Sound Waves and the Doppler Effect

Sound Waves and Properties

The siren's pitch drops the moment the ambulance passes; wavefronts bunch ahead of a moving source and stretch behind it.

Sound is a longitudinal pressure wave propagating through a medium via particle compressions and rarefactions.

The speed of sound depends on the elastic and inertial properties of the medium:

$$v = \sqrt{\frac{B}{\rho}}, \qquad v = 331 + 0.6T_C \text{ m/s (air)}$$

where $B$ is bulk modulus, $\rho$ is density, and $T_C$ is temperature in Celsius.

Sound intensity level ($\beta$) is measured on a logarithmic decibel scale relative to the threshold of hearing:

$$\beta = 10\log_{10}\left(\frac{I}{I_0}\right), \quad I_0 = 10^{-12}\text{ W/m}^2$$

The Doppler Effect

The Doppler effect occurs when a source or observer moves, causing a shift in observed frequency ($f'$):

$$f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)$$

where $v$ is wave speed, $v_o$ is observer speed, and $v_s$ is source speed. Use top signs for approaching, bottom for receding.

Common pitfall: For a source at constant speed, pitch does not slide continuously. You hear a steady high pitch approaching, a steady low pitch receding, and an abrupt drop at passing.

Applications: Radar speed guns, medical ultrasound, and cosmic redshift ($z = \Delta\lambda/\lambda$), proving the universe expands.