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This java applet is a simulation that demonstrates wave motion in a perfectly elastic circular membrane (like a drum head).

An ideal continuous membrane has an infinite number of vibrational modes, each with its own frequency. The lowest-frequency mode is called the fundamental, and involves the entire membrane vibrating up and down at a frequency determined by the membrane's size, tension, and mass. The other modes involve parts of the membrane (called nodal lines) standing still while the rest of the membrane vibrates. When a membrane is vibrating, more than one mode is typically present at once.

At the top of the applet on the left you will see the membrane. To set it in motion, click Fundamental. If you click Clear, it will be at rest again.

Below the membrane you will see a graph showing each normal mode's contribution to the membrane's vibration. The modes are laid out in the following order:

0,1 1,1a 1,1b 2,1a 2,1b ...
0,2 1,2a 1,2b 2,2a 2,2b ...
... ... ... ... ... ...

where m,n refers to the mode with m nodal diameters and n nodal circles. Each mode with nonzero m is doubly degenerate; the difference between the two modes is a rotation. By combining the two degenerate modes in different proportions you can rotate the mode arbitrarily; by combining the two modes with different phases you can get the nodal diameters to rotate with time.

Each element of the grid has a color which indicates the presence or absence of the mode it represents. Black means the mode is not present; green means the mode is present with a positive coefficient, and red means it is present with a negative coefficient.

In addition, each mode may have a phase shift, which indicates that its oscillatory cycle leads or lags the others. This is indicated by a blue line.

You can add or remove a mode by clicking on it. If you click and drag up and down you can fine-tune the magnitude of the mode. If you drag left and right you can alter the phase shift.

The Mouse popup controls what happens when you click on the membrane. If you set it to Poke membrane, the membrane will be poked at the point where you click, affecting the entire membrane. If you set the popup to Strike membrane, the membrane will be struck at the point where you click, affecting only a small area surrounding that point.

If you set it to Adjust view angle or Adjust view zoom, you can adjust the orientation or size of the 3-d view.

The Display popup can be used to control whether the 3-d or 2-d view is shown. By default it is set to Display 3-d Only. To show a 2-d representation, you can select Display 2d Only or Display 3d+2d. The 2-d representation often gives you a clearer picture of what is going on.

The 3d View popup can be used to control the look of the 3-d view. By default it is set to Solid, but you can also see a wireframe view by selecting one of the other options.

The Stopped checkbox can be used to stop the motion of the membrane for a moment.

The Show Frequencies checkbox can be used to show or hide the frequency grid. By default, it is shown.

The Color checkbox can be used to turn off color. This may look better and/or it may make the applet run faster.

The Sound checkbox allows you to hear the sound the membrane would make. High frequencies (above 4000Hz) will not be heard. This checkbox is only present if you have Java 2 installed.

The Simulation Speed slider controls how fast the simulation will proceed.

The Damping slider controls how much damping there is. Damping is a force that slows the membrane down. High harmonics are damped more than lower ones. The default setting is set to zero, so you may want to set this higher in order to get more realistic behavior.

The Brightness slider controls the brightness, just like on a TV set. This can be used to view faint waves more easily.

The Resolution slider will adjust the fineness of the rendering. This can be set lower to speed up the applet.

The Base Frequency slider will adjust the fundamental frequency of the membrane, when the Sound checkbox is on. Normally this would be determined by the tension, size, and mass of the membrane.

Back to the applet.
Acknowledgement: This applet is courtesy of Paul Falstad.

Version 1.6a, posted 21/11/05