Doppler Effect

The Doppler effect changes the observed frequency when a source and observer move relative to each other. For a stationary observer and a source approaching at speed $v_s$, the apparent frequency is $f^{\prime }=f\left(\frac{v}{v-v_s}\right)$, where $v$ is the speed of sound. The denominator shrinks because successive wavefronts are emitted from closer positions, compressing the wavelength. Substituting $f=660\text{ Hz}$, $v=330\text{ m/s}$, and $v_s=30\text{ m/s}$ gives $f^{\prime }=660\left(\frac{330}{330-30}\right)=660\times \frac{330}{300}=660\times 1.1=726\text{ Hz}$. The value $600\text{ Hz}$ wrongly adds $v_s$ in the denominator, which applies to a receding source. The value $660\text{ Hz}$ ignores the motion entirely. The value $693\text{ Hz}$ comes from putting $v_s$ in the numerator using the observer-motion formula by mistake. This is the NCERT Doppler formula for a moving source. As a plausibility check, an approaching source must raise the pitch, so the heard frequency must exceed 660 Hz, and the modest 10 percent increase to 726 Hz is consistent with a source speed that is one-eleventh of the sound speed.