Doppler puzzle

[Mark Schecter]

Hi, John.

Amy is right, by my lights, however dim or unfocused they surely are. 
Ten cents, or whatever, is a _constant_, or perhaps better, a _ratio_, 
assuming velocity is constant and direction is directly toward or away 
from the source, and speed of sound is constant.

We are starting from a given, that the shift is 10 cents. We are not 
asking how that was determined. My point is, 
_no_matter_what_the_frequency_, a ten cent shift can result from only 
one velocity approaching, and a slightly different velocity departing.

Think about the example I gave, that a source of music shifts in a 
coordinated way, all frequencies within the music shift ten cents 
_together_ at some (fixed) velocity. The variable is not the frequency, 
because _all_ frequencies shift together (multiplied by Amy's constant, 
quoted below) - the variable is the velocity of the bicycle in the 
original poster's puzzler. Maybe the doppler equation needs to be 
altered to work with cents instead of hertz, but given all the 
information, someone with better math than mine, such as you or Amy, 
should be able to substitute a temporary frequency into <some> equation 
to determine the one unknown.

Here's Amy's work from a previous thread on this subject:

Amy Zilk wrote:
 > Vladen's answer is the one I get.
 >
 > f is the frequency of the source
 > f_o is the frequency heard by the observer
 > c is the difference in cents (10)
 > Vs is speed of sound in air (1100 ft/s)
 > V is speed of observer (speed of the guy on bike -- the answer)
 >
 > Doppler shift equation for stationary sound source; observer moving
 > directly to or from the source:
 >
 >      f/f_o – 1 = V/Vs
 >
 > You have to get f/f_o from the 10 cents difference.
 >
 >  f/f_o = 2^(c/1200) = 1.005793
 >
 > Plug it in and get
 >
 > 1.005793 – 1 = V/Vs
 >
 >  0.005793 = V/Vs
 >
 >  0.005793 * Vs = V
 >
 >  V = 0.005793 * (1100 ft/s) =  6.3723 ft/s
 >
 > 6.3723 ft/s (3600s/hr) / (5280 ft/mi ) = 4.3 mph
 >
 > az

John Delmore wrote:
>  
> 
> No dice, Amy, you can’t use this dimensionless ratio in the Doppler 
> equation.  You have to have a frequency.  It DOESN’T matter what pitch 
> (frequency) you measure.  The problem comes when you try to use a 
> logarithmic scale (cents) in a linear equation (Hz in Doppler).
> 
> JD

    and

> *From:* pianotech-bounces at ptg.org [mailto:pianotech-bounces at ptg.org] *On 
> Behalf Of *Amy Zilk
> *Sent:* Wednesday, June 21, 2006 5:46 PM
> *To:* Pianotech List
> *Subject:* Re: Doppler puzzle
> 
>  
> 
> Try dividing any two frequencies that differ by 10 cents.  It's a 
> constant.  Ten cents difference anywhere in the spectrum is a constant 
> frequency ratio of ~1.0579.  The doppler shift is found in terms of this 
> ratio and you can convert the 10 cents difference into the ratio of 
> frequencies.  It's counterintuitive for me (but not for Mark), that it 
> doesn't matter which pitch you are measuring but that's the way it works.
> az