Vibrations on a spherical membrane using GNU Scientific Library and gnuplot

Introduction

I’ve always been fascinated by the idea that there are a set of normal vibrational modes orthogonal to each other, and that any arbitrary vibration of some object can be written as a linear combination of those modes. It’s very akin, in my mind, to writing a time-domain signal alternatively as a frequency domain signal in that writing and manipulating the signal in a different domain can lead to some very profound insights.

Not to mention that the patterns are very pretty :)

I decided that I wanted to explore the patterns made by combining these different vibrational modes in an interactive way. While I figure that Mathematica very likely has the tools needed to build this sort of thing, I thought this might be a good way to flex some of my other programming skills. As such, I decided to code it up in C/C++ using some very helpful libraries:

• The Gnu Scientific Library (GSL) for calculating the values of the spherical harmonics along the surface
• gnuplot for plotting the surface

Brief primer on spherical harmonics

Spherical harmonics are special functions which can arise in solving differential equations in spherical coordinates. These functions form a set of orthogonal functions on the sphere, much like Hermite polynomials or simple sine waves might form a set of orthogonal functions for their own spaces.

\[Y_\ell^m\,(θ, \varphi) = e ^{i m \varphi} P_\ell\,(cos{θ})\]

The code proper

Note that you can find a linke to this blah

In order to make this program interactive, I included a very simple interactive prompt for manipulating the amplitudes of each vibrational mode.

int interactive_prompt()
{
    while (true)
    {
	char* line = readline( "sph-harmonics> ");

	char command[100];
	size_t l, m;
	double scale;

	int n_results = sscanf(line, "%s %d %d %lf", command, &l, &m, &scale);

	if ( n_results != 4 )
	{
	    printf("That's not a valid command!\n");
	    free(line);
	    continue;
	}

	std::unique_lock<std::mutex> guard(sph_harm::scales_mutex);
	printf("Settings harmonic with l=%d, m=%d to %f\n", l, m, scale);
	sph_harm::scales[l][m] = scale;
	free(line);
    }
    
    return 0;
}

Resources

https://stackoverflow.com/questions/6803395/child-process-receives-parents-sigint

http://www.dreamincode.net/forums/topic/242335-unix-pipes-and-bidirectional-communication/