This java applet is a simulation that demonstrates waves on a loaded string. A loaded string is a very light (massless) string with a number of small masses attached to it at equal intervals along the string. As the number of masses increases, the behavior of the loaded string more closely approximates the behavior of a real string where the mass is distributed continuously along its length.

A loaded string with n loads has n normal modes of oscillation. The first mode is called the fundamental, and involves the entire string vibrating up and down at a frequency determined by the string's length, tension, and mass. The other modes are called harmonics, and involve parts of the string (called nodes) standing still while the rest of the string vibrates.

The frequency of the harmonics is some multiple of the fundamental frequency. With a real (continuous) string, the frequency of the second harmonic is twice the frequency of the fundamental, and the third harmonic is three times the fundamental, and so on. With a loaded string with a finite number of masses, that's not quite correct, but it's reasonably close, at least for the first few harmonics.

When a string is vibrating, more than one mode is typically present at once, but the fundamental is typically the loudest.

At the top of the applet on the left you will see the string. By default, the number of loads is set to 28. You won't see any loads on the string, because if the number of loads is more than 18, the applet does not show them, to avoid clutter.

To set the string in motion, click "Center Pluck" or "Fundamental". If you click "Fundamental" then the string will vibrate at the fundamental frequency. If you click "Center Pluck", the string will be plucked in the center; this will cause some of the harmonics to be excited as well, although the fundamental will still dominate. If you click "Clear", the string will be at rest again.

Below the function you will see a graph showing each normal mode's contribution to the string's vibration. There are two sets of terms; on top are the magnitude terms, which shows the amplitude of each normal mode, and on the bottom are the phase terms. The fundamental is on the left and higher harmonics are on the right. Since the higher harmonics 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 harmonics, it will turn yellow, and the corresponding harmonic will be drawn on top of the function in yellow (unless it's too small to see). So if you move the mouse over all the harmonics, you can see each of the terms individually.

(One thing to keep in mind when looking at the magnitude of each harmonic is that the scale is not linear. If it were linear, it would look like the higher harmonics 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 string in one of two ways. You can click on it directly; in this case, it will be plucked at that point. Or, you can modify the normal modes.

The "Mouse" popup controls what happens when you click on the string. The default setting is "Pluck string", which causes the string to be plucked where you click. If you set the popup to "Shape string", you can edit the shape of the string directly. If you set it to "Touch string", the string will be touched at the point you click; this can be used to quiet the fundamental and cause one of the harmonics to dominate. And if you set it to "Apply force", a force will be applied to the string at the point you click, for as long as you hold the mouse button down. This works best with some damping.

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 harmonics. 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 Phasors", it will show phasors for the first three modes. If you set it to "Display Modes", it will display graphs of the first twelve active modes.

If you set it to "Display Left+Right", it will decompose the wave on the string into two components, one travelling left and one travelling right. The wave on the string is called a standing wave because it is not moving in any direction. But if you add up the displacements of the travelling waves at each load, you get the standing wave. Notice that the travelling waves always have equal and opposite displacements at the edges because they have to add up to zero there.

The travelling waves will often have a fixed shape, but in some cases they will change shape slightly as they move. Part of this is damping, but even without damping there may be a shape change. The shape change is caused by the fact that not all the harmonics move at the same speed along the string. The lower modes move at about the same speed but the higher modes move slower. This is called dispersion. If the string were continuous, with the mass distributed evenly along its entire length, there would be no dispersion. But the mass is localized at a fixed number of points (at the loads), so the string is dispersive. To see an example of this, turn off damping, turn on "Display Left+Right", and turn up one of the middle harmonics and one of the highest harmonics (by "turn up" I mean set the magnitude of that harmonic to the maximum). Notice that the travelling waves are changing shape as they move. If you increase the number of loads without changing anything else, this effect will be reduced. (Dispersion will still be present, but you will need to select higher modes to see it.)

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

The "Driving Force" checkbox allows you to push the string with a periodic driving force that acts at the center of the string. The magnitude of the force is shown with an arrow that oscillates back and forth. The frequency of the driving force's oscillation can be controlled with the "Force Frequency" slider. The driving force won't accomplish very much unless the force frequency is close to the string's resonance frequency (the fundamental). To set it there, click the "Resonance Frequency" button.

The "Sound" checkbox allows you to hear the sound the string would make. To make this more realistic, move the damping slider to the middle. Many of the higher frequencies will not be heard because Java cannot play sounds with frequencies that high.

The "Touched" checkbox simulates the effect of touching a string in the middle in order to quiet the fundamental. If this is on, and you pluck the string, the first harmonic will dominate.

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 string down. It is proportional to the speed of the string, so high harmonics are damped more than lower ones. The default setting is fairly low so you may want to set this higher in order to get more realistic behavior.

The "Number of Loads" slider will adjust the number of loads on the string. This can be set as low as one. If you change the number of loads then you also change the number of normal modes.

The "Tension" slider will adjust the tension. Lowering the tension will reduce the frequency of the string's vibration but will increase the amplitude.