This java applet is a simulation that demonstrates standing flexural waves in a bar.

At the top of the applet on the left you will see the bar. To set the bar in motion, click on it and drag around to deform it, or click "Fundamental". If you click "Clear", the bar will be at rest again. (The vertical motion of the bar is greatly exaggerated so you can see it clearly.)

Below the bar you will see a graph showing each normal mode's contribution to the bar's vibration. There are two sets of indicators; on top are the magnitude indicators, which shows the amplitude of each normal mode, and on the bottom are the phase indicators. The fundamental is on the left and the higher modes are on the right. Since the higher modes oscillate at a higher frequency than the fundamental, the phase terms will move faster on the right.

If you move the mouse over one of the modes, it will turn yellow, and the corresponding mode will be drawn on top of the bar in yellow (unless it's too small to see). So if you move the mouse over all the modes, you can see each of the terms individually.

(One thing to keep in mind when looking at the magnitude of each mode is that the scale is not linear. If it were linear, it would look like the higher modes all had zero magnitude because their contribution is so small. For small magnitudes, the scale is logarithmic; then about 1/4 of the way up the scale it switches to linear. The same is true for negative magnitudes. There is a horizontal gray line drawn at the point where the scale switches.)

You can modify the bar in one of two ways. You can click on it directly to deform it, or you can modify the normal modes.

The Setup popup can be used to view some interesting cases. The choices are:

• bar, free: this is a bar with free ends.
• bar, hinged: this is a bar with hinged ends, so it can rotate but not move at the ends. This is a bar with simply supported edges.
• bar, clamped: this is a bar with clamped ends, so it can't move or rotate at the ends.
• bar, clamped/free: this is a bar clamped at the left end and free at the right end, similar to one tine of a tuning fork.
• bar, hinged/clamped: this is a bar hinged at the left end and clamped at the right end.
• string, pinned: this is a perfectly flexible string under tension, pinned at both ends.
• string, pinned/free: this is a perfectly flexible string under tension, fixed at the left end and pinned to a moveable support at the right end so it can move up and down while still remaining under tension.
• string, free/free: this is a string pinned to a moveable support at both ends.
• stiff string, pinned: this is a string with adjustable stiffness, pinned at both ends.
• stiff string, clamped: this is a string with adjustable stiffness, clamped at both ends.

The Mouse popup controls what happens when you click on the bar. The default setting is Shape bar, where you can edit the shape of the bar directly. If you set it to Apply static force, then a force will be applied to the bar at the point you click. (If the bar has free edges, then it will not be allowed to translate or rotate as the force is applied.)

The Display popup can be used to display some additional information. By default it is set to "Display Phases", which shows the magnitudes and phases of the modes. If you set it to "Display Phase Cosines", you will see the cosines of the phases rather than the phases themselves. If you set it to "Display Modes", it will display graphs of the first twelve active modes.

The Stopped checkbox allows you to stop or start the simulation.

The Sound checkbox allows you to hear the sound the bar would make. Every time you disturb the bar, it will play the noise made by the bar (but only for one second). Any frequency over 4000 Hz will not be heard because Java cannot play sounds with frequencies that high. (Unless you try the Java 2 version.)

By turning on the sound checkbox and varying the parameters, you can make a variety of different realistic sounds. For example with the bar, free setup, click on the bar to make a notch in it, making it as deep as possible, and then release the mouse. This will make a sound like a metal bar being struck with a hard mallet. A narrow notch will make a sharper sound, and a wide notch (especially a curved one) will make a duller sound. A notch on the end will make a sharper sound as well, because higher modes will be more involved.

Varying the base frequency will simulate bars of different lengths. If the fundamental frequency is around 1000 Hz, then striking the bar near the center will make a sound like a bell or glockenspiel. Varying the damping will also change the character of the sound.

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 bar down. High modes are damped more than lower ones.

The Resolution slider will adjust the accuracy of the simulation. The higher the resolution, the slower the applet may go, but the more accurate the mode frequencies and shapes will be.

The Base Frequency slider adjusts the frequency of the fundamental.

Click here to go to the applet.