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I am attempting to understand the mechanics of a longitudinal sound wave.

I understand that as a material vibrates, it pushes air molecules forward as it moves towards the direction of the wave. For example, a tuning fork "swings" back and forth as it vibrates, and as it moves forward the air is compressed as the air molecules come in contact with the molecules in front of them. This causes the compression to propagate through the medium by pushing forward successive molecules.

However, what I do not understand is what causes the molecules to move back to their initial position after the compression of the wave. My intuition tells me that the vibration would continue to push the particles forward as a new pulse is created, but there is an equal reverse motion after the first half of the period shown in videos and diagrams of longitudinal waves, such as this one by Khan Academy (https://www.youtube.com/watch?v=-_xZZt99MzY). What is the phenomenon that causes these particles to move backwards? Is it the result of the collision with the particles in front of them and the "equal and opposite reaction" of Newton's third law of motion? Is it an equalization of the pressure differential created when the molecules are moved forward and spread out? Is it the air rushing to fill the vacuum created when the vibrating object moves backwards again? Either way I would expect a net forward movement of the molecules - a return to their precise starting position seems too perfect to me.

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  • $\begingroup$ I think you should not think of the membrane as a boundary condition in space but in the pressure. When the membrane moves back there is a drop in pressure locally. I.e. in the air there is a gradient in pressure leading to a flow towards the membrane. $\endgroup$ – Bort Dec 17 '15 at 17:59
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The question has more degrees of complexity.

The first approximation could be the model of many points of mass connected by springs. In this case the point of mass returns to its equilibrium position as the wave passes, cause it's the state of minimal energy. The derivation of wave equation is simply a proof, that the air molecules are governed by such a "spring forces" as well.

The more detailed answer could be very broad, too broad for just one question. Let me just show some cases:

  • If we consider viscosity as well than there are cases (especially in boundary layers) where the wave is polarized and particles are "orbiting" around the equilibrium state.

  • In many cases there are complex motion of the air convection and local oscillation (wind instruments, sound waves in tubes with flow...). These cases are highly nonlinear and I would suggest you further reading before detailed question will be formulated. Start with the Aeroacoustics in wikipedia.

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