Superposition of Harmonic Motions

Introduction

Superposition of SHMs in same direction produces beats (different frequencies) or modified amplitude (same frequency). Perpendicular SHMs produce Lissajous figures. Superposition principle applies because SHM equations are linear.

Same Frequency

x₁ = A₁cos(ωt), x₂ = A₂cos(ωt + φ). Sum: x = Acos(ωt + δ) where A = √(A₲ + A₂² + 2A₁A₂cosφ), tanδ = A₂sinφ/(A₁ + A₂cosφ). Result is SHM with same frequency, new amplitude and phase. Constructive interference (φ = 0): A = A₁ + A₂. Destructive (φ = π): A = |A₁ - A₂|.

Different Frequencies - Beats

x₁ = Acos(ω₁t), x₂ = Acos(ω₂t). Sum: x = 2Acos((ω₁-ω₂)t/2)cos((ω₁+ω₂)t/2). Amplitude modulation at beat frequency ω_b = |ω₁ - ω₂|. Envelope frequency is difference, carrier is average. Audible as volume oscillation when two sound frequencies are close.

Perpendicular SHMs

x = Acos(ωt), y = Bcos(ωt + φ). Path is ellipse in general. Special cases: φ = 0: straight line y = (B/A)x; φ = π/2 and A = B: circle; φ = π/2: ellipse x²/A² + y²/B² = 1.

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Introduction

Superposition of SHMs in same direction produces beats (different frequencies) or modified amplitude (same frequency). Perpendicular SHMs produce Lissajous figures. Superposition principle applies because SHM equations are linear.

Same Frequency

x₁ = A₁cos(ωt), x₂ = A₂cos(ωt + φ). Sum: x = Acos(ωt + δ) where A = √(A₲ + A₂² + 2A₁A₂cosφ), tanδ = A₂sinφ/(A₁ + A₂cosφ). Result is SHM with same frequency, new amplitude and phase. Constructive interference (φ = 0): A = A₁ + A₂. Destructive (φ = π): A = |A₁ - A₂|.

Different Frequencies - Beats

x₁ = Acos(ω₁t), x₂ = Acos(ω₂t). Sum: x = 2Acos((ω₁-ω₂)t/2)cos((ω₁+ω₂)t/2). Amplitude modulation at beat frequency ω_b = |ω₁ - ω₂|. Envelope frequency is difference, carrier is average. Audible as volume oscillation when two sound frequencies are close.

Perpendicular SHMs

x = Acos(ωt), y = Bcos(ωt + φ). Path is ellipse in general. Special cases: φ = 0: straight line y = (B/A)x; φ = π/2 and A = B: circle; φ = π/2: ellipse x²/A² + y²/B² = 1.

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Lissajous Figures

x = Acos(ω_xt), y = Bcos(ω_yt + φ) with frequency ratio ω_x/ω_y = rational number. Produces closed curves whose shape depends on ratio and phase. Used historically to compare frequencies and determine phase relationships. Frequency ratio revealed by counting tangent points.

Fourier Analysis

Any periodic function can be written as sum of harmonics: f(t) = a₀ + Σ[a_ncos(nωt) + b_nsin(nωt)]. SHM is fundamental building block. Even non-periodic functions have Fourier transform representations. Essential for signal processing, acoustics, and quantum mechanics.

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Solved Example: Beat Frequency

Two tuning forks produce frequencies 440 Hz and 445 Hz. A listener hears beats. Find: (a) Beat frequency, (b) Time between successive maxima of loudness, (c) Number of beats heard in 10 seconds. Solution: (a) ω₁ = 2π×440 = 880π rad/s, ω₂ = 2π×445 = 890π rad/s. Beat frequency f_b = |f₂ - f₁| = 445 - 440 = 5 Hz. Or ω_b = |ω₂ - ω₁| = 10π rad/s → f_b = 5 Hz. (b) Time between maxima = 1/f_b = 1/5 = 0.2 s = 200 ms. (c) In 10 seconds: number of beats = f_b × 10 = 5 × 10 = 50 beats. The ear perceives 50 loudness variations in 10 seconds. Each beat corresponds to the two waves going in and out of phase. When in phase (φ = 0), amplitudes add constructively. When out of phase (φ = π), destructive interference. The average frequency heard is (440+445)/2 = 442.5 Hz, with 5 Hz amplitude modulation.

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