Derivation for doppler's effect?
Doppler's effect :-
Relative motion, between a source of sound and a listener, causes the apparent frequency of the sound heard by the listener to be different from the frequency of the sound emitted by the source.
Consider,
S → Source of sound
O → Observer
ν → Frequency of sound emitted by the source
λ → Wavelength of the emitted sound
Source and observer both are at rest.
Velocity of sound in air, v = νλ
Where,
vs → Velocity of the source
vm → Velocity of the medium
v0 → Velocity of the observer
Let the distance between the source and observer = v (That is, ν waves reach the observer from source in 1 s)
∴ Frequency of the sound heard by the listener = ν
When medium is in motion, the distance travelled by the sound wave = v + vm
Distance moved by source in one second, 
Distance covered by the sound in 1 s = (v + vm) − vs (relative to source)
Frequency (ν) remains the same.
∴Apparent wavelength, 
Distance travelled by the observer 
(in 1 s)
Distance available in 1 s to 
Apparent frequency of sound heard, 
From equation (1),
Sign Conversion
From Source → Listener (+ ve velocities)
Listener → Source (− ve velocity)
Medium at rest, vm = 0
Special Cases
-
Source moving towards observer − Observer stationary
∴vs = + ve and v0 = 0
-
Source moving away from observer − Observer stationary
vs = − ve and v0 = 0
-
Source stationary − Observer moving away from source
∴ vs = 0 and v0 = + ve
-
Source stationary − Observer moving towards source
vs = 0 and v0 = − ve
-
Source and observer approaching each other
vs = + ve and v0 = − ve
-
Source and observer moving away from each other
vs = − ve and v0 = + ve
