5) Waves in tubes Bryan is hearing the thuds of Zack fighting back against the a

Question

5) Waves in tubes Bryan is hearing the thuds of Zack fighting back against the aliens (they eventually pounced on him). He is hearing this through an air duct . a "tube". a)He knows the tube with both ends open is 2.5 meters long. What are the lowest three resonant frequencies that Bryan could hear of the wide range of thudding noises/frequencies from Zack. (What are the frequencies that be excited in the tube?) The speed of sound is 331 m/sec (so, in 0 degree air-yes, the aliens have knocked out some of the life support systems, so the ship is starting to freeze!)? b) Two adjacent thud frequencies heard by Bryan in a completely different tube are 90 Hz and 150 Hz. Does the second tube/air-shaft have both ends open, both ends closed, or one open/one closed- explain. And what is the length of this tube (if the speed of sound is the same).

Explanation / Answer

a) Frequencies excited in the tube with both ends open or both ends closed,
f = n V/ 2L ,.....1 where n is any interger, V is speed of wave and L is length of tube
Hence lowest three frequencies are
V/2L , 2V/2L and 3v/2L
= 66.2, 132.4 and 198.6 Hz

b) Frequencies excited in the tube with one end open and one closed,
f = mV/4L ,.......2    where m is odd integer
From equation 1 we get difference of two successive frequencies = (n+1) V/ 2L - nV/2L
= V/2L = minimum frequency
From equation 2 , difference of two successive frequencies = (m+2) V/ 4L - mV/2L ( as m can take only odd values)
= 2V/4L = V/2L = 2 times minimum frequency.

difference of 2 successive frequencies 90 and 150 = 60
But 60 can't be minimum frequency, as 90 and 150 are not mutiples of 60,
minimum frequency is 60/2 = 30
Hence second tube has one open one end closed.

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