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Doppler Effect and Math

Date: 6/1/96 at 14:55:52
From: Anonymous
Subject: Doppler Effect

Dear Dr. Math,

I am doing a research project for Algebra class on the Doppler Effect.  
My partner and I have done extensive research and gathered much 
information, but we are stumped on how the Doppler Effect is related 
to mathematics.  The point of this project is to find the relevance 
between these two subjects, and we just can't figure it out. Please 
help us!
Brian and Tim

Date: 6/1/96 at 19:27:39
From: Doctor Sarah
Subject: Re: Doppler Effect

Hi Brian and Tim -

Here's part of an answer we sent out a while back to a question about 
the Doppler effect.  Maybe you can use it to figure out more examples.

"As an ambulance approaches, the sound waves from its siren
are compressed towards the observer. The intervals between
waves diminish, which translates into an increase in frequency
or pitch. As the ambulance recedes, the sound waves are
stretched relative to the observer, causing the siren's pitch to
decrease. By the change in pitch of the siren, you can determine
if the ambulance is coming nearer or speeding away. If you
could measure the rate of change of pitch, you could also
estimate the ambulance's speed."

Can you work out how this can be done?

The whole answer can be found at:   

-Doctor Sarah,  The Math Forum
 Check out our web site!   

Date: 6/3/96 at 11:18:50
From: Doctor Mike
Subject: Re: Doppler Effect

Hello Doppler Questioner,
Let me add to Doctor Sarah's answer.

I'm glad you asked that question about what the Doppler
Effect has to do with mathematics.  Except for details,
the answer is the same for any scientific subject area.
Whenever you use an equation or a function to express
the relationship between related things, you are using
mathematics.  The value in Dollars of a pile of Pennies
is expressed by the equation D = P/100.  Often this is
expressed using the idea of a function, like D = H(P) 
where H is the "divide by 100" function. 
Science research results in theories about how things
relate to each other.  For the Doppler Effect, if the
distance changes between you and a sound wave source,
like the horn of a car driving past as you walk on a
road, the sound frequency Fh that you "hear" depends on
both the "true" sound frequency Ft, and also the Speed
of the passing car.  So, you get a conversion function

              Fh = C( Ft , Speed )  

Your research should tell you what this equation is.
You can then do interesting variants, such as how the
equation changes if you aren't at the side of the road,
but a block away.  Graph these functions.  Try to do a
Science Fair demonstration showing Doppler Effect for
water waves, which are the ones we can see directly. 
Try to get equations for other wave situations, like 
Red-Shift in astronomy, sonar, or Police radar speed 
I hope this helps.
-Doctor Mike,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Physics/Chemistry

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