- Mach number is the
velocity in units
of the sound speed
of the ambient medium.
- The sound waves
from point source are
spherical waves with
respect to the medium.
The spherical waves in the animations are seen in cross section.

- To analyze the system, consider the
relationship between the time the source emits a sound signal and the time an observer observes it.
The diagram illustrates our setup.

We take the observer to be at x = 0 a perpendicular "a" from the trajectory of the moving source. This sort of distance is called impact parameter, in fact.

We take time zero to be when the source is at x = 0 and let the velocity of the source be v. Without loss of generality, we assume v > 0.

The position of the source at any time t is x = vt. Note negative/positive time gives negative/positive position x.

The sound speed is the phase velocity v

_{ph}of the medium. - The time observation t' is given by
### t' = t + sqrt(x**2+a**2)/v

_{ph}### = t' + sqrt[(vt)**2+a**2]/v

_{ph},Now it is is convenient to switch to reduced quantities: τ = vt/a, τ' = vt'/a, β = v/v

_{ph}.In reduced quantities, the observation time is

### τ' = τ + β*sqrt(*tau;**2+1)

and the delay time

### Δτ = τ' - τ = β*sqrt(*tau;**2+1)

We note that Δτ > 0 always. So the signal is always observed after it is sent---which is a darn good thing.

- In order understand the behavior of τ', we
take the derivatives
of τ' and Δτ with respect to τ to get
### dτ'/dτ = 1 + βtau;/sqrt(*tau;**2+1) and

### dΔτ/dτ = βtau;/sqrt(*tau;**2+1) .

Note that Δτ has a single stationary point: minimum of β at τ = 0.

This is just as it should be: for τ = 0, the source is at x = 0 and is just a distance "a" from the observer.

The We now consider the behavior cases different β regimes.

Behavior cases:

- Subsonic case with β < 1:
In this case, dτ'/dτ > 0 always and τ' is
strictly increasing
with τ.
The only thing to say is the observer receives a signal for all times from τ' = -&infin to τ = +∞.

- Supersonic case with β ≥ 1: In this case, τ' has one stationary point which is minimum
- Mach 0 case has the
point source at rest in the
medium.
In this case, the spherical waves
expand from a fixed point in space and there is
**NO**Doppler shift for an observer at rest in the medium. - Mach 0.7 case has the
point source
in subsonic motion
in the medium.
Since the motion is subsonic, the waves outrun the point source in the forward diection (i.e., its direction of motion) as well as all other directions.

An observer at rest in the medium, observes a Doppler shift, except along a line perpendicular to the direction of motion of the point source.

If the point source is negative/positive of the observer, the observer observes a blueshift/redshift Mach number is less than 1. When the Mach number ≥ 1, the situation is trickier. We discuss below.

- We can get the
Doppler shift formula
for an observer at rest in the medium
for all cases with a little bit of analysis.
We say there is an observer at point on the x = 0 line a general perpendicular distance "a" from the trajectory of the point source and define time zero to be when the point source is at x = 0.

The point source has general position x = vt where v is the point source velocity which without loss of generality we take to be positive.

If a wavefront starts out from the point source at time t', it arrives at the observer at time t given by

### t = t' + sqrt(x**2+a**2)/v

_{ph}### = t' + sqrt[(vt)**2+a**2]/v

_{ph},where v

_{ph}is phase velocity (really phase speed) of the sound wave in the medium frame.Taking the differential with respect to t' gives

### dt = dt' + (v/v

_{ph})*vt*dt'/sqrt[(vt)**2+a**2] = dt'*[1 + βz/sqrt(z**2+1)] ,where β = v/v

_{ph}is Mach number and z = vt/a.Now frequency f = 1/p, where p is the time for a spatial wave cycle to pass a point. If we take dt' to be period for the point source wave emission, then dt is the period for the observer.

Thus we get the Doppler shift formula

### f = f '/[1 + βz/sqrt(z**2+1)] or S = 1/[1 + βz/sqrt(z**2+1)] ,

where f is the observer frequency, f ' is the point source frequency, and S = f '/f is a ratio we call the relative Doppler shift.

