15.17 A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m s^–1).

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Vishal Baghel | Contributor-Level 10

5 months ago

Length of the pipe, l = 20 cm = 0.2 m

Source frequency = $n^{t h}$ normal mode frequency is given by the relation

$ν_{n}$ = (2n-1) $\frac{ν}{4 l}$ , n is an integer = 0,1,2,3, ….

430 = (2n-1) $\times \frac{340}{4 \times 0.2}$

n = 1

Hence, the first mode of vibration frequency is resonantly excited by the given source. In a pipe open at both ends, the $n^{t h}$ mode of vibration frequency is given by the relation:

$ν_{n} = \frac{n v}{2 l}$ , n = $\frac{2 l ν_{n}}{v}$ = $\frac{2 \times 0.2 \times 430}{340}$ = 0.5

Since the number of the node of vibration (n) has to be an integer, the given source does not produce a resound vibration in an open pipe.

Length of the pipe, l = 20 cm = 0.2 mSource frequency = <math><msup><mrow><mrow><mi>n</mi></mrow></mrow><mrow><mrow><mi>t</mi><mi>h</mi></mrow></mrow></msup></math> normal mode frequency is given by the relation<math><msub><mrow><mrow><mi>ν</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub></math> = (2n-1) <math><mfrac><mrow><mrow><mi>ν</mi></mrow></mrow><mrow><mrow><mn>4</mn><mi>l</mi></mrow></mrow></mfrac></math> , n is an integer = 0,1,2,3, ….430 = (2n-1) <math><mo>×</mo><mfrac><mrow><mrow><mn>340</mn></mrow></mrow><mrow><mrow><mn>4</mn><mi></mi><mo>×</mo><mn>0.2</mn></mrow></mrow></mfrac></math>n = 1Hence, the first mode of vibration frequency is resonantly excited by the given source. In a pipe open at both ends, the <math><msup><mrow><mrow><mi>n</mi></mrow></mrow><mrow><mrow><mi>t</mi><mi>h</mi></mrow></mrow></msup></math> mode of vibration frequency is given by the relation:<math><msub><mrow><mrow><mi>ν</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub><mo>=</mo><mi></mi><mfrac><mrow><mrow><mi>n</mi><mi>v</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>l</mi></mrow></mrow></mfrac></math> , n = <math><mfrac><mrow><mrow><mn>2</mn><mi>l</mi><msub><mrow><mrow><mi>ν</mi></mrow></mrow><mrow><mrow><mi>n</mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><mi>v</mi></mrow></mrow></mfrac></math> = <math><mfrac><mrow><mrow><mn>2</mn><mo>×</mo><mn>0.2</mn><mo>×</mo><mn>430</mn></mrow></mrow><mrow><mrow><mn>340</mn></mrow></mrow></mfrac></math> = 0.5Since the number of the node of vibration (n) has to be an integer, the given source does not produce a resound vibration in an open pipe.