
1 
  2 2
 t
2

2

Similarly, substituting potentials into the S-wave equation:
2
2 2








     
2
t
 

 

Here we take advantage of another vector identity:


 

         2   2 
Rearranging:
 2  2 2 
  2     0
 t

2
And hence:

1  
  2 2
 t
2
2
So what’s the point of this? We want to find some
solution, e.g. for P-wave displacement potential,
that allows for separation of variables:
x,t Xx1Y x2 Zx3 T t
The eigenfunctions for a partial differential equation
of this form (i.e., functions which, if plugged into the
equation, will yield solutions of similar form) are:

 x ,t  Ae

i t k  x

(called the “d’Alembert solution”). Here,
i is the imaginary number 1
A is amplitude
 is angular frequency 2/T (& T is time period)

k is spatial wavenumber
2/ (&  is wavelength)
