When a piston generates
a sinusoidal wave in the tube, the new volume of the
air originally located between x_{1} and x_{2} becomes

2-3 Traveling sound waves

Additional material - Derivation of
displacement and pressure amplitudes formula

Consider a tube of cross-sectional area A. The volume of air located between
positions x_{1} and x_{2} is

For

Thus

Using and

Let us take the limit when
x_{2} becomes very close to x_{1}

From the definition of the Bulk modulus

Thus the change in volume ΔV due to the sound wave is