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Waves in one dimension

A nice simulation showing how standing waves arise from the superposition of incident and reflected waves is given in the applet from Walter Fendt's .

For an interactive exercise on standing waves on a string run the Halliday, Resnick and Walker simulation .

For a simulation of longitudinal waves in a pipe see Walter Fendt's applet on .

Waves in two dimensions

For interactive simulations involving waves in two and three dimensions you may like to try some of Paul Falstad's .

For standing waves in two dimensions run both of the following applets

    (Vibrational modes in a 2-d membrane)

     (Vibrational modes in a 2-d circular membrane (drum head))

These applets simulate waves in membranes (rectangular and circular), showing the various vibrational modes. The various possible modes are represented by a square on a grid located beneath the applet. Double-click on one of the grid squares to select a mode (the fundamental mode is in the upper left). You can select any mode, or you can click once on multiple squares to combine modes. Also you may click the membrane itself and drag up or down and release.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. So in the example on the right, mode 1,2 has a positive coefficient, mode 3,1 has a negative one and the rest of the modes are absent.

Waves in three dimensions

Standing waves on a three-dimensional cavity can be studied using Paul Falstad's (Acoustic standing waves in a 3-d box.) 

This uses the same ideas as in the two-dimensional examples above except that, because three dimensions are involved, it requires multiple grids to specify all the possible modes. See detailed for full explanation of how this applet works.

Note that, for performance reasons, only ten modes can be active at a time.

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Chapter13 Subsection 13.8.1

Click on above link to open Subsection 13.8.1 (downloadable pdf)