Doppler Cents Puzzler

[Amy Zilk]

You are right that the bicyclist slows down after he passes the ice 
cream truck, but not because ten cents flat isn't the same as ten cents 
sharp.  It's because the bicyclist is traveling the same direction as 
the sound waves away from the truck and in the opposite direction of the 
sound waves toward the truck.  Ten cents flat is the same frequency 
ratio as ten cents sharp. 
az



V T wrote:
> Hi Mark,
>
> It is a good puzzler because it "tests" for three
> pieces of knowledge:
>
> 1. The definition of the "cent"
> 2. The understanding of the Doppler effect, and
> 3. A subtle hitch that happens when using cents
> instead of Hertz with the Doppler formula
>
> Item three may be what Robert was trying to bring up
> for discussion.  When we use the Doppler formula with
> frequencies given in Hz, for constant bicycle velocity
> the pitch is sharp by the same number of Hertz when
> approaching as it is flat when retreating from the ice
> cream truck.
>
> However - as you mentioned in a previous post - when
> working in cents there is a slight difference in
> velocity between approaching and retreating, since we
> have the rule that the sharp/flat amount is +/- 10
> cents.
>
> In fact, in our puzzler there is a 0.579% difference
> in speed between approach and retreat.  That's about
> 41.7 m/hr difference.  So he did actually experience a
> sort of a sudden but small "ice cream pull" as he
> passed.  I jokingly said "he didn't even slow down" -
> not true, he did!
>
> Here is all the math needed for the puzzler:
>
> f'=frequency heard by moving Robert (Hz)
> f=frequency emitted by ice cream truck (Hz)
> v=bicycle speed (positive for approaching, negative
> for retreating)
> c=speed of sound, about 345 m/sec
>
> f'=f(1+v/c) (Doppler's formula)
>
> You can rearrange this:
>
> f'/f=1+v/c
>
> Note now that f'/f is equal to our ratio of 1.00579
> (10 cents), or 2^(1/120)
>
> Vladan
>
> ========================================
>
> Mark wrote:
>
> Thanks for your thorough explanation. This puzzler
> continues to 
> fascinate, in part because I feel close, but no cigar,
> to having enough
> math to figure it out myself. I do understand your
> point about 2^1/12 
> and 2^1/1200. Out of context, I knew that. Where I get
> into trouble is 
> connecting a mathematical representation to a
> "real-world" problem. I 
> appreciate your (and Amy Z's and John D's) abilities
> in that regard. 
> Sorry it took me so long to reply.
>
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