Class 11 > Unit # 12:Acoustics > Doppler Effect

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Doppler’s Effect & Derivations for Apparent Frequency

Video Lecture: Doppler’s Effect and All Cases

Watch the complete step-by-step video lecture explaining Doppler’s Effect and the derivations of apparent frequency for all motion cases:

Definition of Doppler’s Effect

“The change in the pitch (frequency) of sound caused by the relative motion between the source and observer is called Doppler’s Effect.”

Case 1(A): Listener Moves Towards Stationary Source

Let a listener move with velocity $v_L$ towards a stationary source of sound emitting waves of real frequency $f$, with the speed of sound being $v$. Let the apparent frequency heard by the listener be $f'$.}

As we know that:

$\lambda = \frac{v}{f} \implies f = \frac{v}{\lambda} \quad \text{--- (i)}$

Since the listener moves towards the source with velocity $v_L$, the relative velocity of sound with respect to the listener is $(v + v_L)$. Thus, the apparent frequency is:

$f' = \frac{v + v_L}{\lambda}$

Substituting the value of $\frac{1}{\lambda} = \frac{f}{v}$ from equation (i):

$f' = \left(\frac{v + v_L}{v}\right) f \implies f' = f \left(1 + \frac{v_L}{v}\right)$

This expression shows that the apparent frequency is greater than the real frequency ($f' > f$).

Fig: Listener moving towards a stationary source of sound.

Case 1(B): Listener Moves Away From Stationary Source

Let a listener move with velocity $v_L$ away from a stationary source of sound emitting frequency $f$, with the speed of sound being $v$.

As we know that:

$\lambda = \frac{v}{f} \implies f = \frac{v}{\lambda} \quad \text{--- (ii)}$

Since the listener moves away from the source, the relative velocity of sound for the listener is $(v - v_L)$. Therefore, the apparent frequency is:

$f' = \frac{v - v_L}{\lambda}$

Substituting the value of $\frac{1}{\lambda} = \frac{f}{v}$ from equation (ii):

$f' = \left(\frac{v - v_L}{v}\right) f \implies f' = f \left(1 - \frac{v_L}{v}\right)$

This expression shows that the apparent frequency is less than the real frequency ($f' < f$).

Fig: Listener moving away from a stationary source of sound.

Case 2(A): Source Moves Towards Stationary Listener

Let a source of sound move with velocity $v_S$ towards a stationary listener. The wave crests are compressed because the source moves in the direction of the outgoing waves, resulting in a shortened wavelength.

We know that:

$\lambda = \frac{v}{f} \quad \text{(Distance occupied by one wave)} \quad \text{--- (i)}$

During each vibration (time period $T = \frac{1}{f}$), the source travels a distance equal to $v_S T = \frac{v_S}{f}$ towards the listener. Thus, the apparent wavelength $\lambda'$ is shortened:

$\lambda' = \lambda - \frac{v_S}{f} = \frac{v}{f} - \frac{v_S}{f} \implies \lambda' = \frac{v - v_S}{f} \quad \text{--- (ii)}$

The apparent frequency $f'$ is given by:

$f' = \frac{v}{\lambda'}$

Substituting the value of $\lambda'$ from equation (ii):

$f' = \frac{v}{\left(\frac{v - v_S}{f}\right)} \implies f' = f \left(\frac{v}{v - v_S}\right)$

This expression shows that the apparent frequency is greater than the real frequency ($f' > f$).

Fig: Source moving towards a stationary listener.

Case 2(B): Source Moves Away From Stationary Listener

Let a source of sound move with velocity $v_S$ away from a stationary listener. The wave crests are spread out because the source moves opposite to the outgoing waves, resulting in an increased wavelength.

We know that:

$\lambda = \frac{v}{f} \quad \text{--- (i)}$

During each vibration, the source travels a distance equal to $\frac{v_S}{f}$ away from the listener, increasing the apparent wavelength:

$\lambda' = \lambda + \frac{v_S}{f} = \frac{v}{f} + \frac{v_S}{f} \implies \lambda' = \frac{v + v_S}{f} \quad \text{--- (ii)}$

The apparent frequency $f'$ is given by:

$f' = \frac{v}{\lambda'}$

Substituting the value of $\lambda'$ from equation (ii):

$f' = \frac{v}{\left(\frac{v + v_S}{f}\right)} \implies f' = f \left(\frac{v}{v + v_S}\right)$

This expression shows that the apparent frequency is less than the real frequency ($f' < f$).

Fig: Source moving away from a stationary listener.

Case 3(A): Both Source and Listener Move Towards Each Other

When both the source and the listener move towards each other, combining the effects of listener motion and source motion yields the general expression for apparent frequency:

$f' = f \left(\frac{v + v_L}{v - v_S}\right)$

This expression shows that the apparent frequency increases rapidly.

Case 3(B): Both Source and Listener Move Away From Each Other

When both the source and the listener move away from each other, combining their relative velocities gives the apparent frequency:

$f' = f \left(\frac{v - v_L}{v + v_S}\right)$

This expression shows that the apparent frequency decreases rapidly.

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