2.4 Doppler Effect

Doppler Effect and Sound Waves

The Doppler effect describes how the perceived frequency of a sound wave changes when the source, the observer, or both are in motion relative to the medium. It's the reason an ambulance siren sounds higher-pitched as it approaches you and lower-pitched after it passes. Beyond everyday experience, this effect underpins technologies ranging from medical ultrasound to weather radar.

Fundamental Concepts

When a sound source moves toward you, each successive wave crest is emitted from a position slightly closer than the last. This compresses the wavefronts in front of the source, shortening the wavelength and raising the frequency you hear. Behind the source, the wavefronts spread apart, lengthening the wavelength and lowering the frequency.

A few key points to keep in mind:

• The effect depends on the relative velocities of the source and observer with respect to the medium (usually air), not just their distance apart.
• The Doppler shift is asymmetric: a source moving toward a stationary observer produces a different frequency shift than an observer moving toward a stationary source at the same speed. This is because the source's motion physically changes the wavelength in the medium, while the observer's motion changes only how quickly they encounter existing wavefronts.
• Either the source, the observer, or both can be in motion. The general equation (below) handles all of these cases.

Factors Influencing the Doppler Shift

• Speed of sound in the medium ($v$): A higher speed of sound (for example, in warmer air) means the wavefronts travel faster, which changes the magnitude of the shift for a given source or observer velocity.
• Direction of motion: Motion along the line connecting source and observer produces the full Doppler shift. Motion perpendicular to that line produces no shift (you need the velocity component along the line of sight).
• Medium properties: Temperature and density affect $v$. For air at room temperature (~20 °C), $v \approx 343 \text{ m/s}$.
• Relative speed compared to $v$: The shift grows larger as the source or observer speed approaches the speed of sound. At and beyond $v$, shock waves (sonic booms) form, and the standard Doppler formula no longer applies.

Calculating Frequency Change

General Doppler Effect Equation

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

Sign convention (this is the part most students mix up):

The easiest way to remember the signs is to think about what should make $f'$ larger or smaller.

• Numerator (observer term): Use $+\, v_o$ when the observer moves toward the source (they encounter wavefronts more often, so frequency goes up). Use $-\, v_o$ when moving away.
• Denominator (source term): Use $-\, v_s$ when the source moves toward the observer (wavefronts are compressed, so frequency goes up). Use $+\, v_s$ when moving away.

A quick sanity check: if source and observer approach each other, $f'$ should be greater than $f$. If they move apart, $f'$ should be less than $f$.

Simplified Equations for Common Cases

Stationary observer, moving source:

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

Use $-\, v_s$ for an approaching source, $+\, v_s$ for a receding source.

Moving observer, stationary source:

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

Use $+\, v_o$ for an observer approaching, $-\, v_o$ for an observer receding.

Applying Doppler Effect Equations

Example: Moving Source, Stationary Observer

A fire truck with a siren at $f = 700 \text{ Hz}$ drives toward you at $v_s = 30 \text{ m/s}$. The speed of sound is $v = 343 \text{ m/s}$.

1. Identify the scenario: source approaching, observer stationary. Use $f' = f\left(\frac{v}{v - v_s}\right)$.

2. Substitute: $f' = 700\left(\frac{343}{343 - 30}\right) = 700\left(\frac{343}{313}\right)$.

3. Calculate: $f' \approx 767 \text{ Hz}$. The pitch is noticeably higher.

After the truck passes and moves away, switch the sign:

$f' = 700\left(\frac{343}{343 + 30}\right) = 700\left(\frac{343}{373}\right) \approx 643 \text{ Hz}$

That drop from ~767 Hz to ~643 Hz is the characteristic pitch change you hear as a siren passes.

Example: Moving Observer, Stationary Source

You're in a car moving at $v_o = 25 \text{ m/s}$ toward a stationary speaker emitting $f = 500 \text{ Hz}$.

1. Use $f' = f\left(\frac{v + v_o}{v}\right)$.
2. Substitute: $f' = 500\left(\frac{343 + 25}{343}\right) = 500\left(\frac{368}{343}\right)$.
3. Calculate: $f' \approx 536 \text{ Hz}$.

Notice that at typical walking speeds (~1.5 m/s), the shift would be only a few hertz, which is barely perceptible.

More Complex Scenarios

• Both source and observer moving: Use the full general equation. Be careful with signs; decide a positive direction first and stick with it.
• Motion at an angle: Only the velocity component along the line connecting source and observer matters. If the source moves at speed $v_s$ at angle $\theta$ to that line, use $v_s \cos\theta$ in the equation.
• Reflections: When sound bounces off a moving object (like a wall or vehicle), treat the reflector as both a moving observer receiving the wave and a moving source re-emitting it. The Doppler shift effectively applies twice, which is why radar-based speed measurements use a doubled shift formula.

Practical Applications of the Doppler Effect

Radar and Speed Measurement

Doppler radar sends out a signal of known frequency, then measures the frequency of the reflected signal. The shift between the two reveals the target's velocity.

• Police radar guns bounce microwaves off vehicles and calculate speed from the frequency shift.
• Weather radar (like NEXRAD) tracks the motion of raindrops and ice particles to map wind speeds, detect rotation in storm cells, and issue tornado warnings.
• Traffic monitoring systems use the same principle to measure highway speeds and detect congestion in real time.

Medical Imaging and Diagnostics

Doppler ultrasound sends sound waves (typically 2–18 MHz) into the body and detects shifts caused by moving blood cells.

• It measures blood flow velocity and direction in arteries and veins, helping diagnose conditions like arterial stenosis (narrowing) or deep vein thrombosis.
• In obstetrics, it assesses blood flow in the umbilical cord and fetal heart.
• Doppler echocardiography maps blood flow through heart chambers and across valves, detecting problems like valve regurgitation (backflow) or septal defects.

Astronomical Applications

The Doppler effect applies to all waves, including light. Astronomers use it extensively:

• Redshift and blueshift of spectral lines reveal whether a star or galaxy is moving toward or away from Earth, and how fast.
• The radial velocity method detects exoplanets by measuring tiny periodic Doppler shifts in a star's light caused by the gravitational tug of an orbiting planet.
• Cosmological redshift provides evidence that the universe is expanding. The farther away a galaxy is, the greater its redshift, a relationship described by Hubble's law.

Other Scientific and Industrial Uses

• Acoustic Doppler current profilers (ADCPs) measure water current velocities in oceans and rivers by bouncing sound pulses off suspended particles.
• In aviation, Doppler navigation systems measure an aircraft's ground speed and drift angle.
• Industrial flow meters use ultrasonic Doppler shifts to monitor fluid flow rates in pipes without physically contacting the fluid.