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# 2D Vibrations of a Membrane

The dispersion relation (6.3) can also be obtained by using separation of variables.

Example 6. .1Consider the vibrations of a square membrane inside the region and . The wave equation is describing the displacement, , of the membrane. Suppose the boundary conditions are taken as on the sides of the square , and , . Try a solution of the form so that the wave equation may be expressed in the separable form as Since the left hand side of the equation is only a function of time and the right hand side depends only on spatial variations, they must both be equal to a constant. This separation process can be continued to show that both terms on the right hand side are constants. Hence, where the separation constant is chsoen as in order to satisfy the boundary conditions in . Here is an integer. Similarly, where and is an integer so that the boundary conditions in are satisfied. Combining these results we obtain the equation for as Thus, we have where the frequency satisfies the dispersion relation Thus, the general solution is obtained by adding together all the possible solution to give The constants and are determined by initial conditions on and . Each value of and corresponds to a different possible mode of oscillations. The fundamental mode is given by and with and has It corresponds to the whole membrane oscillating up and down. The mode and has a spatial form given by and has These two normal modes are illustrated in Figure 6.1   Note that we could have obtained exactly the same result by assuming the solution could be expressed, in complex form, as This is, of course, a separable solution since Expressing this in terms of real and imaginary parts gives and so one solution is where  is a constant and and in order to satisfy the boundary conditions in and .   Next: Sound Waves Up: Waves Previous: Waves
Prof. Alan Hood
2000-11-06