#sine wave

Equations of SHM, Phase and Angular Frequency

A particle in SHM has \( x = 7 \sin (3t + \frac{\pi}{6}) \) (in m). What is its acceleration at \( t = 0 \, \text{s} \)?

**General equation of SHM** x = A sin(ωt + φ) or A cos(ωt + φ) includes amplitude A (m), angular frequency ω = √(k/m) (rad/s) for spring system, and initial phase φ (rad) setting t=0 position. Phase (ωt + φ) determines instantaneous state, phase difference Δφ governs interference of two SHM motions. Acceleration: a = -ω² x . ω = 3 s⁻¹, x(0) = 7 sin (π/6) = 7 × 0.5 = 3.5 m . a = -3² × 3.5 = -9 × 3.5 = -31.5 m/s² . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and

Ref: NCERT > Physics Book > Oscillations > Equations of SHM, Phase and Angular Frequency

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Doppler Effect

Two waves \( y_1 = 3 \sin (10x - 20t) \) and \( y_2 = 3 \sin (10x - 20t + \frac{\pi}{4}) \) interfere. What is the ampli

**Relative motion** changes effective wavelength encountered. For moving source approaching stationary observer, wavelength ahead λ' = (v - v_s)/f, so f' = v/λ' = f·v/(v - v_s) > f, basis for calculating apparent pitch shift in sound. Amplitude: A = 2a cos (Φ/2) , a = 3 m , Φ = (π/4) . A = 2 × 3 cos (π/8) ≈ 6 × 0.9239 ≈ 5.54 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 5.5 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Superposition and Interference of Waves

Two waves \( y_1 = 6 \sin (8x - 16t) \) and \( y_2 = 6 \sin (8x - 16t + \frac{2\pi}{3}) \) interfere. What is the amplit

**Interference of waves** produces enhancement or cancellation based on phase. Two equal amplitude waves out of phase by π cancel completely, A = 0, while in-phase superposition doubles amplitude to 2a, demonstrating energy redistribution without violation of conservation. Amplitude: A = 2a cos (Φ/2) , a = 6 m , Φ = (2π/3) . A = 2 × 6 cos (π/3) = 12 × (1/2) = 6 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 6 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Superposition and Interference of Waves

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Superposition and Interference of Waves

Two waves \( y_1 = 4 \sin (5x - 15t) \) and \( y_2 = 4 \sin (5x - 15t + \pi) \) interfere. What is the amplitude of the

**Superposition principle** states resultant displacement equals algebraic sum of individual waves, y = y₁ + y₂. For coherent waves with phase difference φ, resultant amplitude A = √(a₁² + a₂² + 2a₁a₂ cosφ), equal amplitudes give A = 2a cos(φ/2), constructive when φ = 2nπ, destructive when φ = (2n+1)π. Amplitude: A = 2a cos (Φ/2) , a = 4 m , Φ = π . A = 2 × 4 cos (π/2) = 8 × 0 = 0 m (destructive interference). Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Superposition and Interference of Waves

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Superposition and Interference of Waves

Two waves \( y_1 = 3 \sin (4x - 10t) \) and \( y_2 = 3 \sin (4x - 10t + \pi/2) \) interfere. What is the amplitude of th

**Superposition principle** states resultant displacement equals algebraic sum of individual waves, y = y₁ + y₂. For coherent waves with phase difference φ, resultant amplitude A = √(a₁² + a₂² + 2a₁a₂ cosφ), equal amplitudes give A = 2a cos(φ/2), constructive when φ = 2nπ, destructive when φ = (2n+1)π. Amplitude: A = 2a cos (Φ/2) , where a = 3 m , Φ = π/2 . A = 2 × 3 cos π/4 = 6 × (1/√(2)) = 3√(2) ≈ 4.24 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 4.2 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Superposition and Interference of Waves

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Superposition and Interference of Waves

Two waves \( y_1 = 5 \sin (6x - 12t) \) and \( y_2 = 5 \sin (6x - 12t + \frac{\pi}{3}) \) interfere. What is the amplitu

**Interference of waves** produces enhancement or cancellation based on phase. Two equal amplitude waves out of phase by π cancel completely, A = 0, while in-phase superposition doubles amplitude to 2a, demonstrating energy redistribution without violation of conservation. Amplitude: A = 2a cos (Φ/2) , a = 5 m , Φ = (π/3) . A = 2 × 5 cos (π/6) = 10 × (√(3)/2) = 5√(3) ≈ 8.66 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 8.7 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Superposition and Interference of Waves

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