Sound Waves and the Doppler Effect

The siren’s pitch drops the moment the ambulance passes — you have heard the Doppler effect your whole life. Wavefronts bunch ahead of a moving source and stretch behind it. The same compression of waves, applied to starlight, is how we learned the universe is expanding.

Sound is a longitudinal pressure wave in a medium.

Speed of sound depends on the medium:

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

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

Sound intensity level (decibels):

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

Doppler effect — The observed frequency shifts when source or observer moves:

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

Upper signs when approaching, lower when receding.

Applications: radar speed guns, medical ultrasound, redshift of galaxies.

Physics link: The Doppler effect for light gives $f' = f\sqrt{(1\pm\beta)/(1\mp\beta)}$ (relativistic). The cosmic redshift $z = \Delta\lambda/\lambda$ tells us the universe is expanding.

Common pitfall: For a source moving at constant speed the pitch does not slide continuously — you hear one steady raised pitch approaching, one steady lowered pitch receding, with the drop concentrated at the moment of passing.

The Doppler Effect

When a wave source moves relative to an observer, the observed frequency shifts:

$$f' = f \frac{v \pm v_o}{v \mp v_s}$$

where $v$ is the wave speed, $v_s$ the source speed, and $v_o$ the observer speed.

• Source approaching: wavefronts compress → higher frequency (blueshift)
• Source receding: wavefronts stretch → lower frequency (redshift)

The Doppler effect is used in radar, medical ultrasound, and measuring the expansion of the universe.