Respuesta :
Answer:
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)
