20.4 THE WAVE EQUATION
20.4 The wave equationWe have already found that the general solution of the one-dimensional wave
equation is
u(x, t)=f(x−ct)+g(x+ct), (20.26)wherefandgare arbitrary functions. However, the equation is of such general
importance that further discussion will not be out of place.
Let us imagine thatu(x, t)=f(x−ct) represents the displacement of a string attimetand positionx. It is clear that all positionsxand timestfor whichx−ct=
constant will have the same instantaneous displacement. Butx−ct= constant
is exactly the relation between the time and position of an observer travelling
with speedcalong the positivex-direction. Consequently this moving observer
sees a constant displacement of the string, whereas to a stationary observer, the
initial profileu(x,0) moves with speedcalong thex-axis as if it were a rigid
system. Thusf(x−ct) represents a wave form of constant shape travelling along
the positivex-axis with speedc, the actual form of the wave depending upon
the functionf. Similarly, the termg(x+ct) is a constant wave form travelling
with speedcin the negativex-direction. The general solution (20.23) represents
a superposition of these.
If the functionsf andg are the same then the complete solution (20.23)represents identical progressive waves going in opposite directions. This may
result in a wave pattern whose profile does not progress, described as astanding
wave. As a simple example, suppose bothf(p)andg(p) have the form§
f(p)=g(p)=Acos(kp+).Then (20.23) can be written as
u(x, t)=A[cos(kx−kct+)+cos(kx+kct+)]=2Acos(kct)cos(kx+).The important thing to notice is that the shape of the wave pattern, given by the
factor inx, is the same at all times but that its amplitude 2Acos(kct) depends
upon time. At some pointsxthat satisfy
cos(kx+)=0there is no displacement at any time; such points are callednodes.
So far we have not imposed any boundary conditions on the solution (20.26).The problem of finding a solution to the wave equation that satisfies given bound-
ary conditions is normally treated using the method of separation of variables
§In the usual notation,kis the wave number (= 2π/wavelength) andkc=ω, the angular frequency
of the wave.