17.7 The doppler effect  (Page 2/8)

We know that wavelength and frequency are related by $v=f\lambda ,$ where v is the fixed speed of sound. The sound moves in a medium and has the same speed v in that medium whether the source is moving or not. Thus, f multiplied by $\lambda$ is a constant. Because the observer on the right in case (b) receives a shorter wavelength, the frequency she receives must be higher. Similarly, the observer on the left receives a longer wavelength, and hence he hears a lower frequency. The same thing happens in case (c). A higher frequency is received by the observer moving toward the source, and a lower frequency is received by an observer moving away from the source. In general, then, relative motion of source and observer toward one another increases the received frequency. Relative motion apart decreases frequency. The greater the relative speed, the greater the effect.

The Doppler effect occurs not only for sound, but for any wave when there is relative motion between the observer and the source. Doppler shifts occur in the frequency of sound, light, and water waves, for example. Doppler shifts can be used to determine velocity, such as when ultrasound is reflected from blood in a medical diagnostic. The relative velocities of stars and galaxies is determined by the shift in the frequencies of light received from them and has implied much about the origins of the universe. Modern physics has been profoundly affected by observations of Doppler shifts.

Derivation of the observed frequency due to the doppler shift

Consider two stationary observers X and Y in [link] , located on either side of a stationary source. Each observer hears the same frequency, and that frequency is the frequency produced by the stationary source.

Now consider a stationary observer X with a source moving away from the observer with a constant speed $v_{\text{s}}<v$ ( [link] ). At time $t=0$ , the source sends out a sound wave, indicated in black. This wave moves out at the speed of sound v . The position of the sound wave at each time interval of period $T_{\text{s}}$ is shown as dotted lines. After one period, the source has moved $\text{Δ}x=v_{\text{s}}{T}_{\text{s}}$ and emits a second sound wave, which moves out at the speed of sound. The source continues to move and produce sound waves, as indicated by the circles numbered 3 and 4. Notice that as the waves move out, they remained centered at their respective point of origin.