Normal modes of air columns (a) A culvert at a highway construction site acts li

Question

Normal modes of air columns (a) A culvert at a highway construction site acts like an open pipe of length L. On a day when the air temperature in the culvert is 4.5 C an acoustical engineer standing at one end of the culvert is using a wave generator to probe the response of the culvert to pure acoustical tones. The engineer notes a resonance in the culvert at 115.5 Hz, and successive resonance occur at 121.0 Hz and 126.5 Hz. What are (i) the length of the culvert and (ii) the harmonic numbers associated with the three frequencies.

(b) The same culvert is now suddenly blocked at one end by roadfill rip-rap (large rocks), and acts like a closed pipe. The engineer (standing at the open end) now tries to make the pipe resonate at the same three frequencies mentioned in part (a). Assuming the length of the culvert and speed of sound are unchanged, at which of the three frequencies will the pipe resonate, if any?

Explanation / Answer

The temperature of the air is T = 4.5 C The speed of the sound at this temperature is vs = 334.245 m/s The length of the culvert is L a) The resonating frequencies are f1 = 115.5 Hz f2 = 121.0 Hz f3 = 126.5 Hz The resonating frequencis for the open pipe occurs at f = nvs/2L with n = 1,2,3... Let the three frequencis occur for n = m-1, m, m+1 respectively (m-1)vs/2L = 115.5 Hz (m)vs/2L = 121 Hz (m+1)vs/2L = 126.5 Hz (i) on subtracting (2)-(1) or (3)-(2) vs/2L = 5.5 Hz L = 334.245/2*5.5 L = 30.386 m The length of the culvert id 30.386 m (ii) mvs/2L =121.5 Hz m*334.245/2*30.386 = 121.5 m = 22 (nearest integer) The frequncies correspond to 20,21 and 22 harmonics B) The resonating frequencis for the pipe closed at one end occurs at f = (2n+1)vs/4L with n = 0,1,2,3,.. f = (2n+1)334.245/4*30.386 f = (2n+1)*2.75 For f1 = 115.5 Hz, 115.5 = (2n+1)*2.75 n = 20.5 which is not possible For f2 = 121.0 Hz, 121 = (2n+1)*2.75 n = 21.5 which is not possible For f2 = 126.5 Hz, 126.5 = (2n+1)*2.75 n = 22.5 which is not possible So he can form resonance at neither of the three frequencies when the culvert is closed.

Related Questions

Navigate

• Browse (All)
• Browse N
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.