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9 Wave Equations
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Then,attimet,thefunctionw(x)istransportedintou(x, t).uiscalledatravelling
wave with velocity v and profile w. When we set:

w k (x) = exp(ik · x) , (9.2)

where k is a given n-dimensional parameter vector, then by the above transport
process we obtain the so called plane wave:

u k (x, t) = exp ik · (x − vt) (9.3)

representing harmonic oscillations. The parameter vector k is called wave vector
of theplanewave u k , its j-th component k j determines the periodicity

2π
p j =
k j

of thewaveprofile w k in direction x j .
Note that a travelling wave of the form (9.1) solves the first (differential)
order linear transport equation

u t = −v · grad x u , (9.4)

out of which by differentiation we can easily obtain the second order linear
anisotropic wave equation:

u tt = a j,l u x j x l , (9.5)
j,l

whereinthiscasea j,l = v l v j .Forgeneralwavemotions,thereal-valuedcoefficient
matrix A = (a j,l ) j,l is assumed to be symmetric and non-negative definite such
that the total wave energy
1
E(u) = (u t ) +(grad u) A grad u dx (9.6)
2
T
2
n
is a time-conserved quantity, with two nonnegative contributions stemming
from the kinetic and potential energies. Equations of the form (9.5) model,
for example, the motion of thin elastic chords (in one dimension), of thin
membranes (two dimensions) and of three dimensional elastic objects under the
assumption of small oscillation amplitudes (which allows to use linear models).
In these applications the solution u represents the displacement and u t the
velocity. Clearly, appropriate initial-boundary conditions have to be imposed.
Other applications include propagation of small amplitude sound waves in gases
and fluids.