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#wave equation

23 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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Superposition and Interference of Waves

A stationary wave is given by \( y = 0.06 \sin (\frac{\pi x}{2}) \cos (120\pi t) \), where \( x \) and \( y \) are in me

**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. Form: y = A sin (kx) cos (ω t) , k = (π/2) rad/m . λ = (2π/k) = (2π/(π/2)) = 4 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 8 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.04 \sin (30x - 60t) \) reflects at an open boundary. What is the equation of the reflected wave?

**Longitudinal vibrations in rods** clamped at middle have fundamental with node at clamp and antinodes at ends, f₁ = v/(2L), v speed of sound in material (m/s). This relation allows v extraction from measured f₁ and length L, e.g., v = 2L·f₁. At open boundary, no phase change. Incident: y_i = 0.04 sin (30x - 60t) . Reflected: y_r = 0.04 sin (30x + 60t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = 0.04 sin (30x + 60t), illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Sound Waves, Reflection and Characteristics

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Wave Equation and Displacement Relation

A wave is described by \( y(x, t) = 0.05 \sin (45x - 90t) \), where \( x \) and \( y \) are in meters and \( t \) in sec

**Sinusoidal wave form** represents harmonic wave where each particle executes SHM. Coefficients of x and t give spatial and temporal periodicities, allowing wavelength and period extraction, basis for wave analysis in NCERT. Compare with y = a sin (kx - ω t) , k = 45 rad/m , ω = 90 rad/s . Speed: v = (ω/k) = (90/45) = 2 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 2 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Wave Equation and Displacement Relation

A wave is described by \( y(x, t) = 0.06 \sin (30x - 90t) \), where \( x \) and \( y \) are in meters and \( t \) in sec

**Sinusoidal wave form** represents harmonic wave where each particle executes SHM. Coefficients of x and t give spatial and temporal periodicities, allowing wavelength and period extraction, basis for wave analysis in NCERT. Compare with y = a sin (kx - ω t) . k = 30 rad/m , ω = 90 rad/s . Speed: v = (ω/k) = (90/30) = 3 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 3 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.07 \sin (20x - 60t) \) reflects at a rigid boundary. What is the equation of the reflected wave?

**Reflection at boundaries** follows phase change rules: rigid boundary (fixed end) introduces π phase shift, inverting displacement y → -y, while free boundary reflects without phase change. Reflected wave derived by reversing propagation direction kx → -kx and applying phase shift, preserving k = 2π/λ and ω = 2πf. At rigid boundary, phase changes by π . Incident: y_i = 0.07 sin (20x - 60t) . Reflected: y_r = -0.07 sin (20x + 60t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = -0.07 sin (20x + 60t), illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Sound Waves, Reflection and Characteristics

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.03 \sin (20x - 40t) \) reflects at an open boundary. What is the equation of the reflected wave?

**Reflection at boundaries** follows phase change rules: rigid boundary (fixed end) introduces π phase shift, inverting displacement y → -y, while free boundary reflects without phase change. Reflected wave derived by reversing propagation direction kx → -kx and applying phase shift, preserving k = 2π/λ and ω = 2πf. At open boundary, no phase change. Incident: y_i = 0.03 sin (20x - 40t) . Reflected: y_r = 0.03 sin (20x + 40t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = 0.03 sin (20x + 40t), illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Sound Waves, Reflection and Characteristics

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Wave Equation and Displacement Relation

A wave is described by \( y(x, t) = 0.02 \sin (25x - 75t) \), where \( x \) and \( y \) are in meters and \( t \) in sec

**Displacement relation** encodes λ = 2π/k and f = ω/2π. Comparing given equation y = a sin(kx - ωt) with standard form yields k and ω, hence λ = 2π/k and v = ω/k, essential for identifying propagation characteristics and phase. Compare with y = a sin (kx - ω t) , ω = 75 rad/s . Period: T = (2π/ω) = (2π/75) ≈ 0.0838 s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0.08 s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Wave Equation and Displacement Relation

A wave is described by \( y(x, t) = 0.01 \sin (40x - 80t) \), where \( x \) and \( y \) are in meters and \( t \) in sec

**Displacement relation** encodes λ = 2π/k and f = ω/2π. Comparing given equation y = a sin(kx - ωt) with standard form yields k and ω, hence λ = 2π/k and v = ω/k, essential for identifying propagation characteristics and phase. Compare with y = a sin (kx - ω t) , ω = 80 rad/s . Frequency: v = (ω/2π) = (80/2π) ≈ 12.73 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 13 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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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 interdependence and quantization by boundaries.

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

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.05 \sin (12x - 36t) \) reflects at a rigid boundary. What is the equation of the reflected wave?

**Sound wave reflection** at rigid wall behaves like string fixed end, displacement inverted. Equation of reflected wave includes sign change and direction reversal, amplitude unchanged but sign may flip, explaining standing wave formation with incident wave. At rigid boundary, phase changes by π . Incident: y_i = 0.05 sin (12x - 36t) . Reflected: y_r = -0.05 sin (12x + 36t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = -0.05 sin (12x + 36t), illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Sound Waves, Reflection and Characteristics

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Sound Waves, Reflection and Characteristics

A wave \( y = 0.02 \sin (40x - 120t) \) reflects at an open boundary. What is the equation of the reflected wave?

**Reflection at boundaries** follows phase change rules: rigid boundary (fixed end) introduces π phase shift, inverting displacement y → -y, while free boundary reflects without phase change. Reflected wave derived by reversing propagation direction kx → -kx and applying phase shift, preserving k = 2π/λ and ω = 2πf. At open boundary, no phase change. Incident: y_i = 0.02 sin (40x - 120t) . Reflected: y_r = 0.02 sin (40x + 120t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = 0.02 sin (40x + 120t), illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Sound Waves, Reflection and Characteristics

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