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150 3. Further applications

Q3.6 A nonlinear wave equation. A wave is described by the equation

where Seek a solution (for right-running waves only) in the form

find the first two terms and demonstrate the existence of a breakdown when
Now introduce and show that the leading term
in the expansion valid for satisfies the equation

This calculation is now extended: define

and then determine f and g so that

where satisfies (*) and satisfies the corresponding equation for left-going
waves. (You may assume that both f and g possess Taylor expansions about
x + t and x – t, respectively.)
Q3.7 A multi-wave speed equation. A particular wave profile, u(x, t; with as
is described by the equation

where and c are constants (independent of Show that, if
then on the time scale the wave moving at speed and the wave
moving at speed each decay exponentially (in time), to leading order as
Show, also, that on the time scale the wave moving at speed
c has diffused a distance about the wave front and, to leading order,
it satisfies a Burgers equation (see Q3.5).
Q3.8 Water waves with weak nonlinearity, damping and dispersion. The propagation of a
one-dimensional wave on the surface of water can be modelled by the equations