Geology Reference
In-Depth Information
3.2.6 The dispersion relation.
The reader should repeat our analysis using separation constants with
different signs, in order to show that trivial solutions are obtained. With our
selection, it is interesting to observe that the resulting right side of Equation 3.42
can be written as k
x
2
- (
2
/c
2
- k
z
2
) = - k
y
2
or
2
/c
2
= k
x
2
+ k
y
2
+ k
z
2
(3.46)
This is
exactly
the dispersion relation that would have resulted had we directly
substituted elementary solutions of the form
p(x,y,z,t) = exp i( t ± k
x
x ± k
y
y ± k
z
z) (3.47)
in Equation 3.27. Thus, the solution given by Equation 3.45 can be interpreted
as the correct superposition of elementary solutions that satisfies the assumed
rigid wall boundary conditions. Other models, e.g., elastic, attenuative or
porous wall constraints, are likewise modeled. Now, Equation 3.46 can be
manipulated to give
k
z
= ± {
2
/c
2
- k
x
2
- k
y
2
} (3.48)
Here the ± sign indicates the existence of two wave families propagating in
opposite directions along the z axis. Consider the argument of the square root
function. When /c > {k
x
2
+ k
y
2
}, k
z
is real: the wave advances in the z
direction and the eigenfunction in Equation 3-45 is called a “propagating mode.”
The limiting frequency for which k
z
is real is the “cutoff frequency” given by
cut-off
= c {k
x
2
+ k
y
2
} (3.49)
for the (
l,m
) mode (the meaning of
mode
is addressed in the next section).
However, if the input frequency is below cutoff, the argument of the square root
in k
z
= {
2
/c
2
- k
x
2
- k
y
2
} becomes negative, and k
z
is pure imaginary with
k
z
= ± i {k
x
2
+ k
y
2
-
2
/c
2
} (3.50)
If our waveguide is defined on the half-space z > 0, with the excitation source at
z = 0, the sign in Equation 3.50 is chosen so that p(x,y,z,t) physically tends to 0
as z approaches infinity. In this limit, Equation 3.45 becomes
p(x,y,z,t) = P
l,m
cos
l
x/L
x
cos
m
y/L
y
X exp i
t exp(-
{k
x
2
+ k
y
2
-
2
/c
2
} z)
(3.51)
which is a
standing wave
that decays exponentially with z.
3.2.7 Physical interpretation.
To understand how the contrasting solutions given by Equations 3.45 and
3.51 might arise physically, consider a “lumpy” piston located at z = 0, whose
face contains surface irregularities. A theory of “double Fourier series,”
analogous to that for single Fourier series in Chapter 1, can be used to handle
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