Suppose a stationary source is generating sound waves with frequency f₀ = 240 Hz (middle C) and wavelength l₀ = v/f₀.  A stationary observer a certain distance from the source will hear a sound with pitch f₀.  240 time each second the observer’s eardrum will be pushed in and pulled out as pressure crest and pressure trough reach the ear.  The time period between two consecutive crests is T = 1/f₀ = (1/240)s.  Assume the observer gets onto a motorcycle and starts driving away from the source.  Assume that at time t₁ a pressure crest reaches the ear at position x.  The next crest will be at position x at time t₁ + T.  But the ear is no longer there.  The observer has moved.  The crest has to travel an extra distance before it reaches the ear.  This takes an extra time interval Dt.  The time interval between successive crests reaching the ear of the observer now is T' = T + Dt.

While the observer has traveled a distance Dx = v_o(T + Dt),
the wave has traveled a distance Dx + l₀ = v(T + Dt).
Therefore, using l₀ = v/f₀ = vT,  we have  v_oT + v_oDt + vT = vT + vDt, or Dt = v_oT/(v - v₀)
T' = T + v₀T/(v - v₀) = vT/(v - v₀),
f' = f₀(v - v₀)/v.

The period has increased, the apparent frequency of the wave has decreased, the pitch has decreased.  The observer hears a note lower than middle C.  This is an example of the Doppler effect.  If the observer is driving towards the source, then the time interval between successive crests reaching the ear will be shorter than T.  Assume that at time t₁ a pressure crest reaches the ear at position x.  The next crest will be at position x at time t₁ + T.  But it will reach the ear before it reaches position x, because the observer is moving towards the source.  The observer hears a note higher than middle C.

The apparent frequency of the sound wave reaching the observer is

f = f₀(v + v_o)/v.

Here v is the speed of sound and v_o is the component of the velocity of the observer towards the source.  (v_o is negative if the observer moves away from the source.)  We usually do not notice the Doppler effect when moving around on foot, because the speed of sound is so much greater than our speed.  But moving on a motorcycle at 55 miles/h = 24.6m/s towards a source, we have f = f₀(340+24.6)/340 = 1.07f₀.  Moving away from the source we have f = f₀(340 - 24.6)/340 = 0.93f₀.  As we drive by the source, the perceived pitch therefore changes by approximately 14%, a noticeable change.

Moving Sources

The perceived pitch of a sound wave also changes if the observer is stationary and the source is moving.  Then the apparent frequency of the sound wave reaching the observer is

f = f₀v/(v - v_s).

Here v_s is the component of the velocity of the source towards the observer.  (v_s is negative if the source moves away from the observer.)

In the above figure the rings denote successive crests of the sound wave.  The time interval between the emission of successive crests is T.  The larger the ring, the earlier is the time of emission.  All the rings expand with the same speed, the speed of sound in the medium.  If the observer is stationary, then the time interval between the arrival of successive rings at the ear is T.

In this figure the source is moving towards the observer.  The center of each ring is at the position of the source at the time it emits the crest.  Since the source is moving towards the right, the center of successive rings moves towards the right.  If the observer is stationary, then the time interval between the arrival of successive rings at the ear is less than T.

In the time interval  T the wave travels a distance l₀ and the source travels a distance Dx.
l₀/v = Dx/v_s.
The wavelength at the observer is l' = l₀ - Dx = l₀ - l₀v_s/v = l₀ (v - v_s)/v.
f' = v/l' = (v/l₀)[v/(v - v_s)] = f₀v/(v - v_s).

In this figure the source is moving away from the observer.  Since the source is moving towards the left, the center of successive rings moves towards the left.  If the observer is stationary, then the time interval between the arrival of successive rings at the ear is greater than T.

Whenever the source and the observer move with respect to each other, the wavelength of the sound reaching the ear will be Doppler shifted.  But the formula for the Doppler shift depends on who is moving, the source or the observer.  If the source is moving towards the observer with a speed close to the speed of sound, then the wavelength of the sound reaching the ear becomes very short and the pitch becomes very high.  In the formula, f = f₀v/(v - v_s), the denominator gets very small.  When v_s =v , the denominator is zero, so f becomes infinite.  A sonic boom is produced at the location of the observer.

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Problem:

If both source and observer are in motion, then the apparent frequency of the sound wave reaching the observer is

f = f₀(v + v_o)/(v - v_s)

if the observer and the source are approaching each other, and

f = f₀(v - v_o)/(v + v_s)

If the observer and the source are receding from each other.

The Doppler effect is a phenomenon observed with all waves.  It is named for the Austrian scientist Christian Doppler (1803-1853).  Whenever a source generating a wave moves relative to an observer or an observer moves relative to a source, the frequency of the wave at the location of the observer is shifted relative to the frequency of the source.  The frequency at the location of the observer increases when the source and observer are approaching each other, and decreases when they are moving away from each other.

For waves traveling in a medium the frequency of a wave at the location of an observer is given by the two formulas above, with v being the speed of the wave in the medium, v₀ the speed of the observer relative to the medium, and v_s the speed of the source relative to the medium.  Electromagnetic waves can travel through empty space.  For EM waves traveling through empty space, the frequency of a wave at the location of an observer is given by

,

if the observer and the source approach each other with relative speed v.   If the observer and the source recede from each then their relative velocity v in the above formula is a negative number.

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