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[parent] solving the wave equation due to D. Bernoulli (Example)

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

$\displaystyle \frac{\partial^2u}{\partial t^2} = c^2\cdot\frac{\partial^2u}{\partial x^2}$ (1)

satisfied by the ordinates $ u(x,\,t)$ of the points of the string with the abscissa $ x$ on the time moment $ t\,(\geqq 0)$. The boundary conditions are thus
$\displaystyle u(0,\,t) = u(p,\,t) = 0.$
We suppose also the initial conditions
$\displaystyle u(x,\,0) = f(x),\quad u_t'(x,\,0) = g(x)$
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

$\displaystyle u(x,\,t) := X(x)T(t).$
The boundary conditions are then $ X(0) = X(p) = 0$, and the partial differential equation (1) may be written
$\displaystyle c^2\cdot\frac{X''}{X} = \frac{T''}{T}.$ (2)

This is not possible unless both sides are equal to a same constant $ -k^2$ 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:
$\displaystyle X'' = -\left(\frac{k}{c}\right)^2 X,\quad T'' = -k^2T$ (3)

The solutions of these are, as is well known,
\begin{align*}\begin{cases}X = C_1\cos\frac{kx}{c}+C_2\sin\frac{kx}{c}\\ T = D_1\cos{kt}+D_2\sin{kt}\\ \end{cases}\end{align*} (4)

with integration constants $ C_i$ and $ D_i$.

But if we had set both sides of (2) equal to $ +k^2$, we had got the solution $ T = D_1e^{kt}+D_2e^{-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 $ C_1 = 0$, and the one for $ X(p)$ that

$\displaystyle C_2\sin\frac{kp}{c} = 0.$
If one had $ C_2 = 0$, then $ X(x)$ were identically 0 which is naturally impossible. So we must have
$\displaystyle \sin\frac{kp}{c} = 0,$
which implies
$\displaystyle \frac{kp}{c} = n\pi \quad (n \in \mathbb{Z}_+).$
This means that the only suitable values of $ k$ satisfying the equations (3), the so-called eigenvalues, are
$\displaystyle k = \frac{n\pi c}{p} \quad (n = 1,\,2,\,3,\,\ldots).$
So we have infinitely many solutions of (1), the eigenfunctions
$\displaystyle u = XT = C_2\sin\frac{n\pi}{p}x \left[D_1\cos\frac{n\pi c}{p}t+D_2\sin\frac{n\pi c}{p}t\right]$
or
$\displaystyle u = \left[A_n\cos\frac{n\pi c}{p}t+B_n\sin\frac{n\pi c}{p}t\right] \sin\frac{n\pi}{p}x$
$ (n = 1,\,2,\,3,\,\ldots)$ where $ A_n$'s and $ B_n$'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
$\displaystyle u(x,\,t) := \sum_{n=1}^\infty\left(A_n\cos\frac{n\pi c}{p}t+B_n\sin\frac{n\pi c}{p}t\right)\sin\frac{n\pi}{p}x$ (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
$\displaystyle \sum_{n=1}^\infty A_n\sin\frac{n\pi}{p}x = f(x),$
$\displaystyle \sum_{n=1}^\infty B_n\frac{n\pi c}{p}\sin\frac{n\pi}{p}x = g(x)$
on the interval $ [0,\,p]$. But the left sides of these equations are the Fourier sine series of the functions $ f$ and $ g$, and therefore we obtain the expressions for the coefficients:
$\displaystyle A_n = \frac{2}{p}\int_{0}^p\!f(x)\sin\frac{n\pi x}{p}\,dx,$
$\displaystyle B_n = \frac{2}{n\pi c}\int_{0}^p\!g(x)\sin\frac{n\pi x}{p}\,dx.$

Bibliography

1
K. V¨AISÄLÄ: Matematiikka IV. Hand-out Nr. 141.    Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).



"solving the wave equation due to D. Bernoulli" is owned by pahio.
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See Also: example of solving the heat equation, eigenvalue problem

Other names:  vibrating string

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Cross-references: coefficients, expressions, Fourier sine series, interval, order, converges, series, sum, functions, eigenfunctions, eigenvalues, equations, implies, solutions, second order, linear differential equations, negative, positive, sides, partial differential equation, variables, initial conditions, boundary conditions, abscissa, ordinates, wave equation, points
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This is version 9 of solving the wave equation due to D. Bernoulli, born on 2007-01-01, modified 2007-01-15.
Object id is 8708, canonical name is SolvingTheWaveEquationByDBernoulli.
Accessed 2474 times total.

Classification:
AMS MSC: 35L05 (Partial differential equations :: Partial differential equations of hyperbolic type :: Wave equation)
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