When I was a commissioning engineer I adjusted the frequency of
transmitter 'A' (about 140 MHz) using an accurate frequency meter.

Then I drove to transmitter 'B' (about 30 miles away) where I adjusted
the frequency to be the same as transmitter 'A' (to within a few
Hertz).

Then I jumped into the car and headed back towards transmitter 'A'. 
In the car was a receiver tuned to the same frequency.  I turned-on
the receiver and was surprised to hear a howling beat frequency.  The
faster I drove, the higher the pitch.  When I slowed-down the pitch
reduced, becoming inaudible when I was stationary.

How can this effect be explained ?  It cannot be the Doppler Effect
because Mr. Doppler made all his experiments with sound waves.  Sound
waves must travel in a medium, typically air.  Hence the incident
velocity relative to an observer varies according to the speed of the
observer passing through the medium.

This is not true with radio waves whose incident velocity is constant
at the speed of light and independent of the speed of the observer
(according to Einstein).

Can anyone provide a formula to calculate the value of the beat
frequency for any given car velocity ?  I think Doppler's equations
won't work.

---

Request for Question Clarification by pafalafa-ga on 14 Jan 2006 10:22 PST

I'm not really qualified to answer this, but I can't help but ask: 
Are you sure you're not just picking up interference from the car
engine?

---

Clarification of Question by sorwin-ga on 16 Jan 2006 15:51 PST

I now know that the answer is provided by:
        http://www.mathpages.com/rr/s2-04/2-04.htm

Actually, have you ever heard of the famous "red shift" the doppler
"effect" isn't limited to sound.

I would expect you would have to be given a ticket for far exceeding
the speed limit-- by tens of thousands of kilometers per second-- for
the frequency shift you describe.

Did your car have ABS, an electronic speedometer, or some sort of
sensor system in the vehicle's wheels or final drive?

Here are two equivalent ways to think about this.

1) As you travel toward transmitter A the signal you see from that
tower is "blue shifted". The amount of the shift is f*v/c, where f is
140 MHz, v is your speed (let's say 30 meters/second, or ~100
km/hour), and c is the speed of light (300 million meters/second). So
in this case the shift would be about 50 Hz. At the same time, the
signal from transmitter B is "red shifted" by the same amount. Thus,
you would experience a beat between these two of about 100 Hz. The
pitch will be directly proportional to your speed.

2) Alternatively, think about the interference between the two waves.
At some points the waves interference constructively, at other points
they interfere constructively. This creates a stationary pattern of
waves, so-called standing waves. As you drive, you are traveling
through this pattern of standing waves and, the faster you drive, the
faster you pass the crests and the troughs. Again, you hear a tone
whose pitch increases in direct proportion to your velocity
perpendicular to the wave fronts.

This effect is identical to the sound phenomenon originally studied by
Doppler. It is a classical effect and it makes no difference that
there is no propagation medium and that the speed of light in a vacuum
is constant. It is simply the fact that you go by more wave crests per
seond if you are traveling toward the source and less if you are
traveling away. As your speed gets close to the speed of light, you
have to consider relativistic effects and it gets more complicated.

Kottekoe has the right idea.  It is not the Doppler effect but rather
"wave interference".  You can search with that term on Google, but
maybe these sites will surfice:
http://www.colorado.edu/physics/2000/schroedinger/index.html
http://mysite.verizon.net/vzeoacw1/wave_interference.html
http://online.cctt.org/physicslab/content/Phy1/lessonnotes/waves/lessoninterference.asp

Wave interference is similar to what makes wheels in films appear to
stand still or turn backwards.

Yes, it is wave interference. But it is also a very nice demonstration
of the Doppler effect.

Another nice demonstration of the Doppler effect for radio waves is
police radar. The frequency of the beats between the transmitted and
received signal is given by 2*f*v/c. Someone in the car would see the
signal blue-shifted by f*v/c, but when it bounces back to the
policeman, it gets blue-shifted again by the same amount. Again the
beat frequency is proportional to speed, so it is a great way to
measure speed. If they used a 140 MHz radio, it would give exactly the
same shifts as our questioner experienced.

As Silicon Samurai pointed out, Doppler shifts are used in astronomy
to measure the speed with which objects are moving toward or away from
us. Since the universe is expanding, most of these are "red shifts".

I am grateful for all comments from my respondents.

I do like the concept of "wave interference", but still have a problem
with a Doppler explanation.

The formula:    "frequency shift"  =  2 * f * v/c
is Doppler's equation for sound waves and even I can prove it is true.

But I am not clever enough to prove that it is also true for electromagnetic waves.

Can anyone demonstrate the proof, please ?  This must presume that c
is constant for electromagnetic waves received at the car irrespective
of its velocity.

