solving the wave equation due to D. Bernoulli

A string has been strained between the points  (0, 0)  and  (p, 0)  of the x-axis.  The vibration of the string in the xy-plane is determined by the one-dimensional wave equationMathworldPlanetmath

2ut2=c22ux2 (1)

satisfied by the ordinates  u(x,t)  of the points of the string with the abscissa x on the time   t(0). The boundary conditionsMathworldPlanetmath are thus


We suppose also the initial conditions

u(x, 0)=f(x),ut(x, 0)=g(x)

which give the initial position of the string and the initial velocity of the points of the string.

For trying to separate the variables, set


The boundary conditions are then  X(0)=X(p)=0,  and the partial differential equationMathworldPlanetmath (1) may be written

c2X′′X=T′′T. (2)

This is not possible unless both sides are equal to a same constant -k2 where k is positive; we soon justify why the constant must be negative.  Thus (2) splits into two ordinary linear differential equations of second order:

X′′=-(kc)2X,T′′=-k2T (3)

The solutions of these are, as is well known,

{X=C1coskxc+C2sinkxcT=D1coskt+D2sinkt (4)

with integration constants Ci and Di.

But if we had set both sides of (2) equal to  +k2, we had got the solution  T=D1ekt+D2e-kt  which can not present a vibration.  Equally impossible would be that  k=0.

Now the boundary condition for X(0) shows in (4) that  C1=0,  and the one for X(p) that


If one had  C2=0,  then X(x) were identically 0 which is naturally impossible.  So we must have


which implies


This means that the only suitable values of k satisfying the equations (3), the so-called eigenvalues, are

k=nπcp(n=1, 2, 3,).

So we have infinitely many solutions of (1), the eigenfunctions




(n=1, 2, 3,) where An’s and Bn’s are for the time being arbitrary constants.  Each of these functions satisfy the boundary conditions.  Because of the linearity of (1), also their sum series

u(x,t):=n=1(Ancosnπcpt+Bnsinnπcpt)sinnπpx (5)

is a solution of (1), provided it converges.  It fulfils the boundary conditions, too.  In order to also the initial conditions would be fulfilled, one must have


on the interval[0,p].  But the left sides of these equations are the Fourier sine seriesMathworldPlanetmath of the functions f and g, and therefore we obtain the expressions for the coefficients:



  • 1 K. Väisälä: Matematiikka IV.  Hand-out Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
Title solving the wave equation due to D. Bernoulli
Canonical name SolvingTheWaveEquationDueToDBernoulli
Date of creation 2013-03-22 16:31:41
Last modified on 2013-03-22 16:31:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Example
Classification msc 35L05
Synonym vibrating stringPlanetmathPlanetmath
Related topic ExampleOfSolvingTheHeatEquation
Related topic EigenvalueProblem