# What relations between sound frequencies of a sound wave in motion that gets reflected from the wall occur?

Let's say you're driving forward by car and when you press the horn, the sound is reflected from the building in front of us. If we mark the frequency of the sound sent by $$f_1$$, frequency of the sound received by an observer located in the building by $$f_2$$ and frequency of the reflected sound which is received by the observer in the car by $$f_3$$,

What relations can we show for these frequencies?

• How do we describe such wave behavior? is it an example of double doppler-effect? Should any of the frequencies be equal to each other here? – Undergraduate Wannabe Oct 11 '19 at 14:20

Assuming that you are driving straight into the building, the observer in the building hears the sound coming from your moving car with frequency shifted due to the Doppler effect. The frequency is given by:

$$f_2=\frac{c_s}{c_s-v} f_1$$

Here, $$c_s$$ is the speed of sound in air and $$v$$ is the speed of your car. You can find the derivation of the frequency shift due to a moving source online (e.g. http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/DopplerEffect.htm, my expression is slightly rearranged, but essentially the same).

When that sound with frequency $$f_2$$ gets reflected from the building back to the passenger of the car, it can be treated as a case of stationary source and moving receiver. This further Doppler shift is given by:

$$f_3= \frac{c_s+v}{c_s}f_2=\frac{c_s+v}{c_s}\frac{c_s}{c_s-v} f_1 =\frac{c_s+v}{c_s-v} f_1$$

Derivation for a moving observer is again given in the same resource.

I also found a video witch explains an opposite scenario, where the source is stationary and the 'reflector' is moving. It uses the same equations and the final reflected frequency is the same: https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/doppler-effect/v/doppler-effect-reflection-off-a-moving-object

The perceived frequency of the sound is dependent on the frame of reference. If the frame of reference is the car, the barn can be assumed to have a velocity (vector). If you take the projection of the barns velocity vector, onto the vector from the car to the barn at the instantaneous moment the sound hits the barn, this will give you the correct velocity with which to calculate the Doppler effect. In this way it is not really a "double" effect. It is just a single reflection in the stationary frame of reference that is the car, with a moving barn.

• So what are the frequencies relations, would you like to elaborate? – Undergraduate Wannabe Oct 11 '19 at 15:10
• @UndergraduateWannabe, f2 and f3 will be the same. The relation between f1 and f2/3 will be a function of the velocity vector and the location vector. – TallBrian Oct 16 '19 at 22:19