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#node spacing

3 public questions tagged with this topic.

Beats Phenomenon

A stationary wave on a string is given by \( y = 0.08 \sin (2\pi x) \cos (100\pi t) \), where \( x \) and \( y \) are in

**Beat formation** is interference in time with time-varying amplitude. Frequencies close together generate slow modulation, count in given duration obtained by multiplying beat frequency by duration, e.g., 6 Hz × 5 s = 30 beats. Form: y = A sin (kx) cos (ω t) , k = 2π rad/m . Wavelength: λ = (2π/k) = (2π/2π) = 1 m . Distance between nodes: (λ/2) = (1/2) = 0.5 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0.5 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Beats Phenomenon

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Stationary Waves and Standing Waves in Strings

A stationary wave is given by \( y = 0.08 \sin (\frac{\pi x}{3}) \cos (60\pi t) \), where \( x \) and \( y \) are in met

**Stationary waves** form when identical progressive waves traveling opposite directions interfere, y = 2A sin(kx) cos(ωt), nodes where sin(kx)=0, antinodes where |sin(kx)|=1. For string fixed at both ends, allowed wavelengths λₙ = 2L/n, frequencies fₙ = n·v/(2L), n = 1,2,3… harmonic number. Form: y = A sin (kx) cos (ω t) , k = (π/3) rad/m . Wavelength: λ = (2π/k) = (2π/(π/3)) = 6 m . Distance between node and antinode: (λ/4) = (6/4) = 1.5 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1.5 m, illustrating frequency-length-speed interdepen

Ref: NCERT > Physics Book > Waves > Stationary Waves and Standing Waves in Strings

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Wave Equation and Displacement Relation

A stationary wave is given by \( y = 0.07 \sin (\frac{\pi x}{5}) \cos (90\pi t) \), where \( x \) and \( y \) are in met

**Wave equation** y(x,t) = A sin(kx - ωt + φ) describes displacement of progressive harmonic wave, where k = 2π/λ wave number (rad/m), ω = 2πf angular frequency (rad/s), v = ω/k wave speed (m/s). Sign of ωt indicates direction, amplitude A is maximum displacement. Form: y = A sin (kx) cos (ω t) , k = (π/5) rad/m . Wavelength: λ = (2π/k) = (2π/(π/5)) = 10 m . Distance between nodes: (λ/2) = (10/2) = 5 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 5 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Equation and Displacement Relation

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