When sound waves propagate through air, the layers of air are subjected to mechanical disturbances. We can build up the wave equation by thinking carefully.
There are 3 interconnected quantity in a block of air that oscillate as sound waves travel through it : the displacement, the density and the pressure exerted on it. We can start the analysis with any one of these 3 quantities.
We’ll begin with pressure. As a block of air undergoes alternate compression and rarefaction, the pressure change causes an acceleration at each point :
The changing pressure is brought on by a change in density :
The density change is brought on by a displacement change, which was caused by the pressure change in the first place :
Combining the three sets of equations and we obtain the wave equation :
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