Geology Reference
In-Depth Information
c
p
= c
(k
x
2
+ k
y
2
+ k
z
2
) /k
z
(3.52)
which is clearly not the “expected” c of Equation 3.27.
In order to understand Equation 3.52, we introduce (in standard Cartesian
notation) a
wavenumber vector
k
= k
x
i
+ k
y
j
+ k
z
k
whose magnitude is k =
(k
x
2
+ k
y
2
+ k
z
2
). First, we explore its physical significance. Consider a
harmonic component p(x,y,z,t) = sin ( t - k
x
x - k
y
y - k
z
z). Surfaces of constant
p (or phase) are described by f(x,y,z) = t - k
x
x - k
y
y - k
z
z = constant, where t
is a parameter. From vector calculus, normals to surfaces are proportional to -
f. But this gradient function is just k
x
i
+ k
y
j
+ k
z
k
. Thus, the wavenumber
vector
k
is perpendicular to the wave front.
The phase velocity.
Now, it is easy to see that the propagation vector
k
for each of the traveling waves (inferred from Equation 3.47) makes an angle
with the z axis given by
cos = k
z
/k = k
z
/ (k
x
2
+ k
y
2
+ k
z
2
) (3.53)
Since Equation 3.52 states that c
p
= c k/ k
z
or c
p
/c = k/k
z
, combination with
Equation 3.53 shows that the phase velocity is related to “c” by
c
p
= c/cos (3.54)
The group velocity.
Note that each component wave carries energy down
the waveguide by the process of
continual reflection
from the walls (via suitable
combinations of Equation 3.47). The energy of a wave is propagated with speed
c in the direction
k
, but the speed with which energy travels
in the z direction
(defined as the group velocity c
g
) must be given by the component of the plane
wave velocity c along the waveguide axis, namely,
c
g
= c cos = c k
z
/ (k
x
2
+ k
y
2
+ k
z
2
) (3.55)
From Equation 3.46, we have = c ( k
x
2
+ k
y
2
+ k
z
2
)
1/2
. Note that the partial
derivative of Equation 3.55 with respect to k
z
is
/ k
z
= c (1/2) (k
x
2
+ k
y
2
+ k
z
2
)
-1/2
(2k
z
)
= ck
z
/ (k
x
2
+ k
y
2
+ k
z
2
)
(3.56 )
Thus, on comparing Equations 3.55 and 3.56, we find that
c
g
=
/ k
z
(3.57)
which is consistent with the general theory of Chapter 2.
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