Wavelength and Doppler (was Re: Basic Acoustic Derivation/Proof Needed)

From: Jim Carr (jim_at_azwebpages.com)
Date: 10/09/04 

Date: Sat, 9 Oct 2004 15:37:53 -0700

"RP" <no_mail_no_spam@yahoo.com> wrote in message
news:2sqt5tF1nlorkU1@uni-berlin.de...

> Of course direction cannot be discounted, which is why the equations
> that I posted refer to the velocities rather than to the scalar speeds.
> In hindsight, my last statement above should have read:
>
> wavelength = u/freq
>
> is only valid when the frequency is that measured by a detector or
> source that is at rest wrt the medium, or that is in motion at 90 or 270
deg
> wrt the projected ray (in which case cos_theta = 0).
> Or IOW, when (cos v) = 0.

Which agrees with what I was saying, though I admit to never seeing your
formulas with cos before. Like I said, this is not my field. If it's okay
I'm gonna assume the medium doesn't move for the purposes of this
discussion.

> But I have no idea what "apparent" wavelength and frequency is supposed
> to mean, except maybe that the observer was "apparently" wrong about one
> or the other :)

Which, again, agrees with my point. I'll attempt to address "apparent
wavelength" if you'll bear with me to the end before responding. You and I
are not in disgreement. I'm ignoring Porky.

We take the simple wavelength = u/freq formula with everything at rest and
we see an inverse relationship between frequency and wavelength. I never
disputed that. It's not unique to waves. If automobiles driving at 10 feet
per second pass us at a rate of two per second, we can calculate the
distance between the cars. It's just grade school algebra that even I can
do. Of course, we'll have to measure the same point on each car such as the
front bumper (gotta be precise with you physics guys).

So, as we discussed Doppler shift, I got to thinking about what was really
happening. If my source is moving, then the physical distance between two
waves is the speed of the waves times the time interval between waves plus
or minus the speed of the source times the time interval. With a stationary
receiver and medium we can mathematically derive the frequency at which the
receiver encounters the waves. Again, I believe we agree on this part.

But as I was thinking about the receiver moving with a stationary medium and
source, it occured to me that while his movement alters the frequency at
which the waves arrive over time, the distance between the waves really
didn't change. Please keep reading as I address this.

Therefore, our formula didn't apply anymore. This agrees with your
assertation that the formula only works for everything being still. All I
did was add to the formula some compensation for the movement of the
receiver. With that new formula I ended up calculating the same physical
distance for the wavelength at the stationary source regardless of whether
the receiver was moving or not. I also used Doppler to get the correct
frequency as measured at either source.

It's easy to visualize. Drop pebbles into a still pond at a constant rate.
We watch the rings radiate from the center. From above we can watch them
stay the same distance apart. We see a bee fly over the pond on a bee-line
for the point where the pebbles are dropped. How do we calculate the
frequency at which the bee passes over each wave? If we know the speed of
the waves, the speed of the bee and the frequency of the waves, we can
calculate that frequency. This is the Doppler shift formula simply stated.

But if we use the simple wavelength formula, we arrive at a number that
disagrees with what we see from above. If we use mine, we get the same
number. If we use the simple formula (which assumes everything to be still)
in this situation with the receiver moving, it REALLY tells us the how far
Wave B traveled after the bee crossed over Wave A. Is that wavelength?

No. We we know wavelength is defined as the distance between the same points
on two waves, NOT the distance traveled. A subtle but important difference.
Only in the case of everything being still are the wavelength and the
distance traveled the same. For most discussions of audio that assumption
suffices.

Thus if we use my formula we can arrive at the correct wavelength in all
cases of the source or receiver moving in a stationary medium. Am I totally
whacked or is my theory, pardon the pun, sound?

Just to comment on the semantics: Apparent can mean either readily seen *or*
readily clear/understood *or* appearing but not necesarily so. If you say
the apparent wavelength changes using the third definition, then we can
ignore the movement of the receiver when calculating that number. But why
would we do that when we know that formula is only measuring the distance
the second wave traveled, not the distance between the same points on the
wave? As we've seen we can easily calcuate the correct wavelength.

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