Difficulty in understanding non-oscillatory waves

[tsienni]

I understand that a wave is most often oscillatory in character. That said, it does not have to be the case, for a wave is simply not the same as an oscillation: the former refers to a spatial pattern whereas the latter to a variation in time. We may think of a wave formed by an infinite number of motions (typically oscillatory), one at every point in space and all generally different. Now, for a non-oscillatory wave, there's just a single big disturbance that passes anyone point for merely a short time.

Examples of a non-oscillatory wave:
a) the wave thrown off by the bow of a speedboat
b) the sonic boom from a supersonic plane
c) the sound wave emitted from a single gunshot.

If you take a snapshot at any given time, a non-oscillatory wave pattern consists of only one localized disturbance plus tiny motions seen anywhere else. How could the wave, then, be properly propagated in time? A moment later when you do an observation yet again, you see the point where the last disturbance takes place is now virtually motionless. How could this happen at all?

 

[Claude Bile]

The term wave applies to all solutions of the wave equation.

Solutions to the wave equation are all in the form f(x-vt), i.e. solutions to the wave equation all have one thing in common, they propagate with a velocity, v.

These propagating solutions include, but a certainly not limited to sinusoidal wave solutions, however it is possible, via Fourier theory to express any waveform as a linear combination of sinusoidal waves. Even aperiodic waveforms can be expressed in this fashion.

The wave equation is not a fundamental equation, it is derived from other, more fundamental laws. Asking why waves behave as they do is like asking why the laws of mechanics are as they are, or why Maxwell's equations are true.

Claude.