多普勒效应

The Doppler Effect is the name of the phenomenon in which the perception of a wave's frequency differs depending on the observer's velocity relative to the source of the wave. This is a classic effect observed whenever one hears a siren pass by, and it is what gives racecars their classic perceived sound as they drive by.

Suppose that we have a source that is stationary in the frame of reference of our choice. Let us say that the waves emitted by the source have a frequency $f_1$, wavelength $\lambda$, and velocity $v_1$.

Also, let us say that the observer is moving directly toward or away from the source with a velocity $v_2$. We will say that $v_2>0$ if the observer is moving away from the source and that $v_2<0$ if the observer is moving toward the source.

Of course, in real scenarios, this is almost never true, but the calculations are reasonably similar. Our assumption will simplify our lives in this case, and it should be good enough for a good understanding.

Denote the perceived frequency $f_2$. We would like to find the value of $f_2$.

First think about the case in which $v_2=0$. This is the most typical case.

The wave hits the observer at a speed of $v_1$, but the length of the wave is $\lambda$. Thus, the rate of the oscillations is $\frac{v_1}{\lambda}$. Indeed, this is the value of the frequency, $f_1$. This is exactly the derivation of the fact that the velocity is the product of the wavelength and the frequency.

Now, let us factor in the observer's velocity $v_2$.

The length of the wave is still $\lambda$, but the wave is now hitting the observer at a speed of $v_1-v_2$! Thus, the perceived frequency is quite simple: $$f_2=\frac{v_1-v_2}{\lambda}$$ We know that $f_1=\frac{v_1}{\lambda}$, so we obtain the following: $$f_2=f_1\frac{v_1-v_2}{v_1}$$ This equation represents the Doppler shift!

Now, for additional generality, we may factor in the velocity of the source.

Let us redefine the following three velocities relative to the medium. We will say that $v_w$ is the speed of the wave, $v_o$ is the speed of the observer, and $v_s$ is the speed of the source.

Now look up at our above equation: $$f_2=f_1\frac{v_1-v_2}{v_1}$$ We see something quite nice! The velocity $v_1$ is equal to the difference in velocities of the wave and source, $v_w-v_s$! Moreover, the velocity $v_2$ is simply the difference in the velocities of the observer and source, $v_o-v_s$!

Plugging these two in, we get our familiar expression for the Doppler shift: $$f_2=f_1\frac{v_w-v_o}{v_w-v_s}$$ Both of the above equations, whether in the reference frame of the medium or the source, are essentially the same, and both are useful.

For example, it is now easy to see that the perceived frequency is higher than the emitted frequency when the observer is moving toward the source; similarly, the perceived frequency is lower than the emitted frequency when the observer is moving away from the source.

This also explains such occurrences as a sonic boom, when the source velocity meets and exceeds the wave velocity. Notice the frequency becomes infinite and then negative.

Also, notice that when the observer velocity is greater than or equal to the source velocity, the frequency is zero or negative, corresponding to the fact that the wave is not received at all.