Question

A source of sound and a listener are moving in the same direction, not parallel to each other, with unequal velocities ${\text{v}}_{\text{S}}$ and ${\text{v}}_{\text{L}}$ respectively. If the velocity of sound in the medium is $\text{v}$ and its true frequency is $\text{f}$, the frequency perceived by the listener ($\text{f *}$) is given by:

• A. $\text{f *}=\left(\frac{\text{v}-{\text{v}}_{\text{L}}}{\text{v}+{\text{v}}_{\text{S}}}\right)\text{f}$
• B. $\text{f *}=\left(\frac{\text{v}+{\text{v}}_{\text{L}}}{\text{v}-{\text{v}}_{\text{S}}}\right)\text{f}$
• C. $\text{f *}=\left(\frac{\text{v}+{\text{v}}_{\text{L}}}{\text{v}+{\text{v}}_{\text{S}}}\right)\text{f}$
• D. $\text{f *}=\left(\frac{\text{v}-{\text{v}}_{\text{L}}}{\text{v}-{\text{v}}_{\text{S}}}\right)\text{f}$

Answer 1

Answer: (C), (D)

The correct options are
C $\text{f *}=\left(\frac{\text{v}+{\text{v}}_{\text{L}}}{\text{v}+{\text{v}}_{\text{S}}}\right)\text{f}$
D $\text{f *}=\left(\frac{\text{v}-{\text{v}}_{\text{L}}}{\text{v}-{\text{v}}_{\text{S}}}\right)\text{f}$

The source and the listener are moving with unequal directions in the same direction, but not in parallel to each other. Two cases are possible in this scenario:

1. The source following the listener
2. The listener following the source

In both the cases, the apparent frequency depends on the velocities of the source and the listener.

• When the source is following the listener, the apparent frequency is given by:

$\text{f *}=\left(\frac{\text{v}-{\text{v}}_{\text{L}}}{\text{v}-{\text{v}}_{\text{S}}}\right)\times \text{f}$

• When the listener is following the source, the apparent frequency is given by:

$\text{f *}=\left(\frac{\text{v}+{\text{v}}_{\text{L}}}{\text{v}+{\text{v}}_{\text{S}}}\right)\times \text{f}$