A) A longitudinal wave propagating in a water-filled pipe has intensity 2.90*10^

Question

A) A longitudinal wave propagating in a water-filled pipe has intensity 2.90*10^(-6) W/m^2 and frequency 3450 Hz. Find the amplitude A of the wave. Water has density 1000 kg/m^3 and bulk modulus 2.18*10^9 Pa.

B) Find the wavelength of the wave.

C) If the pipe is filled with air at pressure 1.0*10^5 Pa and density 1.20 kg/m^3, what will be the amplitude A and wavelength of a longitudinal wave with the same intensity and frequency as in part A?

D) If the pipe is filled with air at pressure 1.00*10^5 Pa and density 1.20 kg/m^3, what will be wavelength of a longitudinal wave with the same intensity and frequency as in part A?

Explanation / Answer

I = 2.90 * 10-6 W/m2, f = 3450 Hz

Density of water , p = 1000  kg/m3

Bulkmodulus is B = 2.18 * 109 Pa

(a) The speed of the wave is v = sqrt B / p

= sqrt (2.18 * 109 Pa) / (1000  kg/m3 ) = 1.47 * 103 m/s

The relation between the intensity and the amplitude is I = (1/2 ) B w k A2

= (1/2 ) B w (w/v )A2 = (1/2 ) B w2A2 / v  -------------- (1)

The angular frequency, w = 2 pi f, w2   = 4 pi2f2 = 4 pi2 (3450 Hz )2 = 4.699 * 108 rad2/s2

From equation (1) and substituting the related values

I = (1/2 ) B w2A2 / v

(2.90 * 10-6 W/m2) = (0.5) (2.18 * 109 Pa) ( 4.699 * 108 rad2/s2 ) A2 / (1.47 * 103 m/s )

amplitude A = 9.12 * 10-11 m

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b) Wavelength of the wave is, lambda = v / f = (1.47 * 103 m/s ) / 3450 Hz

lambda = 0.426 m    

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c) Given pressure is, P = 1.0 *105  Pa

Density of air is, p' = 1.2 kg/m3   

The speed of the wave v = sqrt[ y P / p]

For air (diatomic ) , the value of y =1.4

v = sqrt [(1.40) (1.0 *105  Pa ) / 1.2 kg/m3]

= 3.416 * 102 m/s

I = (1/2 ) B w2A2 / v = (1/2 ) (y P ) w2A2 / v

(2.90 *10-6 W/m2) = (0.5)(1.40) (1.0 * 105 Pa) ( 4.699 * 108 rad2/s2 ) A2 / (3.416 * 102 m/s )

A = 5.48 * 10-9 m

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d) Wavelength of the wave = v / f

= (3.416 * 102 m/s ) / 3450 Hz

Wavelength = 0.099 m = 0.1 m

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