solving the wave equation due to D. Bernoulli
A string has been strained between the points
and
of the
-axis. The transversal vibration of the string in the
-plane is determined by the one-dimensional wave equation
 |
(1) |
satisfied by the ordinates
of the points of the string with the abscissa
on the time moment
.
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
 |
(2) |
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:
 |
(3) |
The solutions of these are, as is well known,
 |
(4) |
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
which implies
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
or
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
 |
(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
. But the left sides of these equations are the Fourier sine series of the functions
and
, and therefore we obtain the expressions for the coefficients:
- 1
- K. V AIS AL A: Matematiikka IV. Hand-out Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
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