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where Write find
and then determine in the case On the basis of this evi-
dence, confirm that you have a two-term, uniformly valid asymptotic expansion
in
Q3.4 The classical (model) Boussinesq equations for water waves. These equations are writ-
ten as the coupled pair

where is the horizontal velocity component in the flow, the
surface displacement i.e. the surface wave, and is a constant independent of
Find the first terms in the asymptotic expansions
for x = O(1), t = O(1), as (the near-field). Then introduce
the far-field variables: for right-running waves, and hence
find the equations defining the leading order; show that the equation for u
takes the form

the Korteweg-de Vries equation.
Show that a solution of this equation is the solitary wave

where is a free parameter.
Q3.5 Long, small-amplitude waves with dissipation. A model for the propagation of long
waves, with some contribution from dissipation (damping), is

where and are positive constants independent of Follow the same
procedure as in Q3.4 (near-field then far-field, although here the right-going
characteristic will be Show that, in the far-field, the leading term,
for u (say), satisfies an equation of the form

the Burgers equation. Show that this equation has a steady-state shock-profile
solution

for suitable constants C and (> 0), which should be identified. (This
solution is usually called the Taylor shock profile; Taylor, 1910.)