Question

Discuss analytically, the composition of two S.H.M.'s of the same period and
            parallel to each other​

Answer 1

Question : Discuss analytically, the composition of two S.H.M's of the same period and parallel to each other.

Solution : let equations of simple harmonic waves are y₁ = A₁sin(ωt) and y₂ = A₂sin(ωt + θ).

here it is clear that waves has same period and parallel to each other.

now from superposition principle,

composition of waves, y = y₁ + y₂

= A₁sin(ωt) + A₂sin(ωt + θ)

= A₁sin(ωt) + A₂sin(ωt)cosθ + A₂cos(ωt)sinθ

= sin(ωt){A₁ + A₂cosθ} + cos(ωt){A₂sinθ}

we know, asinx + bcosx = √(a² + b²)sin{x + tan¯¹(b/a)}

by using we get,

= √{(A₁ + A₂cosθ)² + (A₂sinθ)²} sin[(ωt) + tan¯¹(A₂sinθ)/(A₁ + A₂cosθ)]

= √{A₁² + A₂² + 2A₁A₂cosθ} sin[(ωt) + tan¯¹(A₂sinθ)/(A₁ + A₂cosθ)]

let A = √{A₁² + A₂² + 2A₁A₂cosθ}

tan¯¹(A₂sinθ)/(A₁ + A₂cosθ) = Φ

so the composition of waves , y = Asin(ωt + Φ)

it is clear that the composition of waves is also a simple harmonic wave.

where amplitude, A = √{A₁² + A₂² + 2A₁A₂cosθ}

and phase , Φ = tan¯¹[(A₂sinθ)/(A₁ + A₂cosθ)]