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a
2
/ t +
r
k
(k)
a
2
/ x =
(2.58)
{2
i
-
r
kk
f '[x -
r
k
(k) t]/{1 + t
r
kk
(k) f '[x -
r
k
(k) t]}}a
2
The 2
i
term describes amplitude changes due to damping or instability,
while the second term in the { } brackets describes geometric changes due to ray
convergence or divergence. When kinematic shocks form, the above
denominator vanishes, and the right side of Equation 2.58 becomes infinite.
Thus, singularities in k(x,t) are accompanied by singularities in the energy. If
shocks do not form, we can obtain a closed form solution describing the
variation of a
2
along a ray. From elementary calculus, the total differential da
2
satisfies da
2
= a
2
/ t dt + a
2
/ x dx. Upon division by dt, we obtain da
2
/dt =
a
2
/ t + dx/dt a
2
/ x. If we compare this result with Equation 2.58, we find
da
2
/dt = {2
i
-
r
kk
f '[x-
r
k
(k) t]/{1+t
r
kk
(k) f '[x-
r
k
(k) t]}}a
2
(2.59)
provided that we follow the rays defined by Equations 2.44 and 2.45.
Again, recall that along each such ray, k(x,t) remains a constant equal to its
initial t = 0 value. Consequently, since the right side of Equation 2.59 is
constant, it follows that
a
2
(t) = a
2
0
exp{2
i
-
r
kk
f '[x-
r
k
(k)t]/{1+t
r
kk
(k)f '[x-
r
k
(k)t]}}t (2.60)
Unlike the amplitude laws given in the previous section, where either / t = 0 or
/ x = 0, the generalized results derived here apply to wave motions that vary
unsteadily in both t and x.
2.3.4 Example 2-6. Modeling dynamically steady motions.
Analytical solutions can be obtained for another broad class of problems,
those in which a wavemaker excites a system with uniform properties at
constant frequency
0
. For such problems, the wavenumber field k(x) is
uniquely determined by an equation nonlinear in k, i.e.,
0
=
r
(k)
(2.61)
where the real dispersion function
r
(k) is prescribed.
Setting a
2
/ t = 0 in Equation 2.55, we obtain d{
r
k
(k)a
2
}/dx = 2
i
a
2
. If
we now carry out the ordinary differentiation, we have, successively,
r
k
(k) da
2
/dx + a
2
r
kk
(k)dk/dx = 2
i
a
2
(2.62)
r
k
(k) da
2
/dx = (2
i
-
r
kk
(k)dk/dx) a
2
(2.63)
da
2
/a
2
= 2
i
(dx/
r
k
(k)) -
r
kk
(k)dk/dx (dx/
r
k
(k))
(2.64)
da
2
/a
2
= 2
i
dt -
r
kk
(k)/
r
k
(k) dk
(2.65)
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