How Do You Derive the Doppler Effect for Moving Observers and Sources?

[tjaeger]

Homework Statement

I have a problem that asks to derive the doppler effect for the two different cases of a moving observer and a moving source.

Homework Equations

I should get f' = f*(v +/- vo)/(v +/- vs) as my general equation, where f' is the observed frequency and f is the frequency of the wave.

The Attempt at a Solution

I wish I had more work to show, but I'm not really sure how to derive formulas. Where should I begin?

 

[tjaeger]

Nevermind, I figured it out. Wavelength changes with a moving source and velocity of the sound wrt the observer changes when the observer is moving.

 

[Moetasim]

As a scientist, it is important to understand the underlying principles and equations that govern a phenomenon. In this case, the Doppler effect is a result of the relative motion between a source of waves and an observer. To derive the Doppler effect, we need to consider the properties of waves and the relative motion between the source and observer.

First, let's define some variables:

- f: frequency of the wave emitted by the source
- f': frequency of the wave observed by the moving observer
- v: speed of the wave (in a vacuum)
- vo: speed of the observer
- vs: speed of the source

Next, we need to understand how the frequency of a wave is affected by relative motion. The frequency of a wave is defined as the number of complete cycles it makes in a given time. Therefore, if the source is moving towards the observer, the number of cycles per second (frequency) will increase, and if the source is moving away from the observer, the frequency will decrease.

Now, let's consider the case of a moving source. In this case, the source is emitting waves at a constant frequency f, but due to its motion, the distance between consecutive wave crests (wavelength) changes. This results in a change in the observed frequency f'. To calculate this change, we can use the formula for the speed of a wave, v = fλ, where λ is the wavelength. As the source moves, the wavelength changes according to the equation λ' = λ - vs*t, where vs is the speed of the source and t is time. Substituting this into the formula for the speed of the wave, we get v = f(λ - vs*t). Rearranging for f', we get f' = v/(λ - vs*t). Since we are interested in the frequency of the wave observed at a specific time, we can substitute t = d/vo, where d is the distance between the source and the observer and vo is the speed of the observer. This gives us the final formula for the observed frequency in the case of a moving source: f' = f*(v - vs)/(v - vo).

Similarly, we can derive the formula for a moving observer by considering the change in wavelength due to the observer's motion. In this case, the source is emitting waves at a constant frequency f, and the wavelength remains unchanged. However, due to the observer's motion,