The Doppler formula for light is exactly the same as for sound and the
derivation is identical. Relativistic effects only come into play if
the relative velocity between the transmitter and receiver gets close
to the speed of light. The factor of two in my formula applies to the
original problem and to the police radar example, but there is no
factor of two if you are talking about a fixed observer and a moving
transmitter, or vice versa. The derivation is quite trivial in the
nonrelativistic limit. For a fully relativistic derivation, which
shows that the formula is the same for light and sound, see the
following:

http://www.mathpages.com/rr/s2-04/2-04.htm

kottekoe is correct.  His (her?) comments pretty much cover
everything, but here I go just the same...

"How can this effect be explained ?  It cannot be the Doppler Effect
because Mr. Doppler made all his experiments with sound waves.  Sound
waves must travel in a medium, typically air.  Hence the incident
velocity relative to an observer varies according to the speed of the
observer passing through the medium."

Yes, but there is also a Doppler shift when the sound source is moving
and the observer is stationary, as when a whistling train passes
someone standing near the tracks.  The whistle has a higher pitch when
approacing the observer, and a lower one when leaving, even though the
speed of the sound waves relative to the observer is the same in each
case.

In the late 1800s, Michelson and Morley did an experiment to test for
changes in the speed of light during different seasons when the earth
was moving in different directions.  They thought that, like any other
wave, light must travel in a medium (the 'ether') and that its
apparent speed would depend on the observer's motion through this
ether.  Thus, the speed of earth relative to the 'rest frame' of the
universe could be determined by precise measurements of the speed of
light.  Of course, they found no change at all in the speed of light,
and their preliminary conclusion was that the moving earth 'swept
along' the nearby ether.  Then Einstein came along with the correct
explanation, no need for ether, no need for a universal rest frame,
special relativity, blah blah blah.  But then, 60 years later or so, a
couple of radio astronomers or something discovered the cosmic
microwave background radiation (CMB).  This is radiation left over
from the big bang, and it is spread nearly uniformly over the entire
sky.  But, there is a dipole in the frequency of the CMB.  In one half
of the sky the frequency is slightly higher, and in the other, it's
slightly lower.  This is a Doppler shift.  It shows that Michelson and
Morley were actually kind of right--the universe does have a unique
'rest frame'.  And we (our whole galaxy) are moving relative to that
frame at some quite high speed.  600 km/s rings a bell, but I'll have
to check on that.  This we know from the Doppler shift of
electromagnetic radiation.

Enough blabbering by me, I guess.  One last thing though:  some people
seem to be under the impression that you must go extremely fast to
cause a noticible Doppler shift in light.  This is actually true in
most cases.  If your radio is stuck on 95.7 FM, and you'd rather
listen to 100.3, you must drive toward the 100.3 trasmitter at over
10,000 kilometers per second.  Most cars won't do that.  But in this
case, we're not talking about shifting from 95.7 MHz to 100.3 MHz, but
from 140 MHz to 140.00005 MHz.  That 50 Hz change causes the 100 Hz
beat frequency you hear, and you only have to drive 0.03 kilometers
per second, much safer.

Yes, Kottekoe, that's the answer:
        http://www.mathpages.com/rr/s2-04/2-04.htm

I shall mull over this article at length.  It seems there is rather
more to the Doppler Effect than Doppler ever suspected.

I was very pleased, at that time, to be able to experience the effect
of this physical principle through one of my natural senses.

So thanks again to all contributors who have helped to enlighten me.

I'm thinking it could be engine noise... similar to the noise heard on
blank AM frequencies when you rev your engine.  I would be interested
in doing another experiment without a car.  Eliminate that
possibility.

If the pulsating noise could be heard while walking vs. running, you
could try that.  Or, get in the car again, put it in neutral at a high
speed, and turn off the engine.  See if the noise is still there, and
if it changes as the car slows.

In my opinion this is not due to Doppler effect but interference
fringes formed by the two transmitters. If you want more information
on interference you can check any good optics book or website. (e.g.
http://www.optics.arizona.edu/jcwyant/JoseDiaz/MichelsonInterferometerFringes.htm)
. These fringes are formed due to superposition of two electromagnetic
waves and in a 3D space they are spherical. As you were driving you
were crossing these fringes, i.e. alternet high and low intensity
zones. these were causing the beates. when driving fast the rate of
crossing them was high and so was the pitch.
I hope this will solve your purpose.

Go back and read kottekoe's comments.  The concepts in them are all
correct, even though the numbers are wrong. You can think of it as
driving through interference "fringes", but you can just as correctly
think of it as a beat frequency between doppler shifted signals.

I said kottekoe's numbers were wrong.  The doppler shift in a 140 MHz
signal due to a 30 m/s relative velocity is 14 Hz, not 50.  So driving
at this speed would cause a beat frequency of 28 Hz.  In terms of
standing waves, the wavelength of a 140 MHz EM wave is just over 2
meters.  So you get a node spacing (fringe wavelength) of just over a
meter in the standing wave pattern.  Driving though this pattern at 30
m/s, you pass 28 nodes every second.  So again, 28 Hz.