Question

Derive expression for beat

Answer 1

Answer:

frequency of beat

Explanation:

it's mathematical derivation is ∆n =n1 - n2

Answer 2

Expression for beat:

The displacement of one beat is given as:

y₁ = A cos (2πf₁)t

The displacement of another beat is given as:

y₂ = A cos (2πf₂)t

The total is given as:

y = y₁ + y₂ = A cos (2πf₁)t + A cos (2πf₂)t

y = A [cos (2πf₁)t + cos (2πf₂)t] → (equation 1)

Now,

cos (a + b) = cos a cos b - (sin a sin b)

cos (a - b) = cos a cos b + (sin a sin b)

⇒ cos (a + b) + cos (a - b) = 2 cos a cos b → (equation 2)

Now,

a = 2π (f₁t + f₂t)/2

b = 2π (f₁t - f₂t)/2

a + b = 2πf₁t

a - b = 2πf₂t

Now, the equation (2) becomes,

cos (2πf₁t) +cos (2πf₂t) =2 cos (2π (f₁t + f₂t)/2)cos (2π (f₁t - f₂t)/2) → (equation 3)

Now,

(f₁t - f₂t)/2 = Average frequency = fav

(f₁t - f₂t) = Average difference = Δf

y = y₁ + y₂ = 2A (cos (2π Δf/2)t) × (cos (2π fav)t)

∴ y = 2A (cos (2π Δf/2)) × (cos (2π fav)) → (equation 4)