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Typically, the power supplied by a wave is averaged over many periods; since the wave is not at peak amplitude for most of oscillation, it does not make sense for the peak amplitude to be the only important factor. Furthermore, for waves that are not harmonic (not sinusoidal), there may not be a single well-defined peak amplitude.
In these cases, it is more correct to use the root-mean-square amplitude derived by taking the square root of the average of y^2(x,t) over a period. When the waves are harmonic, averaging the square of the sine or cosine function over a period typically contributes a factor of ½.
If the solution to the wave equation describes sound waves, the intensity directly corresponds to the loudness of the wave, as typically measured in decibels. The decibel scale measures the loudness of sounds only as relative to the threshold of human hearing.
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