#density

Superposition and Interference of Waves

A longitudinal wave in a medium has a speed of 1450 m/s and a density of 1100 kg/m³. What is the bulk modulus of the med

**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. Speed: v = √((B/rho)) . 1450 = √((B/1100)) ⇒ 1450² = (B/1100) . B = 1450² × 1100 = 2.31 × 10⁹ Pa . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 2.31 × 10⁹ Pa, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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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.

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

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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.

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

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Transverse and Longitudinal Waves

A longitudinal wave in a medium has a speed of 1500 m/s and a density of 1200 kg/m³. What is the bulk modulus of the med

**Wave classification** depends on particle vibration relative to propagation. Longitudinal waves have particle oscillation parallel to propagation, creating compressions and rarefactions as in sound in air; transverse have perpendicular oscillation. Tuning fork generates longitudinal sound because air cannot sustain shear. Speed: v = √((B/rho)) . 1500 = √((B/1200)) ⇒ 1500² = (B/1200) . B = 1500² × 1200 = 2.7 × 10⁹ Pa . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 2.7 × 10⁹ Pa, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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Transverse and Longitudinal Waves

A longitudinal wave in a medium has a speed of 1600 m/s and a bulk modulus of 2.56 × 10⁹ Pa. What is the density of the

**Nature of wave** determines energy transfer mechanism. Longitudinal nature of sound explains propagation through fluids with density variations carrying energy, unlike transverse waves needing rigidity for restoring force. Speed: v = √((B/rho)) . 1600 = √((2.56 × 10⁹/rho)) ⇒ 1600² = (2.56 × 10⁹/rho) . rho = (2.56 × 10⁹/1600²) = (2.56 × 10⁹/2.56 × 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.

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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Transverse and Longitudinal Waves

A longitudinal wave in a medium has a speed of 1400 m/s and a density of 800 kg/m³. What is the bulk modulus of the medi

**Mechanical waves** require medium and can be transverse or longitudinal. Sound in air is longitudinal since fluids support only compressional motion, while string waves are transverse. Progressive waves transfer energy without net mass transport, distinguishing from standing waves. Speed: v = √((B/rho)) . 1400 = √((B/800)) ⇒ 1400² = (B/800) . B = 1400² × 800 = 1.568 × 10⁹ Pa . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1.57 × 10⁹ Pa, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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Transverse and Longitudinal Waves

A longitudinal wave in a medium has a speed of 1600 m/s and a density of 1250 kg/m³. What is the bulk modulus of the med

**Nature of wave** determines energy transfer mechanism. Longitudinal nature of sound explains propagation through fluids with density variations carrying energy, unlike transverse waves needing rigidity for restoring force. Speed: v = √((B/rho)) . 1600 = √((B/1250)) ⇒ 1600² = (B/1250) . B = 1600² × 1250 = 3.2 × 10⁹ Pa . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 3.2 × 10⁹ Pa, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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Wave Speed, Energy and Power

Which property of a gas affects the speed of longitudinal waves more significantly than its density?

**Wave speed on string** is v = √(T/μ), T tension (N), μ = m/L linear mass density (kg/m). Higher T increases restoring force raising speed, heavier μ lowers speed. Frequency follows f = v/λ, linking mechanical properties to wave dynamics and harmonic series. Speed of sound in a gas is v = √((gamma P/rho)) . The ratio gamma (specific heats) influences v more than rho alone, as P/rho relates to temperature, but gamma varies with molecular structure. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Ratio of specific heats, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Speed, Energy and Power

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Wave Speed, Energy and Power

The bulk modulus of water is 2.2 × 10⁹ Pa and its density is 1000 kg/m³. What is the speed of sound in water?

**Transverse wave velocity** depends on medium not frequency alone. For string under tension, v ∝ √(T/μ), calculation requires μ from mass and length, then square root evaluation, giving v in m/s, then f = v/λ for given wavelength. Speed: v = √((B/rho)) = √((2.2 × 10⁹/1000)) = √(2.2 × 10⁶) ≈ 1483 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1480 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Speed, Energy and Power

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Wave Equation and Displacement Relation

A copper rod has a density of 8900 kg/m³ and a longitudinal wave speed of 3560 m/s. What is its Young’s modulus?

**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. v = √((Y/rho)) . 3560 = √((Y/8900)) ⇒ 3560² = (Y/8900) . Y = 3560² × 8900 ≈ 1.13 × 10¹¹ Pa . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1.13 × 10¹¹ Pa, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Wave Speed, Energy and Power

What is the primary factor affecting the speed of sound in a solid rod compared to a gas?

**Energy transport** in waves scales with amplitude squared A² and frequency squared ω². Wave speed determines propagation rate, and understanding T and μ allows quantitative prediction of v and associated frequencies. In solids, sound speed depends on Young’s modulus ( v = √((Y/rho)) ), which measures stiffness and is much higher in solids than the bulk modulus in gases, leading to faster sound propagation. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Stiffness, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Speed, Energy and Power

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Sound Waves, Reflection and Characteristics

A steel rod of density 7800 kg/m³ has a Young’s modulus of 2 × 10¹¹ Pa. What is the speed of a longitudinal wave in the

**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. Speed: v = √((Y/rho)) = √((2 × 10¹¹/7800)) ≈ √(2.56 × 10⁷) ≈ 5060 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 5060 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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