- Phase velocity
is the propagation velocity
of a wave phenomenon relative
to the (transmission) medium,
and
**NOT**the velocity of oscillation of whatever is oscillating to make the waves.The absolute value of the phase velocity (i.e., the phase speed) in the medium frame of reference is set by the nature of the waves and the (transmission) medium: it is an invariant for a given system.

It is

**NOT**affected by the motion of the source of the waves. - For our derivation, the
phase velocity v
_{ph}of waves relative to the medium reference frame is positive/negative if the waves are traveling to in positive/negative direction (i.e., to the right/left in the diagram).The invariant phase speed is |v

_{ph}|. - The observer of the waves
is moving with velocity v relative
to the medium.
If v > 0 / v < 0, the observer is moving in the positive/negative direction.
Any motion of the observer perpendicular to the propagation direction gives no Doppler effect, and so we do

**NOT**consider such motion further. We are considering just a 1-dimensional case. - The observer observes the
waves
to have observer-frame phase velocity
v
_{ph}'.The prime symbol means the quantity is being measured in the observer frame.

If a quantity has

**NO**prime, it means the quantity is explicitly a medium-frame quantity or is frame-invariant. - Note v
_{ph}' = v_{ph}- v. - The wavelength λ is the
length a complete spatial cycle of the waveform
and is the distance between any two corresponding points on
adjacent waveform cycles.
We require the wavelength to be always positive in the formalism we develop here.

Since our derivation is in the classical limit, length is frame-invariant, and so λ is the same for all observers no matter how they are moving. Thus, we do

**NOT**give λ a prime for the observer frame.Wavelength λ

**DOES**depend on the motion of a source as we show below.Also, since we are in the classical limit, the flow of time is frame-invariant too.

- The observer-frame phase velocity
must obey
### v

_{ph}' = ±λ/p' ,where the upper/lower case of "±" is for waves moving in the positive/negative direction in the observer frame (

**NOT**the medium frame) and p' is the time period it takes for a wavelength to pass the observer. - Clearly, N waves pass an observer in Np' periods.
We define the directional frequency F ' to
be the number of waves passing the observer in the
positive direction to be
### F' = ± N/(Np') = ±1/p' = v

_{ph}'/λ .where the positive/negative case is for the waves moving in the positive/negative direction.

The funky directional frequency is useful in getting nice formulae without the explicit "±" symbol.

- From the last equation follows the formula
### F'λ = v

_{ph}'which is called a dispersion relation.

It is a dispersion relation for a non-dispersive medium since directional frequency F ' and inverse of wavelength are linearly related to each other.

The version of the dispersion relation for an observer at rest in the medium frame

### Fλ = v

where only the v_{ph}_{ph}is, as aforesaid, the medium-frame phase velocity whose absolute value is an invariant for a given system.The version of dispersion relation that everyone remembers just uses absolute values:

### |F'|λ = |v

_{ph}| ,where the absolute value signs are suppressed if you know what you mean.

- Dividing the third to last formula by the
second to last formula gives
### F'/F = v

_{ph}'/v_{ph}= (1 - v/v_{ph})which we rearrange to get

### F' = F(1 - v/v

_{ph}) .Now we that there is a sign change between F' and F only when v/v

_{ph}> 1 which means that the observer is, in fact, overtaking the waves: i.e., plowing into their back ends whether the waves are moving in the positive or negative direction.Here we introduce (ordinary) frequency f '. In the medium frame, we define f = |F| which means means the medium-frame frequency is always positive which is a darn good convention.

Thus, have F = ±f where the upper/lower case is for positive/negative directional frequency.

Substituting for F in the last displayed formula, we get

### F' = ±f(1 - v/v

_{ph}) and then ± F' = f(1 - v/v_{ph}) .We define f ' = ±F' which when you think about it---while jumping up and down on one foot with your hands over your eyes---that f ' is only negative for v/v

_{ph}> 1: i.e., when there is a change of sign from the always positive f: i.e., when the observer is overtaking the waves. This is a darn good result in my view.We now obtain

### f ' = f(1 - v/v

_{ph}) ,This formula is the most basic Doppler effect formula for the classical limit.

Many intro physics textbooks make heavy weather of the derivation of the formula, but it's really easy once you recognize that wavelength is the observer-frame invariant in the classical limit.

Yours truly has made heavy weather of the derivation by worrying about directions of motion and the observer overtaking waves---but really you have to worry about things if you want to be general---which you do.

- Let's explore the behavior of
### f ' = f(1 - v/v

_{ph}) .Behavior:

- f ' > f for v/v
_{ph}< 0 : this is a blueshift. - f ' = f for v/v
_{ph}= 0 : this is a**NO**Doppler shift case. - 0 < f ' < f for 0 < v/v
_{ph}< 1 : this is a redshift without the observer overtaking the waves. - f ' = 0 for v/v
_{ph}= 1 : this a redshift to zero frequency and occurs when with the observer moves at the phase velocity. - -1 < f ' < 0 for 1 < v/v
_{ph}< 2 : this is a redshift with the observer overtaking the waves. - f ' = -f for v/v
_{ph}= 2 : this is a**NO**Doppler shift case in my view since the absolute value of frequency has**NOT**changed. - f ' < -f for v/v
_{ph}> 2 : this is a blueshift in view since the absolute value of frequency has increased.