- We can fully explore the
Doppler effect for our
system using
the Doppler shift formula for S:
### S = 1/[1 + βz/sqrt(z**2+1)] ,

- If we Taylor expand to
1st-order
in small z and small βz, we get
the 1st-order Doppler shift formula
### S_1st_order = 1 - βz

which is a valid approximation to the exact Doppler shift formula when z << 1 and βz << 1. The relative relative error grows as z and βz.

Note the motion of point source can subsonic (β < 1), sonic (β = 1) , or supersonic sonic (β > 1) as long as βz << 1.

- The derivative
### dS/dz = -β/{(z**2+1)**(3/2)*[1 + βz/sqrt(z**2+1)]**2} .

The derivative is negative for all z, except at z = ±∞ where it is zero which means z = ±∞ are the only stationary points and at the infinite discontinuity that exists if 1 + βz/sqrt(z**2+1) goes to zero which it can do for β ≥ 1.

- Consider the case of β < 1 (i.e., the subsonic case).
In this case, there is no
infinite discontinuity.
In this case, S = 1/[1-β] at z = -∞ which is the global maximum, decreases to S = 1 at z = 0, and then decreases to S = 1/[1+β} at z ∞ which is the global minimum.

thu Thus, as stated above in different, If the observer is in front/behind this line, it observes a blueshift/redshift Mach number is less than 1.

# Under construction below

relativeWe now see explicitly if the frequency is to the negative/positve (i.e., t and z are negative/positive), then we have the observer observes a blueshift/redshift at least when β < 1.

- What if β ≥ 1?
In this case, there is a z value for which f ' goes to infinity due to the the denominator in the Doppler shift formula going to 0: i.e., z

_{∞}We can solve for this z

_{∞}:### 0 = 1 + β*z/sqrt(z**2+1)

### -1 = β*z/sqrt(z**2+1)

### -sqrt(z**2+1) = β*z

### z**2+1 = (β*z)**2 .

### Thus, z

_{∞}= -1/sqrt(β**2-1) ,where the positive root obtained in getting the last step is

**NOT**a solution of the original equation and has been suppressed. - What does infinite frequency mean?
For ideal case, the answer might be as follows.

Note first frequency only goes to infinity for an instant in time for the observer which means unobservable in reality and even if ideally if you desire to think of it that way. However, in a time interval around that instant a very high frequency would be observed. The better the time resolution of the observer, the higher frequency it could observe, but no real observer has infinite resolution.

If the high frequency were beyond its time resolution, the observer

**MIGHT**perceive an average of nothing due oscillations that cancel on average.I think real systems might behave as I have just described, but I'm not sure. In real systems, there are usually many complicating effects that can completely negate the predictions of infinities made by ideal cases.

- What about
supersonic aircraft?
I think they are significantly different from ideal system. It produces a continuous stream of wave cycles with wavelength that depends on source velocity and the direction of emission.

Supersonic aircraft produce a single shock wave with a cone shape and a wavelength that depends on the aircraft along with ??? velocity and the direction of emission???.

So an ordinary Doppler effect analysis doesn't apply. The wavelength

**CANNOT**be squeezed to zero.What does apply is that there is time for an observer at rest when the emission from a supersonic aircraft is concentrated. This time is the conical shock wave (which is manifested as a sonic boom) passes the observer.

Say in its own frame, the supersonic aircraft produces constant sound intensity I'. The integrated sound intensity in time Δ t' is I'Δt' assuming no canceling interference

The observer detects the same integrated sound intensity I'Δt' in time time interval Δt.

The observer detects sound intensity I = I'Δt'/Δt

- If we Taylor expand to
1st-order
in small z and small βz, we get
the 1st-order Doppler shift formula

- Subsonic case with β < 1:
In this case, dτ'/dτ > 0 always and τ' is
strictly increasing
with τ.

Caption: Animations illustrating dynamically the sound waves and Doppler effect for an ideal isotropic point source of sound producing a continuous stream of wave cycles moving with a range of velocities (Mach 0, Mach 0.7, Mach 1.0, Mach 1.4) with respect a medium.

Features:

Animation link: Wikimedia Commons.

File: Waves file: doppler_effect_sonic.html.