17.4 Doppler Effect and Sonic Booms

Doppler Effect and Sonic Booms

Doppler Effect and Sound Perception

The Doppler effect is the apparent change in frequency of a wave caused by relative motion between the source and the observer. You've experienced it firsthand: an ambulance siren sounds higher-pitched as it approaches you and lower-pitched as it drives away. The actual frequency the siren emits never changes, but your perceived frequency does because of the motion.

This effect applies to all types of waves, not just sound. Astronomers use the Doppler effect with light waves to measure how fast distant galaxies are moving away from us (called redshift). For this unit, though, we're focused on sound.

Here's how the frequency shifts depending on who's moving:

• Source moving toward observer → observed frequency is higher than the actual frequency (sound waves get compressed)
• Source moving away from observer → observed frequency is lower than the actual frequency (sound waves get stretched)
• Observer moving toward source → observed frequency is higher
• Observer moving away from source → observed frequency is lower
• Both moving → the shift depends on their combined relative velocity

Since frequency and wavelength are inversely related ($v = f\lambda$), a higher observed frequency also means a shorter observed wavelength, and vice versa.

Sound Waves and Relative Motion

Sound waves are longitudinal waves that travel through a medium by compressing and expanding it. The speed of sound in air is approximately 343 m/s at 20°C. This value matters because the Doppler equation uses it directly.

Pitch is the perceptual quality tied to frequency. Higher frequency means higher pitch. So when the Doppler effect increases the observed frequency, you hear a higher pitch, and when it decreases the observed frequency, you hear a lower pitch.

Doppler Shift Equation and Applications

The general Doppler shift equation for sound is:

$f_{obs} = \frac{v \pm v_{obs}}{v \mp v_s} \cdot f_s$

Where:

• $f_{obs}$ = observed frequency
• $f_s$ = source frequency (what the source actually emits)
• $v$ = speed of sound in the medium
• $v_{obs}$ = speed of the observer
• $v_s$ = speed of the source

Choosing the correct signs is the trickiest part. Use this rule:

• Use the upper signs (+ in numerator, − in denominator) when source and observer are moving toward each other. This makes the fraction larger, giving a higher observed frequency.
• Use the lower signs (− in numerator, + in denominator) when they are moving apart. This makes the fraction smaller, giving a lower observed frequency.

For common special cases:

• Stationary observer, moving source: $f_{obs} = \frac{v}{v \mp v_s} \cdot f_s$
• Moving observer, stationary source: $f_{obs} = \frac{v \pm v_{obs}}{v} \cdot f_s$

Steps for solving Doppler problems:

1. List your knowns: $f_s$, $v$, $v_s$, and $v_{obs}$. Set whichever speeds are zero (stationary source or observer).
2. Determine the relative motion: are the source and observer getting closer or farther apart?
3. Pick the correct signs based on that motion (upper for approaching, lower for receding).
4. Plug values into the equation and solve for $f_{obs}$ (or for a speed, if frequency is given).

Quick example: A fire truck emitting a 700 Hz siren approaches you at 30 m/s. You're standing still. The speed of sound is 343 m/s.

$f_{obs} = \frac{343}{343 - 30} \times 700 = \frac{343}{313} \times 700 \approx 767 \text{ Hz}$

The observed frequency is higher, which matches what you'd expect for an approaching source.

Sonic Booms: Causes and Characteristics

A sonic boom is the shock wave produced when an object travels faster than the speed of sound (supersonic speed). At 20°C, that threshold is about 343 m/s (roughly 1,125 ft/s or 767 mph).

How a sonic boom forms:

1. As any object moves through air, it creates pressure waves that spread outward at the speed of sound.
2. When the object moves faster than sound, it outruns its own pressure waves. The waves can't get ahead of it, so they pile up.
3. These overlapping pressure waves merge into a single, intense shock wave that forms a cone-shaped wavefront behind the object, called a Mach cone.
4. When this cone sweeps past an observer on the ground, they hear a loud, explosive boom.

The Mach number describes how fast an object moves relative to the speed of sound: $M = \frac{v_{object}}{v_{sound}}$. An object at Mach 1 is traveling exactly at the speed of sound. Mach 2 is twice the speed of sound.

Characteristics of sonic booms:

• A sudden spike in air pressure followed by a rapid drop, perceived as a sharp, explosive sound
• Intense enough to rattle windows and vibrate building structures
• Not a one-time event at the moment the object "breaks" the barrier; the shock wave is continuous as long as the object stays supersonic

Factors affecting intensity:

• Mach number: higher speeds produce stronger booms
• Altitude: higher altitude means the shock wave spreads over a larger area, reducing intensity at ground level
• Size and shape of the object: larger objects displace more air and create stronger shock waves
• Atmospheric conditions: temperature, humidity, and wind patterns all influence how the shock wave propagates

Supersonic aircraft like fighter jets (e.g., the F-16) and the now-retired Concorde produced notable sonic booms. Because of the disruption and potential structural damage, most countries restrict supersonic flight over populated areas.