Wave Equation and Displacement Relation

Wave Equation and Displacement Relation

A transverse wave on a string has a speed of 24 m/s and a period of 0.02 s. What is its wavelength?

**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. Frequency: v = (1/T) = (1/0.02) = 50 Hz . Wavelength: λ = (v/v) = (24/50) = 0.48 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0.48 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Wave Equation and Displacement Relation

A wave on a string has an amplitude of 0.02 m and a period of 0.01 s. What is the maximum transverse speed of a particle

**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. ω = (2π/T) = (2π/0.01) = 200π rad/s . Max speed: vₘₐₓ = a ω = 0.02 × 200π ≈ 12.57 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 12.6 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Wave Equation and Displacement Relation

A transverse wave on a string has a speed of 30 m/s and a wavelength of 1.5 m. What is the time period of the wave?

**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. Speed: v = v λ , v = (v/λ) = (30/1.5) = 20 Hz . Time period: T = (1/v) = (1/20) = 0.05 s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0.05 s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Wave Equation and Displacement Relation

A longitudinal wave travels in a medium with bulk modulus 1.8 × 10⁹ Pa and density 900 kg/m³. What is its speed?

**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. Speed: v = √((B/rho)) = √((1.8 × 10⁹/900)) = √(2 × 10⁶) ≈ 1414 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1400 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Wave Equation and Displacement Relation

A wave on a string has an amplitude of 0.04 m and a period of 0.025 s. What is the maximum transverse speed of a particl

**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. ω = (2π/T) = (2π/0.025) = 80π rad/s . Max speed: vₘₐₓ = a ω = 0.04 × 80π ≈ 10.05 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 10 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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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.

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Wave Equation and Displacement Relation

Which wave type is most likely to propagate fastest in a solid medium?

**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. In solids, longitudinal waves (P-waves) travel faster than transverse waves (S-waves) because the bulk modulus plus shear modulus exceeds the shear modulus alone, increasing speed. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Longitudinal, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Wave Equation and Displacement Relation

A longitudinal wave in a medium has a speed of 1550 m/s and a density of 1200 kg/m³. What is the bulk modulus of the med

**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. Speed: v = √((B/rho)) . 1550 = √((B/1200)) ⇒ 1550² = (B/1200) . B = 1550² × 1200 = 2.88 × 10⁹ Pa . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 2.88 × 10⁹ Pa, illustrating frequency-length-speed interdependence and quantization by boundaries.

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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.

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Wave Equation and Displacement Relation

A longitudinal wave in a medium has a speed of 1700 m/s and a bulk modulus of 2.89 × 10⁹ Pa. What is the density of the

**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. Speed: v = √((B/rho)) . 1700 = √((2.89 × 10⁹/rho)) ⇒ 1700² = (2.89 × 10⁹/rho) . rho = (2.89 × 10⁹/1700²) = (2.89 × 10⁹/2.89 × 10⁶) = 1000 kg/m³ . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1000 kg/m³, illustrating frequency-length-speed interdependence and quantization by boundaries.

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Wave Equation and Displacement Relation

What happens to a transverse wave when it reflects off a fixed end of a string?

**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. At a fixed end, the displacement must be zero, requiring the reflected wave to be out of phase by π radians (inverted) to cancel the incident wave at the boundary. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields It reflects with a phase change of π radians, illustrating frequency-length-speed interdependence and quantization by boundaries.

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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.

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