Page 153 -

149
9. Wave Equations

Waves occur in many aspects of our daily lives and in the nature which sur-
rounds us. Just take a stone and throw it into a resting water surface: you will
observe a surface wave, spreading in concentric circles around the impact point
on the water. Or think of the high breaking waves in the ocean which are so
highly desirable for surf champions. Less pleasantly, there are the energy waves
generated by potent seaquakes, which travel under the ocean surface with the
speed of about thousand kilometres per hour and turn into deadly tsunami
water waves close to beaches. Other examples are the sound waves, generated
by our speech, propagating in the air to the partner of our conversation, elec-
1
tromagnetic waves described by the Maxwell equations, light propagating in
spherical waves from a source and, even more fundamentally, as established by
2
quantum mechanics [19] , there is a matter-wave duality which basically states
that even particles with positive mass (say, electrons) have wave-like features
(e.g. delocalisation).
So how is wave motion characterized? Webster’s dictionary gives the follow-
ing definition:
a disturbance or variation that transfers energy progressively from point
topointinamediumandthatmaytaketheformofanelasticdeformation
or of a variation of pressure, electric or magnetic intensity, electric
potential, or temperature.

Clearly, this refers to the time-dependent transport of some physical quan-
tity (e.g. energy) in certain spatial directions of a medium, such that typical
characteristics of the quantity are maintained during the transport process.
We remark that the transport of, say, energy is typically affected WITHOUT
significant transport of particles of the medium. 3
As maybe the most simple example, consider a (possibly) complex-valued
n
function w = w(x), defined on R , and set
n
u(x, t) = w(x − vt), x ∈ R , t ∈ R , (9.1)
where v is a given n-dimensional parameter vector, x denotes the spatial variable
and t represents time. Obviously, this function in space-time can be interpreted
inthe following way: take w = w(x) and move it with speed |v| inthe direction v .
|v|
1 http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Maxwell.html
2 see http://www.kfunigraz.ac.at/imawww/vqm/ for a visualisation attempt
3
For enlightening animations of wave motion we refer to the webpage http://
www.kettering.edu/∼drussell/demos.html