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# Please give the derivation of the equation of wave motion

10 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: ${ \partial^2 u \over \partial t^2 } = c(u)^2 \nabla^2 u$