14.3 Doppler Effect and Sonic Booms

The Doppler Effect

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Doppler Effect and Sound Perception

The Doppler effect describes the change in observed frequency when a sound source and an observer are moving relative to each other. The key idea: motion changes the spacing between wave crests as they arrive at your ear, which your brain interprets as a change in pitch.

When the source moves toward you, each successive wave crest is emitted from a position slightly closer than the last. This compresses the wavelength, raising the frequency you hear. Think of an ambulance siren sounding higher-pitched as it approaches.

When the source moves away from you, the opposite happens. Each crest is emitted from farther away, stretching the wavelength and lowering the perceived frequency. That same ambulance siren drops in pitch the moment it passes you and starts moving away.

The observer can be the one moving, too. If you're running toward a stationary siren, you encounter wave crests more frequently, so the pitch sounds higher. Running away from it, you encounter them less frequently, and the pitch drops.

A few additional points:

• The greater the relative speed between source and observer, the larger the frequency shift. A jet flying past at 300 m/s produces a far more dramatic pitch change than a car passing at 15 m/s.
• The Doppler effect isn't limited to sound. It applies to all waves, including light and other electromagnetic waves. Astronomers use the redshift and blueshift of light from galaxies to determine whether they're moving toward or away from Earth.

Applying the Doppler Shift Formula

The general Doppler equation for sound is:

$f_o = f_s \frac{v \pm v_o}{v \mp v_s}$

where:

• $f_o$ = observed frequency (what the listener hears)
• $f_s$ = source frequency (what the source actually emits)
• $v$ = speed of sound in the medium (about 343 m/s in air at 20°C)
• $v_o$ = speed of the observer
• $v_s$ = speed of the source

Getting the signs right is the trickiest part. Here's a reliable method:

1. Draw a quick sketch showing the source, observer, and the direction each is moving.
2. Pick a positive direction (typically from source toward observer).
3. In the numerator ($v \pm v_o$): add $v_o$ if the observer moves toward the source, subtract if the observer moves away.
4. In the denominator ($v \mp v_s$): subtract $v_s$ if the source moves toward the observer, add if the source moves away.
5. Plug in values and solve.

Quick sanity check: If source and observer are getting closer together, $f_o$ should be greater than $f_s$. If they're moving apart, $f_o$ should be less than $f_s$. If your answer violates this, recheck your signs.

Example: A fire truck emitting a 600 Hz siren drives toward a stationary observer at 30 m/s. The speed of sound is 343 m/s. The observer isn't moving, so $v_o = 0$. The source moves toward the observer, so you subtract $v_s$ in the denominator:

$f_o = 600 \times \frac{343 + 0}{343 - 30} = 600 \times \frac{343}{313} \approx 657 \text{ Hz}$

The observed frequency is higher than the source frequency, which makes sense since the truck is approaching.

You can also rearrange the formula to solve for $v_s$ or $v_o$ if you know the observed and source frequencies. Just isolate the unknown algebraically.

Wave Behavior in the Doppler Effect

What's physically happening to the waves ties directly to the pitch change you hear:

• Compression of wavefronts occurs when the source and observer approach each other. The waves bunch up in front of the source, shortening the wavelength and increasing the frequency.
• Stretching of wavefronts occurs when they move apart. The waves spread out behind the source, lengthening the wavelength and decreasing the frequency.

The source itself doesn't change the frequency it emits. A siren always vibrates at the same rate. The Doppler effect is entirely about how the motion rearranges the spacing of the waves as they travel through the medium to your ear.

Sonic Booms

Characteristics of Sonic Booms

A sonic boom is the intense sound produced when an object travels faster than the speed of sound (supersonic speed, above Mach 1). It's not a one-time event that happens only at the moment the object "breaks the sound barrier." The boom is generated continuously for as long as the object remains supersonic.

Here's how it forms:

1. Any object moving through air generates pressure waves that spread outward in all directions at the speed of sound.
2. When the object moves slower than sound, these waves propagate ahead of it without piling up.
3. At Mach 1, the object moves as fast as its own pressure waves. The waves can no longer get ahead and begin stacking on top of each other, forming a compressed shock wave.
4. Above Mach 1, the object outruns all of its pressure waves. The overlapping wavefronts form a cone-shaped shock wave called a Mach cone, with the object at the tip.
5. As this cone sweeps along the ground, anyone it passes over hears a sharp, loud "bang" from the sudden pressure change.

The Mach number is the ratio of the object's speed to the speed of sound: $M = \frac{v_{object}}{v_{sound}}$. Mach 1 is the speed of sound, Mach 2 is twice the speed of sound, and so on. The higher the Mach number, the narrower the Mach cone angle.

Sonic boom characteristics:

• The boom sounds like a sharp crack or thunderclap caused by the abrupt jump in air pressure as the shock front passes.
• Intensity depends on the object's size, shape, altitude, and speed, as well as atmospheric conditions. A large aircraft like the Space Shuttle produced a much more powerful boom than a small fighter jet.
• A sonic boom is not something you hear only once per flight. Every observer along the flight path hears it as the Mach cone reaches them.

Impacts and restrictions:

• Sonic booms can startle people and wildlife, and in strong cases can crack windows or knock objects off shelves.
• Because of these effects, supersonic flight over populated land areas is restricted in many countries. The Concorde, for example, was limited to supersonic speeds only over the ocean.