#second harmonic

Doppler Effect

A string fixed at both ends has a length of 1.4 m and a wave speed of 70 m/s. What is the frequency of its second harmon

**Frequency shift** proportional to source speed relative to wave speed v. Understanding sign convention for approaching versus receding is key, with approaching increasing frequency and receding decreasing, central to Doppler applications. For fixed ends: v_n = (n v/2L) . Second harmonic ( n = 2 ): v₂ = (2 × 70/2 × 1.4) = (140/2.8) = 50 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 50 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Sound Waves, Reflection and Characteristics

A pipe closed at one end has a length of 0.85 m and resonates at its second harmonic with a speed of sound of 340 m/s. W

**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. For closed pipe: v_n = (n + (1/2)) (v/2L) , n = 1 for second harmonic. v₁ = (1 + (1/2)) (340/2 × 0.85) = 1.5 × (340/1.7) = 300 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 300 Hz, 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 pipe closed at one end has a length of 0.6 m and resonates at its second harmonic with a speed of sound of 360 m/s. Wh

**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. For closed pipe: v_n = (n + (1/2)) (v/2L) , n = 1 for second harmonic. v₁ = (1 + (1/2)) (360/2 × 0.6) = 1.5 × (360/1.2) = 1.5 × 300 = 450 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 450 Hz, 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 pipe open at both ends has a length of 0.25 m and a speed of sound of 340 m/s. What is the frequency of its second har

**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. For open pipe: v_n = (n v/2L) . Second harmonic ( n = 2 ): v₂ = (2 × 340/2 × 0.25) = (680/0.5) = 1360 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1360 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.

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

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Vibrations of Air Columns - Open and Closed Pipes

A pipe open at both ends has a length of 0.3 m and a speed of sound of 330 m/s. What is the frequency of its second harm

**Air column vibrations** depend on end conditions. Pipe closed at one end has displacement node at closed end and antinode at open, allowing only odd harmonics, fundamental f₁ = v/(4L). Open pipe has antinodes at both ends, fₙ = n·v/(2L), all harmonics present, v sound speed. For open pipe: v_n = (n v/2L) . Second harmonic ( n = 2 ): v₂ = (2 × 330/2 × 0.3) = (660/0.6) = 1100 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1100 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 vibrates in its second harmonic with a frequency of 80 Hz. If its length is 1 m, what is the

**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. For fixed ends: v_n = (n v/2L) . Second harmonic ( n = 2 ): 80 = (2 × v/2 × 1) . 80 = (v/1) ⇒ v = 80 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 80 m/s, 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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Stationary Waves and Standing Waves in Strings

A string of length 0.8 m is fixed at both ends. If the speed of the wave is 40 m/s, what is the frequency of the second

**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) . Second harmonic: n = 2 . v₂ = (2 × 40/2 × 0.8) = (80/1.6) = 50 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 50 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 0.42 m and a speed of sound of 336 m/s. What is the frequency of its second 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 = 1 for second harmonic. v₁ = (1 + (1/2)) (336/2 × 0.42) = 1.5 × (336/0.84) = 1.5 × 400 = 600 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 600 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

What is the relationship between the wavelengths of the fundamental mode and the second harmonic in a string fixed at bo

**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 a string fixed at both ends, fundamental wavelength is λ₁ = 2L , and second harmonic is λ₂ = L . Thus, λ₁ = 2 λ₂ , or λ₂ = λ₁ / 2 . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Second harmonic is half the fundamental, 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 0.75 m and a speed of sound of 330 m/s. What is the frequency of its second 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) . Second harmonic ( n = 2 ): v₂ = (2 × 330/2 × 0.75) = (660/1.5) = 440 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 440 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 1.2 m and a wave speed of 72 m/s. What is the frequency of its second harmon

**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. For fixed ends: v_n = (n v/2L) . Second harmonic ( n = 2 ): v₂ = (2 × 72/2 × 1.2) = (144/2.4) = 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 > Stationary Waves and Standing Waves in Strings

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