Sound as Pressure Waves

Consider sound traveling through a medium. The longitudinal disturbances create a pressure difference between its successive columns, which then undergo oscillations.

Consider an undisturbed cylinder of the medium, with cross-sectional area A along the x-axis. Its longitudinal displacement, given by y, is a wave function.

As the wave travels, its ends at x-1 and x-2 are displaced by y-1 and y-2, respectively. If the latter is greater, the cylinder expands, and the pressure falls from the surrounding pressure.

Its initial volume is known, and its change in volume is derived. The fractional change in volume is then obtained.

Recall the definition of bulk modulus, from which the gauge pressure is obtained. On simplification, the gauge pressure is observed to be a wave.

The gauge pressure is maximum at points of zero displacement and minimum at points where the displacement is maximum.

Its amplitude is proportional to the displacement amplitude, the bulk modulus of the medium, and the wave number. So, it is inversely proportional to the wavelength.