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194 4. The method of multiple scales
where Determine c (k) and (and confirm that satisfies the
usual condition for the group speed) and find the equation for
Q4.24 A weakly nonlinear wave: NLS II. See Q4.23; repeat all this for the equation
Q4.25 A weakly nonlinear wave: NLS III. See Q4.23; repeat all this for the equation
Q4.26 Kd V NLS. Consider the Korteweg-de Vries (KdV) equation
where is aparameter.Introduce and
(and here and will be corrections to the original c and
because the given KdV equation has already been written in a suitable moving
frame). Seek a solution
where find c(k), and the equation for (This
example demonstrates that an underlying structure of the KdV equation is
an NLS (Nonlinear Schrödinger) equation; indeed, it can be shown that, in
the context of water waves, for example, the relevant NLS equation for that
problem matches to this NLS equation—see Johnson, 1997.)
Q4.27 Ray theory. A wave (moving in two dimensions) slowly evolves, on the scale
so that where
show that
(a) where and
(b) (the eikonal equation);
(c) (so the vector k is ‘irrotational’).
(Given that the energy in the wave motion is E(X, Y, T), it can be shown
that where All this is the basis for ray
theory, or the theory of geometrical optics, which is used to describe the properties
of waves that move through a slowly changing environment.)
Q4.28 Boundary-layer problem I. Use the method of multiple scales to find, completely,
the first term of a uniformly valid asymptotic expansion of the solution of
where