Doppler effect for spacetime and timespace

We follow the presentation of Guy Moore, formerly of McGill University, online here.

The Doppler effect is the phenomenon we have all noticed, that a sound produced by a moving source, or which you hear while you are moving, has its perceived frequency shifted.

Spacetime (3+1)

If a source of sound makes a sudden bang (pressure pulse) at time 0, then the spatial location of the pressure pulse in relation to time will look like successive concentric circles. For a source moving to the right, letting out a series of “bangs,” the location of the successive pressure peaks will be closer together on the right and further apart on the left.

Now remember that a sound is just a series of pressure peaks, which are tightly separated in time, which is the period of the wave. Therefore, instead of thinking of the sound waves as due to “bangs,” you can think of them as pressure peaks in a periodic sound wave. In front of the moving object the peaks are closer together. That means that the wave length is shorter, which means it is a higher frequency wave. Behind, the peaks are farther apart, meaning that it is a longer wavelength sound, at a lower frequency. This is the gist of the Doppler effect.

Let us now actually calculate the size of the effect. Suppose a sound source is moving right at you, at velocity v. At time 0, it emits a pressure peak. At time Δt, it emits a second pressure peak. If its distance from you at time 0 was x, its distance from you at time Δt was x – vΔt (it is nearer, since it is moving towards you).

The times that the two pressure peaks will reach you, are the time they were emitted plus the propagation distime. For the first peak, that means,

t_arrival1 = 0 + x / v_sound

while for the second peak, it is

t_arrival2 = Δt + (x – vΔt) / v_sound.

The actual period T and frequency f of the sound being produced are,

T_produced = t_pulse2 – t_pulse1 = Δt

f_produced = 1 / (T_produced) = 1/Δt.

The period and frequency you perceive, are the time difference of the arrivals of the pulses, and its inverse:

T_observed = t_arrival2 – t_arrival1 = ((v_sound – v) / v_sound) Δt

f_observed = 1 / T_observed = (v_sound / (v_sound – v)) f_produced

or,

Approaching Source: f_observed / f_produced = v_sound / (v_sound – v_source)

= T_produced / T_observed.

Note that the time, distance, and Δt all do not matter to the final result.

The quick way to understand (and derive) this result is as follows. The distime between when you receive two pressure peaks, is the distime between when they were made, plus the time difference in their propagation times. That time difference is the distance away from you that the source moved between the pulses, divided by the speed of sound. The distance is the time times the speed of the source.

Timespace (1+3): f ↔ h, s ↔ t, S ↔ T, u ↔ v, w ↔ x

If a source of sound makes a sudden bang (pressure pulse) at stance 0, then the tritime of the pressure pulse in relation to stance will look like successive concentric circles. For a source moving to the right, letting out a series of “bangs,” the location of the successive pressure peaks will be closer together on the right and further apart on the left.

Now remember that a sound is just a series of pressure peaks, which are tightly separated in stance, which is the wavelength of the wave. Therefore, instead of thinking of the sound waves as due to “bangs,” you can think of them as pressure peaks in a periodic sound wave. In front of the moving object the peaks are closer together. That means that the frequency is higher, which means the wavelength is shorter. Behind, the peaks are further apart, meaning the frequency is lower and the wavelength is longer. This is the gist of the Doppler effect.

Let us now actually calculate the size of the effect. Suppose a sound source is moving right at you, at lenticity u. At stance 0, it emits a pressure peak. At stance Δs, it emits a second pressure peak. If its distime from you at stance 0 was w, its distime from you at stance Δs was w – vΔt (it is nearer, since it is moving towards you).

The stances that the two pressure peaks will reach you, are the stance they were emitted plus the propagation distance. For the first peak, that means,

s_arrival1 = 0 + w / u_sound

while for the second peak, it is

s_arrival2 = Δs + (w – uΔs) / u_sound.

The actual wavelength S and periodicity h of the sound being produced are,

S_produced = s_pulse2 – s_pulse1 = Δs

h_produced = 1 / (S_produced) = 1/Δs.

The wavelength and periodicity you perceive are the stance difference between the arrivals of the pulses, and its inverse:

S_observed = s_arrival2 – s_arrival1 = ((u_sound – u) / u_sound) Δs

h_observed = 1 / S_observed = (u_sound / (u_sound – u)) h_produced

or,

Approaching Source: h_observed / h_produced = u_sound / (u_sound – u_source)

= S_produced / S_observed.

Note that the stance, distime, and Δs all do not matter to the final result.

The quick way to understand (and derive) this result is as follows. The distance between where you receive two pressure peaks, is the distance between where they were made, plus the distance interval in their propagation stances. That distance difference is the time away from you that the source moved between the pulses, divided by the pace of sound. The distime is the stance times the pace of the source.