You are standing on a train station platform as a train goes by close to you. As the train approaches, you hear the whistle sound at a frequency of f1 = 95 Hz. As the train recedes, you hear the whistle sound at a frequency of f2 = 78 Hz. Take the speed of sound in air to be v = 340 m/s.
a) Find an equation for the speed of the sound source vs., in this case it is the speed of the train. Express your answer in terms of f1, f2 and v.

Speed of the sound source,

vₛ = v(f₁ - f₂)/(f₁ + f₂)

Explanation:

The phenomenon can be explained with the Doppler's Effect explanation.

Doppler's Effect explains how relative frequency of a sound source varies with the velocity of the source or the observer.

Generally, the mathematical expression for Doppler's Effect is given below

f' = f [(v + v₀)/(v - vₛ)]

where f' = observed frequency

f = actual frequency

v = velocity of sound waves

v₀ = velocity of observer

vₛ = velocity of sound source

When the train is moving towards the stationary observer,

f' = observed frequency of the sound wave = f₁

f = actual frequency of the sound wave = f

v = velocity of sound waves = v

v₀ = velocity of observer = 0 m/s

vₛ = velocity of sound source = vₛ

f' = f [(v + v₀)/(v - vₛ)]

f₁ = f [(v + 0)/(v - vₛ)]

f₁ = fv/(v - vₛ) (eqn 1)

When the train is moving away from the stationary observer,

f' = observed frequency of the sound wave = f₂

f = actual frequency of the sound wave = f

v = velocity of sound waves = v

v₀ = velocity of observer = 0 m/s

vₛ = velocity of sound source = - vₛ (train is moving away from the observer, hence, the negative sign)

f' = f [(v + v₀)/(v - vₛ)]

f₂ = f [(v + 0)/(v - (-vₛ)]

f₂ = fv/(v + vₛ) (eqn 2)

Since the actual frequency of the sound wave doesn't change, we make f the subject of formula in the two cases and equate that to each other

From eqn 1

f₁ = fv/(v - vₛ)

f = f₁ (v - vₛ)/v

From eqn 2

f₂ = fv/(v + vₛ)

f = f₂ (v + vₛ)/v

f = f

f₁ (v - vₛ)/v = f₂ (v + vₛ)/v

f₁ (v - vₛ) = f₂ (v + vₛ)

f₁v - f₁vₛ = f₂v + f₂vₛ

f₁vₛ + f₂vₛ = f₁v - f₂v

(f₁ + f₂)vₛ = v(f₁ - f₂)

vₛ = v(f₁ - f₂)/(f₁ + f₂)