What's the Difference in the Doppler Equation for a Moving Source vs a Moving Observer? (HL)

The equations differ because a moving source changes the wavelength in the medium, while a moving observer changes the rate at which wavefronts are encountered. Therefore, source speed appears in the denominator, whereas observer speed appears in the numerator.

This is Additional Higher Level (AHL) content from C.5 Doppler effect.

Here, $f$ is the emitted frequency, (f') is the observed frequency, $v$ is the wave speed in the medium, $u_s$ is the source speed, and $u_o$ is the observer speed. All speeds are measured relative to the medium.

For a moving source, the source emits each successive wavefront from a different position. When approaching, the wavelength ahead becomes shorter:

$$\lambda' = \frac{v-u_s}{f}$$

Since $f'=v/\lambda'$, this gives $f'=fv/(v-u_s)$.

For a moving observer, the wavelength in the medium remains unchanged. However, an approaching observer meets wavefronts more rapidly, so their relative speed with the wavefronts is $v+u_o$.

Another way to see the distinction is to track spacing and encounter rate. A source moving towards a stationary observer compresses the wavefront spacing, so the wavelength decreases while the wave speed stays constant. An observer moving towards a stationary source does not alter that spacing; instead, the observer crosses the unchanged wavefronts at a greater rate. This distinction explains the different equation structures rather than merely providing a sign rule.

For example, let $f=500\,\text{Hz}$, $v=340\,\text{m s}^{-1}$, and the relevant moving speed be $20\,\text{m s}^{-1}$. An approaching source gives

$$f'=500\left(\frac{340}{320}\right)=531\,\text{Hz},$$

whereas an approaching observer gives

$$f'=500\left(\frac{360}{340}\right)=529\,\text{Hz}.$$

A common misconception is that motion changes the wave speed. It does not: $v$ remains fixed by the medium.

Exam technique: For a Calculate question, first identify what is moving, then choose signs by checking the physical result: approach must increase (f'), while recession must decrease it. Do not select signs by memorizing an isolated $\pm$ symbol.