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I was recently studying about Longitudinal Waves and I have a little trouble understanding the Displacement versus distance graph for these waves. Firstly, how exactly does one come up with such a graph? I understand about compressions and rarefactions but I don't know how exactly to plot a graph between the displacement of the particles and the distance travelled by the wave. Moreover, why is that these graphs look identical for both transverse and longitudinal waves? I reckon that this is because the individual particles are undergoing simple harmonic motion, but I'm still missing some connection. Please help.

Thank you.

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For small displacements, the solution of the wave equation tells us that each point is moving sinusoidally, and that there is a phase velocity associated with the wave: $$\Delta x=A\cos(\omega t -kx+\phi)$$

At a given time $t$ you see that $\omega t$ adds a constant phase, and you are left with a wave that varies spatially with a wave number $k$.

The math for the transverse wave looks very similar although the relationship between $\omega$ and $k$ will be different (the wave velocity is different). Since the equations look similar, the graphs look similar.

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  • $\begingroup$ I still don't understand. If you have a, say 10 air columns, oscillating to and fro, then how do you come up with the displacement vs distance graph? $\endgroup$ – model_checker Jan 9 '16 at 17:37
  • $\begingroup$ Do you understand the equation I wrote down? $\endgroup$ – Floris Jan 9 '16 at 18:12
  • $\begingroup$ @ShreyAryan What do you mean by your picture of ten air columns? I don't understand what you are trying to say. $\endgroup$ – garyp Jan 9 '16 at 18:28
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A wave is a harmonic oscillation travelling through space at constant speed.

What you call 'displacement' is simply the value of the oscillating property (water displacement, air pressure or an electromagnetic field, to name but three), in time. If the oscillation is harmonic the graphs of the waves will look near-identical, regardless of whether the wave is longitudinal or transverse.

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  • $\begingroup$ I still don't understand. If you have a, say 10 air columns, oscillating to and fro, then how do you come up with the displacement vs distance graph? $\endgroup$ – model_checker Jan 9 '16 at 17:37
  • $\begingroup$ What you call a displacement is simply the air pressure in function of time, which will be a $\sin$ function. $\endgroup$ – Gert Jan 9 '16 at 18:56

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