#fourth harmonic

Doppler Effect

A string of length 2.2 m fixed at both ends has a wave speed of 66 m/s. What is the frequency of its fourth harmonic?

**Relative motion** changes effective wavelength encountered. For moving source approaching stationary observer, wavelength ahead λ' = (v - v_s)/f, so f' = v/λ' = f·v/(v - v_s) > f, basis for calculating apparent pitch shift in sound. v_n = (n v/2L) . Fourth harmonic ( n = 4 ): v₄ = (4 × 66/2 × 2.2) = (264/4.4) = 60 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 60 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Stationary Waves and Standing Waves in Strings

A string of length 2.8 m fixed at both ends has a wave speed of 84 m/s. What is the frequency of its fourth 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) . Fourth harmonic ( n = 4 ): v₄ = (4 × 84/2 × 2.8) = (336/5.6) = 60 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 60 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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Vibrations of Air Columns - Open and Closed Pipes

A pipe closed at one end has a length of 0.28 m and a speed of sound of 336 m/s. What is the frequency of its fourth har

**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 = 3 for fourth harmonic. v₃ = (3 + (1/2)) (336/2 × 0.28) = 3.5 × (336/0.56) = 3.5 × 600 = 2100 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 2100 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 fixed at both ends has a length of 2.5 m and a wave speed of 100 m/s. What is the frequency of its fourth harmo

**Standing wave in fixed string** has nodes at ends, quantizing modes. Fundamental n=1 has λ₁ = 2L, higher harmonics multiples of fundamental f₁. Third harmonic n=3 has three half-wavelengths in length L, f₃ = 3v/(2L), illustrating standing wave condition and boundary enforcement. For fixed ends: v_n = (n v/2L) . Fourth harmonic ( n = 4 ): v₄ = (4 × 100/2 × 2.5) = (400/5) = 80 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 80 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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Vibrations of Air Columns - Open and Closed Pipes

A pipe open at both ends has a length of 0.55 m and a speed of sound of 330 m/s. What is the frequency of its fourth har

**Organ pipe modes** illustrate boundary influence. Closed pipe odd harmonic series contrasts with open pipe full series, affecting timbre. Frequency scales inversely with length, explaining pitch variation with pipe length. For open pipe: v_n = (n v/2L) . Fourth harmonic ( n = 4 ): v₄ = (4 × 330/2 × 0.55) = (1320/1.1) = 1200 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1200 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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