Problem 2.9a
Show that equation (2.9a) relating the potential functions
and
to the vector displacement
requires that
and
[see equations (2.1e) and (2.1g)] be solutions of the P- and S-wave equations, that is, of equation (2.5a) with
replaced by
and
, respectively.
|
|
(2.9a)
|
and
being solutions of the P- and S-wave equations, respectively.
Background
The dilatation
and component of rotation
are defined in equations (2.1e,g).
While solutions of the wave equation (see problem 2.5) furnish values of
or a component of rotation
, we often need to know the displacements
(defined in problem 2.1) which are not easily found given
or
. This difficulty can be avoided by using potential functions that are solutions of the wave equations and from which
, hence
also, can be found by differentiation.
Note that derivatives of a solution of a differential equation are also solutions.
The vector operator
(called “del”) and its properties are discussed in Sheriff and Geldart, 1995, Section 15.1.2c.
Solution
From equation (2.1e) and the definition of
, we get for the dilatation
|
|
(2.9b)
|
since
and
. Because
is a solution of the P-wave equation,
must also be a solution.
We have
[see Sheriff and Geldart, 1995, equations (15.13) and (15.9)]. Since
is a solution of the S-wave equation,
is also a solution.
Problem 2.9b
In two dimensions, the potential function
can be defined as
|
|
(2.9c)
|
Show how to obtain the displacements
, the dilatation
, and rotation
from this equation (see Sheriff and Geldart, 1995, Section 15.1.2c and problem 15.5c).
Solution
From equation (2.1d) we see that
is the
-component of
, that is, of
and
, so
-component of
. From Sheriff and Geldart, 1995, equation (15.13) we have
Thus,
|
|
(2.9d)
|
and
|
|
(2.9e)
|
To get the dilatation
, we use equation (2.1e) and Sheriff and Geldart, 1995, problem 15.5c and obtain
|
|
(2.9f)
|
The rotation
can be obtained by taking the curl of equation (2.9c) but an easier method is to substitute equations (2.9d) and (2.9e) in equation (2.1g). This gives
Thus
has only a
-component given by
Continue reading
Also in this chapter
External links
find literature about Potential functions used to solve wave equations
|
|
|
|
|
|
|
|