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9 Wave Equations
154

Fig. 9.3. Wave Breaking

Then a straightforward application of the formula (9.9) gives the solution of
theinitial valueproblem:
1 1
u(x, t) = δ(x + vt)+ δ(x − vt) , (9.10)
2 2
i.e. the initial delta mass is split into two equal parts, each of which is transported
along a characteristic in x, t-space. In particular this means that every point x
in R ‘feels’ the effect of the point source at only one moment in time. In two
and more dimensions the corresponding solution is not compactly supported
1
anymore, butitdecaystozeroas |x| tends to infinity, in particular as √ in
|x|
two dimensions and as 1 in three dimensions. Note that in a one dimensional
|x|
world a sound emitted from a source can only be heard at one instant of time,
not ‘continuously’ (with loudness decaying with distance from the source and
with time) as in our three dimensional world!
A simple computation shows that the function

k
u k (x, t) = exp ik · x ± v t
|k|
is a special plane wave solution of the n-dimensional wave equation

2
n
u tt = v Δu , x ∈ R , t ∈ R