The Doppler effect is a phenomenon that describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. This effect is not limited to sound waves; it also applies to electromagnetic waves, such as light. When an observer moves towards a light source, the observed frequency increases, while it decreases when moving away. The fundamental equation for the Doppler effect in light is given by:
f_{\(\text{observed}\)} = f_{\(\text{source}\)} \(\left\)(1 \(\pm\) \(\frac{v_{\text{relative}\)}}{c}\(\right\))
In this equation, fobserved is the frequency observed, fsource is the frequency emitted by the source, vrelative is the relative velocity between the observer and the source, and c is the speed of light (approximately \(3 \times 10^8\) m/s). The sign used in the equation depends on the direction of motion: a plus sign is used when the observer and source are moving closer together, while a minus sign is used when they are moving apart.
To illustrate this, consider a distant star emitting light with a wavelength of 630 nanometers. If the star is receding from Earth at a speed of \(3 \times 10^6\) m/s, we can calculate the observed frequency. First, we convert the wavelength to frequency using the relationship:
f_{\(\text{source}\)} = \(\frac{c}{\lambda_{\text{source}\)}}
Substituting the values, we find:
f_{\(\text{source}\)} = \(\frac{3 \times 10^8 \text{ m/s}\)}{630 \(\times\) 10^{-9} \(\text{ m}\)} \(\approx\) 4.76 \(\times\) 10^{14} \(\text{ Hz}\)
Next, we apply the Doppler effect equation to find the observed frequency:
f_{\(\text{observed}\)} = 4.76 \(\times\) 10^{14} \(\left\)(1 - \(\frac{3 \times 10^6}{3 \times 10^8}\)\(\right\))
Calculating this gives:
f_{\(\text{observed}\)} \(\approx\) 4.71 \(\times\) 10^{14} \(\text{ Hz}\)
Finally, to find the observed wavelength, we use the frequency we just calculated:
\(\lambda\)_{\(\text{observed}\)} = \(\frac{c}{f_{\text{observed}\)}} = \(\frac{3 \times 10^8 \text{ m/s}\)}{4.71 \(\times\) 10^{14} \(\text{ Hz}\)} \(\approx\) 636 \(\text{ nm}\)
This example demonstrates that even at significant velocities, the change in wavelength due to the Doppler effect can be relatively small unless the speeds approach a significant fraction of the speed of light. Understanding the Doppler effect is crucial in fields such as astronomy, where it helps in determining the movement of stars and galaxies.
