String Vibration problem

Problem definition

String vibration equation is expressed by the function of space and time $y(x,t)$: $$y(x,t)\; =\; u(x)\phi (t).\; t:\; time,\; x:\; x\; position,\; y:\; y\; position,\; amplitude.$$
Initial string shape is given: $$y=g(x)\; at\; t=0$$
Initial string velocity is zero: $$\frac{\partial y(x,0)}{\partial t}=\; 0$$
Both end of the string is fixed: $$y(0,t)\; =\; 0,\; y(1,t)\; =\; 0$$
Function $y(x,t)$ follows following partial differential equation: $$u\frac{2^{}\phi }{d{2}^{}}=c^2\phi \frac{2^{}u}{2^{}},\; c:\; constant,c^2=\frac{tension}{density}.$$

Solution

Above problem is solved by Jean d'Alembert in 1747. Following more explicit solution is found by Daniel Bernoulli in 1755 : $$y(x,t)\; =\sum _{\mathrm{n=1}}^{\infty }{A}_{\mathrm{n}}sin(n\pi x)cos(n\pi ct),$$ $$g(x)\; =\sum _{\mathrm{n=1}}^{\infty }{A}_{\mathrm{n}}sin(n\pi x)$$ Where $g(x)$ coefficient $A_{\mathrm{n}}$is calculated using Discrete Sine Transform (but Fourier Transform is not known at that time).

Windows App

Windows App: https://sourceforge.net/projects/playpcmwin/files/others/WWStringVibration103.zip/download
Source code : https://sourceforge.net/p/playpcmwin/code/HEAD/tree/PlayPcmWin/WWStringVibration/

How to install

Extract Zip file to create folder contains WWStringVibration.exe and accompanying DLLs.

How to use

Run WWStringVibration.exe. Edit $g(x)$ using mouse. Press Start to simulate string vibration.

How to uninstall

Just delete downloaded files.

License

MIT License.