derrivvation of Dopplers effect of sound
Answers
Moving source and stationary observer
Consider the Doppler Effect when the the observer is stationary and the source of the wavefront is moving tpward it in the x-direction.
Source is moving toward stationary observer
Source is moving toward stationary observer
Note: According to our conventions, the source velocity is constant and less than the wave velocity, the x-direction is positive, and only motion along the x-axis is considered.
Finding observed wavelength
The wave velocity is:
c = λS/T
where:
c is the wave velocity
λS is the wavelength of the source or the distance between crests
T is the time it takes a wave to move one wavelength λS
Solving for T:
T = λS/c
If the source is moving at a velocity vS toward a stationary observer, then the distance that the source moves in time T is:
d = vST
where
d is the distance the source moves in time T
vS is the velocity of the source toward a stationary observer
When the source is moving in the x-direction, it is "catching up" to the previously emitted wave when it emits the next wavefront. This means the wavelength reaching the observer, λO, is shortened.
Note: If the source was moving in the opposite direction, λO would be lengthened.
The observed wavelength λO is then:
λO = λS − d
Observed wavelength as a function of source velocity
Observed wavelength as a function of source velocity
Substitute T = λS/c into d = vST:
d = vSλS/c
Substitute this value for d into λO = λS − d:
λO = λS − vSλS/c
Factoring out λS gives you:
λO = λS(1 − vS/c)
The equation is also often written as:
λO = λS(c − vS)/c
If the source is moving away from the observer, the sign of vS changes.
Change in wavelength
Define the change in wavelength as:
Δλ = λS − λO
Since λO = λS − d:
Δλ = λS − (λS − d)
Also since d = vSλS/c:
Δλ = λS − (λS − vSλS/c)
Δλ = λSvS/c
Moving observer and stationary source
Suppose the source is stationary and the observer is moving in the x-direction away from the source.
Observer moving away from oncoming waves
Observer moving away from oncoming waves
Finding observed wavelength
In this situation, the observed wave frequency is a combination of the wave velocity and observer velocity, divided by the actual wavelength:
fO = (c − vO)/λS
where
fO is the observed frequency
vO is the observer velocity
But also fO = c/λO:
c/λO = (c − vO)/λS
Reciprocating both sides of the equation:
λO/c = λS/(c − vO)
λO = λSc/(c − vO)
Multiply by c:
λO = λS/[(c − vO)/c]
Thus:
λO = λSc/(c − vO)
or
λO = λS/(1 − vO/c)
Change in wavelength
The change in wavelength is defined as:
Δλ = λS − λO
Substitute λO = λSc/(c − vO):
Δλ = λS − λSc/(c − vO)
Multiply λS times (c − vO)/(c − vO):
Δλ =[ λS(c − vO) − λSc]/(c − vO)
Reduce and simplify:
Δλ =[ λSc − λSvO− λSc]/(c − vO)
Thus:
Δλ = −λSvO/(c − vO)
or
Δλ = λS/(1 − c/vO)
General wavelength equation
When both the source and observer are moving in the x-direction, you can combine the individual equations to get a general Doppler Effect wavelength equation.
Let λO1 be the wavelength equation for a moving source and stationary observer:
λO1 = λS(c − vS)/c
For the case when both the source and observer moving, substitute λO1 for λS in the
λO = λSc/(c − vO):
λO = λO1c/(c − vO)
λO = [λS(c − vS)/c]c/(c − vO)
Simplify:
λO = λSc(c − vS)/c(c − vO)
Thus:
λO = λS(c − vS)/(c − vO)
or
λO(c − vO) = λS(c − vS)
Change in wavelength
The general change in wavelength is:
Δλ = λS − λO
Substitute for λO:
Δλ = λS − λS(c − vS)/(c − vO)
Δλ = [λS(c − vO) − λS(c − vS)]/(c − vO)
Δλ = (λSc − λSvO − λSc + λSvS)/(c − vO)
Thus:
Δλ = λS(vS − vO)/(c − vO