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Once the wavenumber distribution k(x) is available, the energy or
momentum density can be obtained by integrating the following steady forms,
( r k (k,x)E)/ x = E{2 i + r t (k,x)/ r } (2.122)
( r k (k,x)M)/ x = M{2 i - r x (k,x)/k} (2.123)
The derivatives in Equations 2.122 and 2.123 are ordinary derivatives, and
standard numerical integration procedures may be used. Later, KWT is used to
model bending waves propagating on drillstrings in the presence of nonuniform
axial strain. E(x) and M(x) are shown to vary inversely with group velocity
r k (k,x), as expected from Equations 2.122 and 2.123. Interestingly, all
solutions to Equation 2.120 for k(x) admit small values of r k near the neutral
point, thus demonstrating the natural tendency toward lateral instability.
2.5.6 Waves in nonuniform moving media.
We defer our discussion of KWT in moving media (e.g., ocean currents,
steady winds, and gusts) until Chapter 7 where offshore water wave applications
are given. For now, we summarize the principal results. If r (k,x,t) and
i (k,x,t) describe the wave propagation problem without any flow, then the
kinematic equations for the problem with a mean flow U(x,t) are
dk/dt =
r x (k,x,t) + U x k
(2.124)
dx/dt = U(x,t) +
r k (k,x,t)
(2.125)
E/ t + ((U(x,t) +
r k (k,x,t))E)/ x =
(2.126)
E{2
i + {
r t (k,x,t) + U
r x (k,x,t) - k
r k (k,x,t) U x }/
r }
M/ t +
((U(x,t) +
r k (k,x,t))M)/ x =
(2.127)
M{ 2 i - r x (k,x,t)/k - U x }
2.5.7 Average Lagrangian formalism.
In this topic, only the simplest “recipes” obtained in kinematic wave theory
appear; in order to understand the complete theory and the full power behind
KWT, training is required in variational calculus (Gelfand and Fomin, 1963).
We give a flavor of these methods only; the reader may ignore this work without
loss of continuity.
2.5.8 Example 2-8. Wave action conservation.
We consider the transverse vibrations of a string, which satisfy l 2 u/ t 2 -
T 2 u/ x 2 = 0, allowing l = l (x) and T = T(t). The kinetic energy density is
1/2 l ( u/ t) 2 , while the potential strain energy density is 1/2 T ( u/ x) 2 . The
Lagrangian L of the dynamical system equals their difference, that is, L = 1/2
l ( u/ t) 2 - 1/2 T ( u/ x) 2 . The uniform plane wave solution is u(x,t) = a sin
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