Compensating for Doppler Effect in Moving Car: Formula and Graphing

[bwinter]

Homework Statement

Trying to find the formula to generate a sin wave that would compensate for the Doppler effect if played from a car moving 50 mph past a stationary observer 1 meter from the car's path.

Homework Equations

$ƒ_{observed} = \frac{v}{v+v_{s}}ƒ_{source}$

The Attempt at a Solution

Tried to work this out using variables first. Say d is the distance from observer to car's path.

First, we want to keep the observed frequency constant, so rewrite Doppler formula for source:
$ƒ_{source} = \frac{v+v_{s}}{v}ƒ_{observed}$

Then, taking the component of the car's velocity towards the observer
$V_{o} = V_{s}cosθ$

Where θ is the angle between the car's path, and the direct line of sight to the observer.

But we want this in terms of d, time t and V_s, so we can rewrite θ thusly

$θ=tan^{-1}(\frac{d}{V_{s}t})$

And then plugging back into V_o, we get
$V_{o}=\frac{V_{s}}{\sqrt{(\frac{d}{V_{s}t})^{2}+1}}$

So plug this back into our Doppler equation.

$ƒ_{s}=\frac{v+\frac{V_{s}}{\sqrt{(\frac{d}{V_{s}t})^{2}+1}}}{v}ƒ_{o}$

I've tried graphing this using ƒ_observed=440 Hz and V_s=22 m/s, and the graph is symmetrical about t = 0, when it obviously should not be. I'm not sure where I'm going wrong.

 

[vela]

Check your expression for $\theta$ in terms of $d$, $V_s$, and $t$. Are you sure that's what you want?

 

[bwinter]

Why wouldn't it be? Tan gives me opposite and adjacent components which are d and tVs. Are you saying it should be sin or cos?

 

[vela]

Consider the sign of your expression for $V_o$ as you pass $t=0$.

 

[bwinter]

Am I approaching this the right way? I don't see what else theta can be written as.

would writing it in terms of cosine make sense? then $V_{o}=\frac{V_{s}^{2}t}{\sqrt{d^{2}+V_{s}^{2}t^{2}}}$

 

[vela]

Your approach is fine, but you have to be a bit careful. Try plotting $V_o$ vs $t$. Is it what you expect?

 

[bwinter]

Yes! this new Vo works. Thanks!