How Do You Calculate the Doppler Shift for Light (Δf/f ≈ v/c)?

For light, when the relative speed is much smaller than the speed of light, calculate the Doppler shift using $\frac{\Delta f}{f} \approx \frac{v}{c}$. Multiply the emitted frequency by (v/c) to obtain the magnitude of the frequency change.

This is part of C.5 Doppler effect and applies to both SL and HL. Here, $f$ is the emitted frequency, $\Delta f=f'-f$ is the change in observed frequency, $v$ is the radial relative speed, and $$.

The mechanism is that motion changes the rate at which successive wavefronts reach the observer. During approach, wavefronts arrive more frequently; during recession, they arrive less frequently. The speed of light remains $c$ in a vacuum: the Doppler effect changes the observed frequency and wavelength, not the propagation speed.

Rearranging gives:

$$\Delta f \approx f\frac{v}{c}$$

Only the component of relative velocity along the line joining the source and observer is used.

If motion is not directly along the line of sight, use the radial component $v_r=v\cos\theta$ rather than the total speed. For transverse motion in this non-relativistic approximation, $v_r=0$, so there is no first-order shift. This prevents overestimating the shift.

The fractional shift is dimensionless, so frequency units cancel before multiplication.

For example, a galaxy emitting light at $f=6.00\times10^{14}\ \text{Hz}$ recedes at $v=3.00\times10^6\ \text{m s}^{-1}$. The magnitude of the shift is

$$|\Delta f|=(6.00\times10^{14})\frac{3.00\times10^6}{3.00\times10^8} =6.00\times10^{12}\ \text{Hz}.$$

Because the galaxy is receding, $\Delta f=-6.00\times10^{12}\ \text{Hz}$, giving

$$f'=f+\Delta f=5.94\times10^{14}\ \text{Hz}.$$

A common misconception is that frequency and wavelength shift in the same signed direction. Since $c=f\lambda$, an increased frequency means a decreased wavelength; for small shifts, $\frac{\Delta f}{f}\approx-\frac{\Delta\lambda}{\lambda}$.

In an IB Calculate response, state the equation, substitute SI values, show the result, and interpret approach or recession. Do not use this approximation when $v$ is comparable to $c$; a relativistic expression is then required.