**Doppler's Effect: **The phenomenon of variation in the pitch of a sound due to the relative motion of the source and the observer (listener) is called the Doppler's effect. Due to this effect, the pitch emitted by the siren of an approaching ambulance appears increased. Similarly, the pitch appears to drop when it is moving away.

**Calculation of apparent frequency when both the source and observer are moving:**

If both source and observer are moving then apparent frequency f' is given by, \(f'=\frac{v'}{\lambda'}\) where v' is the velocity of the sound waves relative to the observer O and λ' is the wavelength of the wave reaching O. This is the general formula to calculate f', when the source and the observer move along a straight line. So, we can use this to find the apparent frequency in any of the cases considered before. Suppose, v is the velocity of sound in air, f is the true frequency of the source, u_{0} is the velocity of the observer O and u_{s} is the velocity of the source S.

**When the source S and observer O are moving in the same direction as shown in the figure:**

The velocity of sound wave v and velocity of then observer u_{0} are in the same direction. Therefore, the velocity of sound waves relative to O is,

v' = v - u_{0}

Since the source is moving towards observer O, the wavelength of the weave reaching O is,

\(\lambda'=\frac{v-u_s}f\)

Or, \(f'=\frac{v'}{\lambda'}=\frac{v-u_0}{\displaystyle\frac{v-u_s}f}\)

∴ The apparent frequency, \(f'=\frac{v-u_0}{v-u_s}f\)

Here, if u_{0} = u_{s}, then f' = f

**If the source and observer are moving in the same direction as shown in the figure.**

If the source and observer are moving in the same direction as shown in the figure, v and u_{0} are in the opposite directions, so the velocity of the sound wave relative to O is, v' = v + u_{0}. Since the source is moving away from the observer, the wavelength of the wave reaching O is

\(\lambda'=\frac{v+u_s}f\)

∴ \(f'=\frac{v'}{\lambda'}=\frac{v+u_0}{\displaystyle\frac{v+u_s}f}\)

Or, \(f'=\frac{v+u_0}{v+u_s}f\)

Here, if u_{o} = u, then f' = f