Waves2D_3D - previous index next Waves in Two and Three...

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previous index next Waves in Two and Three Dimensions 6/2/08 Michael Fowler Introduction So far, we’ve looked at waves in one dimension, traveling along a string or sound waves going down a narrow tube. But waves in higher dimensions than one are very familiar—water waves on the surface of a pond, or sound waves moving out from a source in three dimensions. It is pleasant to find that these waves in higher dimensions satisfy wave equations which are a very natural extension of the one we found for a string, and—very important—they also satisfy the Principle of Superposition , in other words, if waves meet, you just add the contribution from each wave. In the next two paragraphs, we go into more detail, but this Principle of Superposition is the crucial lesson. The Wave Equation and Superposition in One Dimension For waves on a string, we found Newton’s laws applied to one bit of string gave a differential wave equation, 22 1 2 y y x vt = and it turned out that sound waves in a tube satisfied the same equation . Before going to higher dimensions, I just want to focus on one crucial feature of this wave equation: it’s linear , which just means that if you find two different solutions ( ) 1 , yx t and ( ) 2 , t then () ( ) 12 ,, t yx t + is also a solution, as we proved earlier. This important property is easy to interpret visually : if you can draw two wave solutions, then at each point on the string simply add the displacement ( ) 1 , t of one wave to the other ( ) 2 , t you just add the waves together—this also is a solution. So, for example, as two traveling waves moving along the string in opposite directions meet each other, the displacement of the string at any point at any instant is just the sum of the displacements it would have had from the two waves singly. This simple addition of the displacements is termed “interference”, doubtless because if the waves meeting have displacement in opposite directions, the string will be displaced less than by a single wave. It’s also called the Principle of Superposition . The Wave Equation and Superposition in More Dimensions What happens in higher dimensions? Let’s consider two dimensions, for example waves in an elastic sheet like a drumhead. If the rest position for the elastic sheet is the ( x , y ) plane, so when it’s vibrating it’s moving up and down in the z -direction, its configuration at any instant of time is a function ( ) zxyt .
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2 In fact, we could do the same thing we did for the string, looking at the total forces on a little bit and applying Newton’s Second Law. In this case that would mean taking one little bit of the drumhead, and instead of a small stretch of string with tension pulling the two ends, we would have a small square of the elastic sheet, with tension pulling all around the edge. Remember that the net force on the bit of string came about because the string was curving around, so the tensions at the opposite ends tugged in slightly different directions, and didn’t cancel. The 2 / 2 y x ∂∂ term measured that curvature, the rate of change of slope. In two dimensions, thinking
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Waves2D_3D - previous index next Waves in Two and Three...

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