solving the wave equation due to D. Bernoulli
A string has been strained between the points and of the -axis. The vibration of the string in the -plane is determined by the one-dimensional wave equation
satisfied by the ordinates of the points of the string with the abscissa on the time . The boundary conditions are thus
We suppose also the initial conditions
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 , and the partial differential equation (1) may be written
This is not possible unless both sides are equal to a same constant where is positive; we soon justify why the constant must be negative. Thus (2) splits into two ordinary linear differential equations of second order:
The solutions of these are, as is well known,
with integration constants and .
But if we had set both sides of (2) equal to , we had got the solution which can not present a vibration. Equally impossible would be that .
Now the boundary condition for shows in (4) that , and the one for that
If one had , then were identically 0 which is naturally impossible. So we must have
This means that the only suitable values of satisfying the equations (3), the so-called eigenvalues, are
So we have infinitely many solutions of (1), the eigenfunctions
where ’s and ’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
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
- 1 K. Väisälä: Matematiikka IV. Hand-out Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
|Title||solving the wave equation due to D. Bernoulli|
|Date of creation||2013-03-22 16:31:41|
|Last modified on||2013-03-22 16:31:41|
|Last modified by||pahio (2872)|