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        Please give the derivation of the equation of wave motion
8 years ago

							Dear student, ${ \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u$
where $\scriptstyle\nabla^2$ is the (spatial) Laplacian and where c is a fixed constant equal to the propagation speed of the wave. This is known as the  non-dispersive wave equation. For a sound wave in air at 20°C this  constant is about 343 m/s . For the vibration of a string the speed can vary widely, depending upon the linear density of the string and the tension on it. For a spiral spring   it can be as slow as a meter per second. More realistic differential  equations for waves allow for the speed of wave propagation to vary with  the frequency of the wave, a phenomenon known as dispersion. $v_\mathrm{p} = \frac{\omega}{k}.$
Another common correction in realistic systems is that the speed can  also depend on the amplitude of the wave, leading to a nonlinear wave  equation: ${ \partial^2 u \over \partial t^2 } = c(u)^2 \nabla^2 u$

8 years ago
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