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Step-By-Step Derivation of Doppler Effect


In order to derive the Doppler effect, there are two situations that needs to considered, and they are:

Moving Source and Stationary Observer Where the Wave Travels with the Source


Important Derivations: Waves | Physics Class 11 - NEET
Where,
c: wave velocity
λs: wavelength of the source
T: time taken by the wave
Important Derivations: Waves | Physics Class 11 - NEET
Where,
vs: velocity with which source is moving towards a stationary observer
d: distance covered by the source
Important Derivations: Waves | Physics Class 11 - NEET

Moving Observer and a Stationary Source

Important Derivations: Waves | Physics Class 11 - NEET
Where,
f0: observed frequency
v0: observer velocity
Important Derivations: Waves | Physics Class 11 - NEET

One Dimensional Wave Equation Derivation

The wave equation in classical physics is considered to be an important second-order linear partial differential equation to describe the waves. The wave equation arises in fields like fluid dynamics, electromagnetics, and acoustics. D’Alembert discovered the one-dimensional wave equation in 1746, after ten years Euler discovered the three-dimensional wave equation.

One Dimensional Wave Equation Derivation


Consider the relation between Newton’s law that is applied to the volume ΔV in the direction x:
Important Derivations: Waves | Physics Class 11 - NEET
Where,
F: force acting on the element with volume ΔV
Important Derivations: Waves | Physics Class 11 - NEET
Important Derivations: Waves | Physics Class 11 - NEET
(as dt is small, it is not considered and ΔSx is in x direction so ΔyΔz and from Newton Law.
Important Derivations: Waves | Physics Class 11 - NEET
Above equation is known as the equation of motion.
Important Derivations: Waves | Physics Class 11 - NEET
Where,
K: bulk modulus
Rewriting the above equation:
Important Derivations: Waves | Physics Class 11 - NEET
Where,
K: bulk modulus
Rewriting the above equation:
Important Derivations: Waves | Physics Class 11 - NEET
Where,
c: velocity of sound given as
Important Derivations: Waves | Physics Class 11 - NEET
Thus, above is the one-dimensional wave equation derivation.

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