Physics · Waves & Sound

Doppler Effect

The frequency you hear changes when the source or the observer moves. In this lab you can show wavefronts, the Mach cone, the sound of a passing vehicle and the redshift of light on one page, with live measurements.

ENTR

Record strip: each tick marks the moment a wave crest reaches that observer. The denser the ticks, the higher the frequency. To keep it visible, f₀ is kept low (a few waves per second); the frequency ratios are the same for real sound.

Source

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λ₀ –

Mach –

A · on the right

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formula –

B · on the left

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formula –

C · beside the path

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measurement only

Time scale Source frequency f₀ Source speed v_s (+ to the right) Horizontal speed for A and B v_o Air temperature

Presets

Display

Color code (blue compression, red stretching) Show emission centers Measure the wavelength Mach cone (v_s > v) Listen Sound type

The sound is a 440 Hz tone shifted by the heard frequency ratio. Only what the selected observer hears is played (tones, ticks and the sonic boom included). You can also select an observer by clicking it on the canvas; click again to turn it off. Drag the ◆ handle next to each observer in any direction to give it a velocity vector; a longer arrow means a higher speed. Space pauses, R restarts.

Heard frequency

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Highest (approaching)

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f₀·v/(v − v_s)

Lowest (receding)

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f₀·v/(v + v_s)

Sound level

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∝ 1/R, independent of frequency

Time scale Horn frequency f₀ Vehicle speed v_s Distance from the road d Air temperature Sound type

Speed ratio β = v/c

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z –

Selected line

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With the classical formula

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λ₀(1 + v/c)

Color

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Source speed (+ receding)

Tracked line

Suggested order for class

Stationary sourceThe rings are concentric and reach every observer at the same rate. f′ = f₀.
Set the source movingEach ring stays centered where the wave was emitted. Show the centers to prove it.
Mach 0.9 and Mach 1Rings pile up in front and the wavelength shrinks toward zero. At Mach 1, A hears nothing.
Mach 1.5The envelope of the rings forms a cone with sin θ = v/v_s. An observer hears the shock wave when the cone reaches them.
Move the observerSource at rest, observer running: the wavelength stays the same, the arrival rate changes.
Passing vehicleIn tab 2, see and hear how a high pitch turns into a low one, with the graph alongside.
LightIn tab 3 the same idea applies to color and spectral lines.

Formula card

General case (motion along a line)

f′ = f₀ · (v + v_o) / (v − v_s)

v is the speed of sound, v_o the observer's speed toward the source, v_s the source's speed toward the observer. Approaching is positive, receding is negative.

Source passing by

f′ = f₀ · v / (v − v_s cos θ)

θ is the angle between the source's velocity vector and the line to the observer. At θ = 90°, f′ = f₀.

Mach number and shock cone

M = v_s / v sin θ = 1 / M

When M > 1 the source outruns its own waves. θ is the half-angle of the cone.

Speed of sound in air

v ≈ 331.3 + 0.606 · T

T is the air temperature in °C. About 343 m/s at 20 °C.

Light (relativistic Doppler)

λ′ = λ₀ √((1 + β)/(1 − β))

β = v/c, positive when receding. z = (λ′ − λ₀)/λ₀ is the redshift; at low speed z ≈ v/c.

Case	v_s	v_o	Result
Source approaching	+	0	f′ > f₀, wavelength shortens
Source receding	−	0	f′ < f₀, wavelength lengthens
Observer approaching	0	+	f′ > f₀, wavelength unchanged
Observer receding	0	−	f′ < f₀, wavelength unchanged

In daily life and in science

Radar and speed checks

The frequency shift of a wave reflected from a vehicle gives its speed. On reflection the shift is applied twice.

Medical ultrasound

Sound waves reflected from blood cells in a vessel measure the direction and speed of blood flow.

Weather radar

The motion of raindrops reveals wind and the rotation of storms.

Exoplanet hunting

Lines in a star's light shift periodically because an orbiting planet tugs the star slightly.

The expanding universe

Lines from distant galaxies are redshifted. Hubble's law: v = H₀·d, with H₀ ≈ 70 km/s/Mpc.

Bats and dolphins

They use the frequency shift of an echo to sense the speed of prey and obstacles.

Common misconceptions

"The source's frequency changes."

The source vibrates at the same rate. What changes is the number of waves reaching the observer per second.

"The sound gets louder."

Pitch (frequency) and loudness (amplitude) are separate quantities. Both change as an ambulance approaches, but for different reasons.

"The speed of sound adds to the source's speed."

The medium sets the speed of sound. The source's motion only changes the wavelength.

"Sound cannot get through at Mach 1."

The source catches up with its own wave. An observer ahead hears nothing until the source passes by.

Prepared byMehtap Aydoğan