Wave equation
The vertical displacement $u(x,t)$ of the vibrating string of length L is determined from:
\begin{array}{rl}
u_{tt}(x,t) = a^2\,u_{xx}(x,t), & 0 < x < L,\quad t>0; \\
u(0,t) = 0,\; u(L,t)=0, & t>0;\\
u(x,0) = f(x), \;\;u_t(x,0)=g(x), & 0< x < L.
\end{array}
Simulation
In the following simulation, the wave $u(x,t)$ is graphed as a function of $x$ for various times.
Things to try:
- Change the initial conditions $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$.
- Increase $n$, the number of terms in the solution.
- Change the length $L$.
- Click on Play button to start simulation.
Sorry, the simulation is not supported for small screens.
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Download
Reference
[1] Trench, William F., "Elementary Differential Equations with Boundary Value Problems" (2013). Books and Monographs. Book 9.
Link:
The University of Queensland