Dispersive Waves of Physical Systems
Many physical systems support dispersive waves. Examples of commonly encountered dispersive waves are smallamplitude water waves, waves in stratified fluid, elastic waves, and magnetohydrodynamic waves. A common feature of these dispersive waves is that they are governed by linear partial differential equations with constant coefficients. The following are some of these equations.
a.
KleinGordon equation
b. Beam equation
д2ф + 2 д4ф = 0 W + p 9X4 = °
c. Linearized Korteweg de Vries equation for water waves
дф д 3ф
~dt + p~X “ 0
d. Linearized Boussinesq equation for water and elastic waves
Э2ф _ 2 дф _ 2 д4ф = 0 dt2 а дх2 ^ дХ2дt2
Because these equations are linear with constant coefficients, they can be readily solved by FourierLaplace transform. The FourierLaplace transforms of these equations are
(a) (of2 _ (fa2 _ p2)ф = H1 (а, ш).
(b) (of2 _ p2a4)ф = H2(а, ш).
(c) (ш + pa3)ф = H3(a).
(d) (ш2 _ (fa2 + p2a2of2)ф = H4(а, ш).
The righthand side of each of these equations represents some arbitrary initial conditions. The bracket multiplying the transform of the unknown on the left side of each equation is called the dispersion function, which will be denoted by D(a, m). The dispersion relation of the dispersive waves is given by the zeros of the dispersion function; i. e.,
D(a, m) = 0 ^ ш = ш(а).
Eq. (4.1) is a relationship between wave number a and angular frequency ш.
The solution of a dispersive wave system in wave number space, in general, may be written in the following form:
H(а, ш)
и =————
D(a, ш)
The corresponding solution in physical space is found by inverting the transforms of Eq. (4.2) as follows:



Suppose, for real a, the dispersion relation has a simple real zero. That is, ш(а) is a solution of
D(a, ш(а)) = 0.
Thus, for the шintegral of Eq. (4.3), there is a pole lying on real шaxis. The contribution of this pole can be found by evaluating the шintegral by the Residue Theorem. This gives, t > 0
и(х, t)
where Ф = a(x/t) — m(a) is the phase function of the integral. For large t, Eq. (4.5) may be evaluated asymptotically by the method of stationary phase (see Appendix B). The stationary phase point as is given by
Subscript s indicates the evaluation at a = as(x/t). In the following, the subscript s is dropped with the understanding that a is as(x/t) and m is m(as). The asymptotic formula may be rewritten in the compact form as follows:
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