#fifth harmonic

Superposition and Interference of Waves

A string of length 1.5 m fixed at both ends has a wave speed of 45 m/s. What is the frequency of its fifth harmonic?

**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. v_n = (n v/2L) . Fifth harmonic ( n = 5 ): v₅ = (5 × 45/2 × 1.5) = (225/3) = 75 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 75 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Vibrations of Air Columns - Open and Closed Pipes

A pipe closed at one end has a length of 0.45 m and a speed of sound of 360 m/s. What is the frequency of its fifth harm

**Resonance in pipes** occurs when length accommodates standing wave pattern. Closed pipe L = (2n-1)λ/4, so f₁ = v/(4L). Given f₁ and v, length follows L = v/(4f₁), enabling length calculation from measured resonance frequency and sound speed 330-340 m/s. For closed pipe: v_n = (n + (1/2)) (v/2L) , n = 4 for fifth harmonic. v₄ = (4 + (1/2)) (360/2 × 0.45) = 4.5 × (360/0.9) = 4.5 × 400 = 1800 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1800 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Vibrations of Air Columns - Open and Closed Pipes

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Vibrations of Air Columns - Open and Closed Pipes

A closed pipe of length 0.3 m has a speed of sound of 330 m/s. What is the frequency of its fifth harmonic?

**Resonance in pipes** occurs when length accommodates standing wave pattern. Closed pipe L = (2n-1)λ/4, so f₁ = v/(4L). Given f₁ and v, length follows L = v/(4f₁), enabling length calculation from measured resonance frequency and sound speed 330-340 m/s. For closed pipe: v_n = (n + (1/2)) (v/2L) , n = 4 for fifth harmonic. v₄ = (4 + (1/2)) (330/2 × 0.3) = 4.5 × (330/0.6) = 4.5 × 550 = 2475 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 2475 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Vibrations of Air Columns - Open and Closed Pipes

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Stationary Waves and Standing Waves in Strings

A string of length 2.1 m fixed at both ends has a wave speed of 63 m/s. What is the frequency of its fifth harmonic?

**Quantization due to boundaries** leads to discrete harmonic series. Frequency difference between harmonics is f₁, so f₃ - f₁ = 2f₁ = v/L. Understanding node-antinode pattern explains resonance and overtones in strings. v_n = (n v/2L) . Fifth harmonic ( n = 5 ): v₅ = (5 × 63/2 × 2.1) = (315/4.2) = 75 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 75 Hz, 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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