- f ' > f for v/v
- Now say we have a source with motion-independent fixed
frequency f_source.
Our conventions require |f '| = f_source in a general the source frame
moving at general velocity v.
In the medium frame, the frequency is

### f = f '/(1 - v/v

_{ph}) ,f ' = |f '| = f_source if v/v

_{ph}≤ 1 (i.e., the source is**NOT**overtaking the waves it generates) and f ' = -|f '| = -f_source if v/v_{ph}> 1 (i.e., the source is overtaking the waves it generates). The convention for f ' for the source just ensures that medium-frame frequency f obeys the convention of always being positive.The wavelength of the waves is given by

### λ = |v

_{ph}|/f = |v_{ph}(1 - v/v_{ph})/f '| = |(v_{ph}- v)/f '| .This wavelength is, of course, frame invariant and all observers (including one moving with the source) observe it.

Note the λ does depend on |v

_{ph}-v| and |f '|.If v = 0, then λ has the value it would have for the source at rest in the medium which can be considered the source's intrinsic wavelength---but it is

**NOT**the wavelength observed by any observer when the source is moving---that is wavelength is given by the formula above. - Now say the source is observer 1 with
velocity v_1 and
frequency f_1'
and we have observer 2 who is a receiver with
velocity v_2 and
frequency f_2'.
We find the general classical Doppler shift formula (hereafter just the Doppler shift formula)

### f_2' = f_1'(1 - v_2/v

_{ph})/(1 - v_1/v_{ph}) ,where recall v

_{ph}gives the direction of motion of the waves by its sign. - Elucidating all the behavior of the
Doppler shift formula
is tricky, but to
1st-order exactness
in small |v_1/v
_{ph}| and | v_2/v_{ph}| we find the simpler 1st-order Doppler shift formula### f_2'_1st_order = f_1'[1 - (v_2-v_1)/v

_{ph}] ,where we have expanded the denominator in the geometric series and truncated to 1st-order in small |v_1/v

_{ph}| and | v_2/v_{ph}|. The relative error from the exact Doppler shift formula varies as with |v_1/v_{ph}| and |v_2/v_{ph}| for |v_1/v_{ph}| << 1 and | v_2/v_{ph}| << 1.We see that to 1st-order the Doppler shift depends on the only the relative velocities of the source and receiver via the quantity (v_2-v_1)/v

_{ph}**NOT**on the individual sizes of v_1 and v_2 as is the case for the full Doppler shift formula. In contrast, the relativistic Doppler effect in general only depends on the relative velocities.From the 1st-order Doppler shift formula, we see there is a redshift/blueshift when (v_2-v_1)/v

_{ph}is positive/negative: i.e., when displacement is increasing/decreasing if we take the direction of v_{ph}to define the positive direction.Note the use of displacement,

**NOT**distance (which is the absolute value of displacement)in the last statement. As matter of fact, redshift/blueshift when distance is increasing/decreasing too. We give the argument below when considering the Doppler shift formula. - The 1st-order Doppler shift formula
can be written in the simplified memorable form
### Δf/f = -Δv/v

_{ph},where Δf is change in frequency, f is either of f_1' or f_2' since it is all the same to 1st-order in small |v_1/v

_{ph}| and | v_2/v_{ph}| and Δv = v_2-v_1.Actually, the same 1st-order formula fortuitously applies to EMR with vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s replacing v

_{ph}(see Wikipedia: Relativistic Doppler effect: Motion along the line of sight). - To elucidate all the behavior of the
Doppler shift formula
it is best to work in reduced quantities.
We define the reduced quantities S = f_2'/f_1', x = v_1/v

_{ph}, and y = v_2/v_{ph}and obtain the Doppler effect formula### S = (1 - y)/(1 - x) .

We will call S the Doppler shift and note that there is a redshift/blueshift for |S| < 1 / |S| > 1.

Note also x and y are reduced velocities,

**NOT**positions.For clarity, we will say x observer for source and y observer for receiver.

- Using reduced velocities effectively defines
the direction of
wave propagation as the positive direction
and makes the reduced phase velocity 1.
This convention is
**NOT**a limitation since all directions of observer motions are considered by letting x and y go negative.If x < 1 (i.e., the x observer reduced velocity is less than 1), there can only be waves to the right of the source since the x observer cannot outrun the waves it generates.

- The innocent little 2-variable
reduced Doppler shift formula
has a rich structure.
The diagram below illustrates its various domains of behavior in the x-y plane.

The S would be plotted in the third dimension with an axis perpendicular to the x-y plane.

