understand that when a source of sound waves moves relative to a stationary observer, the observed frequency is different from the source frequency (understanding of the Doppler effect for a stationary source and a moving observer is not required)

Doppler Effect for Sound Waves

1. Objective

To understand how the frequency heard by a stationary observer changes when the source of sound moves towards or away from the observer.

2. Basic Idea

When a source emits sound waves while moving, the spacing of the successive wavefronts in the direction of motion is altered. This changes the wavelength that reaches a stationary observer, and because the speed of sound in the medium is essentially constant, the observed frequency differs from the source frequency.

3. Derivation of the Doppler Formula (Moving Source, Stationary Observer)

Let

• \$f_s\$ – frequency of the source (Hz)
• \$v\$ – speed of sound in the medium (m s⁻¹)
• \$u\$ – speed of the source relative to the medium (m s⁻¹), positive when moving towards the observer
• \$\lambda'\$ – wavelength of the waves that reach the observer
• \$f'\$ – frequency heard by the observer (Hz)

When the source is at rest, the wavelength is simply

\$\lambda = \frac{v}{f_s}\$

If the source moves towards the observer, each successive crest is emitted from a point that is closer to the observer by a distance \$u\,T\$, where \$T = 1/f_s\$ is the period of emission. The effective wavelength in front of the source becomes

\$\lambda' = \lambda - uT = \frac{v}{fs} - \frac{u}{fs} = \frac{v-u}{f_s}\$

Because the speed of sound relative to the observer remains \$v\$, the observed frequency is

\$f' = \frac{v}{\lambda'} = \frac{v}{\dfrac{v-u}{fs}} = \frac{v\,fs}{v-u}\$

When the source moves away from the observer, \$u\$ is taken as negative (or the formula is written with \$v+u\$ in the denominator). The general expression is therefore

\$f' = \frac{v\,f_s}{v \mp u}\$

where the upper sign (–) is used when the source approaches, and the lower sign (+) when it recedes.

4. Table of Symbols

5. Example Calculation

Suppose a siren on a moving ambulance emits a sound of \$f_s = 800\ \text{Hz}\$. The speed of sound in air is \$v = 340\ \text{m s}^{-1}\$, and the ambulance approaches the observer at \$u = 30\ \text{m s}^{-1}\$.

1. Insert the values into the approaching‑source formula:

\$f' = \frac{340 \times 800}{340 - 30} = \frac{272000}{310} \approx 877\ \text{Hz}\$

2. Interpretation: The observer hears a higher pitch (877 Hz) than the actual emitted pitch (800 Hz) because the wavefronts are compressed.
3. If the ambulance passes and moves away at the same speed, the observed frequency becomes

\$f' = \frac{340 \times 800}{340 + 30} = \frac{272000}{370} \approx 735\ \text{Hz}\$

4. The pitch now drops below the source frequency, illustrating the symmetric nature of the effect.

6. Key Points to Remember

• The speed of sound \$v\$ is determined by the medium, not by the motion of source or observer.
• When the source moves towards a stationary observer, the observed frequency increases; when it moves away, the frequency decreases.
• The Doppler formula for a moving source is \$f' = \dfrac{v\,f_s}{v \mp u}\$.
• Only the relative motion of the source matters for this section; the observer is assumed stationary.
• For small source speeds (\$u \ll v\$), the fractional change in frequency is approximately \$\frac{u}{v}\$.

7. Practice Questions

1. A train horn emits a sound of \$f_s = 1000\ \text{Hz}\$. The speed of sound is \$340\ \text{m s}^{-1}\$ and the train approaches a platform at \$20\ \text{m s}^{-1}\$. Calculate the frequency heard by a passenger standing on the platform.
2. During the same scenario, after the train passes the platform and recedes at the same speed, what frequency does the passenger hear?
3. If a source moves away from an observer at \$15\ \text{m s}^{-1}\$ and the observer measures a frequency of \$450\ \text{Hz}\$, while the source frequency is known to be \$500\ \text{Hz}\$, determine the speed of sound in the medium.

8. Summary

The Doppler effect for sound demonstrates how motion of the source alone can alter the perceived pitch. By recognizing the change in wavelength while the wave speed stays constant, the relationship \$f' = \dfrac{v\,f_s}{v \mp u}\$ provides a quantitative tool for solving a wide range of A‑Level physics problems.