17.7 The doppler effect  (Page 3/8)

Using the fact that the wavelength is equal to the speed times the period, and the period is the inverse of the frequency, we can derive the observed frequency:

As the source moves away from the observer, the observed frequency is lower than the source frequency.

Now consider a source moving at a constant velocity $v_s,$ moving toward a stationary observer Y , also shown in [link] . The wavelength is observed by Y as ${\lambda }_{\text{o}}={\lambda }_{\text{s}}-\text{Δ}x={\lambda }_{\text{s}}-v_{\text{s}}{T}_{\text{s}}.$ Once again, using the fact that the wavelength is equal to the speed times the period, and the period is the inverse of the frequency, we can derive the observed frequency:

When a source is moving and the observer is stationary, the observed frequency is

where $f_{\text{o}}$ is the frequency observed by the stationary observer, $f_{\text{s}}$ is the frequency produced by the moving source, v is the speed of sound, $v_{\text{s}}$ is the constant speed of the source, and the top sign is for the source approaching the observer and the bottom sign is for the source departing from the observer.

What happens if the observer is moving and the source is stationary? If the observer moves toward the stationary source, the observed frequency is higher than the source frequency. If the observer is moving away from the stationary source, the observed frequency is lower than the source frequency. Consider observer X in [link] as the observer moves toward a stationary source with a speed $v_{\text{o}}$ . The source emits a tone with a constant frequency $f_{\text{s}}$ and constant period $T_{\text{s}}.$ The observer hears the first wave emitted by the source. If the observer were stationary, the time for one wavelength of sound to pass should be equal to the period of the source $T_{\text{s}}.$ Since the observer is moving toward the source, the time for one wavelength to pass is less than $T_{\text{s}}$ and is equal to the observed period $T_{\text{o}}=T_{\text{s}}-\text{Δ}t.$ At time $t=0,$ the observer starts at the beginning of a wavelength and moves toward the second wavelength as the wavelength moves out from the source. The wavelength is equal to the distance the observer traveled plus the distance the sound wave traveled until it is met by the observer: