Sound and the Doppler Effect

By the end of this lesson, you will be able to describe the properties of sound waves, calculate sound levels in decibels, and apply the Doppler effect to predict frequency shifts.

Sound as a Longitudinal Wave

Sound waves are longitudinal mechanical waves — compressions and rarefactions in a medium.

Speed of sound depends on the medium:

• Air (20°C): ~343 m/s
• Water: ~1480 m/s
• Steel: ~5100 m/s

Faster in denser, stiffer media (speed ∝ $\sqrt{B/\rho}$ where $B$ = bulk modulus).

Range of human hearing: 20 Hz – 20,000 Hz.

Infrasound (< 20 Hz): elephants, earthquakes.

Ultrasound (> 20 kHz): bats, medical imaging.

Sound Intensity and Decibels

Sound intensity ($I$): power per unit area (W/m²).

Decibel scale: logarithmic scale to account for the enormous range of human hearing.

$$\beta = 10\log\left(\frac{I}{I_0}\right) \text{ dB}$$

$I_0 = 10^{-12}$ W/m² (threshold of hearing).

Sound	dB	$I$
Whisper	30	$10^{-9}$ W/m²
Normal conversation	60	$10^{-6}$ W/m²
Concert	110	$10^{-1}$ W/m²
Jet engine	140	$10^{2}$ W/m²

Every 10 dB: intensity increases by 10×.

The Doppler Effect

The observed frequency changes when the source and observer move relative to each other.

$$f_{\text{obs}} = f_{\text{source}}\left(\frac{v_{\text{sound}} \pm v_{\text{observer}}}{v_{\text{sound}} \mp v_{\text{source}}}\right)$$

Rules for signs: upper sign when approaching; lower when receding.

Approaching: observed frequency > source frequency (higher pitch).

Receding: observed frequency < source frequency (lower pitch).

Example — Doppler

An ambulance (840 Hz siren) moves at 30 m/s toward a stationary observer. Speed of sound = 343 m/s. Find the observed frequency.

$f_{\text{obs}} = 840 \times \frac{343}{343-30} = 840 \times \frac{343}{313} = 840 \times 1.096 = 921$ Hz

Applications

Medical ultrasound: Doppler shift in reflected ultrasound measures blood flow velocity.