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Dear student,
where 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.
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:
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