To elucidate the figure:

- The y = x, S = 1 line gives no Doppler shift.
- The y = 2 - x, S = -1 line gives no Doppler shift
since |S| = 1.
- The y = 0, S =0, has the
frequency
Doppler shifted to zero.
- The x = 0, S = ±∞ line has the
frequency going to an infinite
discontinuity:
the exact specification is
essential discontinuity.
The point (x=1,y=1) is particularly the pathological.
Consider the line y = s(x-1) + 1 in the
x-y plane, where s is
an general slope.
This line always passes through point (x=1,y=1) and along the line
we have S = s eveywhere except the point (x=1,y=1), where the S is undefined
undefined.
However, clearly the
limit as you go to
point (x=1,y=1) is s.
This means the set of
limit at point (x=1,y=1) includes all
real numbers and ±∞.
- If you go along a vertical line with x < 1 / x > 1, S is always decreasing/increasing with increasing y.
Thus these lines together with the y = x and y = 2- x lines allow us to identify which
the octants have
redshifts
(octants with labels
-1 < S < 0 and 0 < S < 1)
and which have
blueshifts
(octants with labels
S > 1 and S < -1).
- The 2 quadrants with |S| > 1
are the blueshift
quadrants.
- The 2 quadrants with |S| < 1
are the redshift
quadrants.
- Note that the
1st-order
Doppler shift
formula in reduced quantities is
### S_1st_order = 1 - (y-x) ,

which is valid for |x| << 1 and |y| << 1: i.e., the small domain near the origin in the diagram.

For the 1st-order Doppler shift formula in its domain of validity, y > x / y < x is always redshift/blueshift.

Note since x < 1, the x observer cannot outrun the waves it generates. Thus, there are only waves to the right of the x observer.

Thus, the y observer observes

**NO**waves if it is left of the x observer.If y > x, then no matter what the initial conditions, the y observer will eventually (if not initially) be right of the x observer and observe redshifted waves. Since y < 1, the y observer cannot outrun waves once it starts seeing them, the y observer will see them as long as the setup continues.

When the y observer is observing waves, the distance between the observers is increasing.

If y < x, the distance between the observers is always decreasing as long as the y observer is observing waves. Once it gets left of the x observer there are no more waves to observe.

- Now to return to the general
Doppler shift
formula.
The diagram shows that for both y > x and y < x, there are redshift and blueshift domains.

- Consider the y > x domain.
Eventually (if not initially), the y observer will be right of the x observer and will observe redshifted waves only if y < 1 or either redshifted or blueshifted waves if y > 1 and x < 1 or only blueshifted waves if x >1.

If y > 1, the y observer will eventually outrun the waves if it started a finite time in the past.

Note that if x > 1, then the x observer will outrun the waves it generates and so the y observer (who must also obey y >1) can observe them eventually (if not initially) left of the x observer if the y observer started left of the x observer.

- Consider the y > x domain.
Eventually (if not initially), the y observer will be right of the x observer and will observe redshifted waves only if y < 1 or either redshifted or blueshifted waves if y > 1 and x < 1 or only blueshifted waves if x > 1.

If y > 1, the y observer will eventually outrun the waves if it started a finite time in the past.

Note that if x > 1, then the x observer will outrun the waves it generates and so the y observer (who must also obey y >1) can observe them eventually (if not initially) left of the x observer if the y observer started left of the x observer.

This means there can be an observed blueshift with the distance between the two observers decreasing.

- Consider the y < x domain.
If the y observer has y > 1, then eventually (if not initially) the y observer observe redshifted waves generated by the the x observer and then will continue to observe them as long as the setup continues. If the y observer started right of the x observer, then the the distance between the observers will decrease until the x observer passes the y observer and then it will increase. So one can have a redshift with decreasing distance for awhile.

If the y observer has y < 1 and starts left of the x observer then the y observer never observes waves.

If the y observer has y < 1 and starts right of the x observer then the y observer will observe waves eventually (if not initially) until the x observer passes the y observer. In this case, the y observer will observe either redshifted or blueshifted waves if y < 1 and x > 1 or only blueshifted waves if x < 1.

Note that if the y observer is right of the x observer and observes

Credit/Permission: © David Jeffery, 2016 / Own work.

Image links: themselves: Figure 1, Figure 2.

File: Waves file: doppler_effect_derivation.html.

- The y = x, S = 1 line gives no Doppler shift.

Caption: The shown diagrams are used in the derivation below of the Doppler effect for mechanical waves in the classical limit (AKA the nonrelativistic limit) for 1-dimensional motion in a non-dispersive medium (i.e., one in which the phase speed is independent of frequency).

We are treating an ideal special case, but it's useful for understanding the Doppler effect in general and it does apply to good accuracy in some cases (e.g., sound waves in many cases).

Note that the relativistic Doppler effect has somewhat different formulae and derivation. The relativistic Doppler effect applies to electromagnetic radiation (EMR) in vacuum which is intrinsically relativistic since EMR in vacuum travels, of course, at the vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s .

Derivation: