10.4 Simulation of the Wave Equation for a Vibrating String

At the end of this chapter we consider in Fig.10.5 the simulation of a vibrating string as solution of the wave equation. Fig.10.4 shows three snapshots as examples from the simulation in Fig.10.5. The left part of the picture shows the “start impulse”, a Gaussian concentrated in the middle of the string with maximum 1. In the middle of the picture follows the situation shortly after the start: two Gaussian impulses of height 12 run into opposite directions. They are finally reflected at the ends of the string and interfere with each other in the picture on the right, which results in the reconstruction of the original form and amplitude, but with a negative sign after the first reflection


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Figure 10.4: Propagation of an excitation on a string that is fixed on both ends; Gaussian as initial impulse for t = 0 on the left; two pulses run in opposite direction in the middle; on the right the reconstruction with negative amplitude after reflection at the ends is shown

Gaussian impulses and symmetric wave functions propagate on the string unchanged for a long time, as long as no damping is taken into account in the wave equation.


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Figure 10.5: Time development of an original short triangular pulse at the end of the string. The picture shows the state after the first reflection at the end of the string

The interactive figure 10.5 show the situation a short time after the start on triangular impulse, that was originally concentrated at the end of the string. After some time one observes deviations, that are due to the discontinuity of the first derivative at the beginning and end of the impulse. This example also demonstrates the limits of the numerical computation.

A selection menu contains the following start functions for the initial deflection of the string:

There is a parameter a in most of the functions, that can be changed. The formulas for the initial deflections themselves are editable, such that many more start situations can be simulated.

The description pages contain further details and suggestions for experiments. This animation is aesthetically quite pleasing, because it gives the music lover hints about the tone qualities, that are possible due to very different over tone mixtures. More details about this is discussed in the manuals.

This simulation was originally developed by Francisco Esquembre, the pioneer of the EJS program and extended by us.

end of chapter 10