A guitar string of length Ls=0.65m vibrates in its second overtone, producing a frequency of fs=480Hz. The linear mass density of the string is μ=1.2×10−3kgm−1. Neglect air loading on the string.
An organ pipe containing air is positioned nearby. The pipe is open at one end and closed at the other, and its length can be varied. The organist wishes to adjust the pipe so that one of its harmonics exactly matches the 480Hz tone from the string. Assume the speed of sound in the room is initially v0=343ms−1.
(a) Determine the speed of the transverse wave on the guitar string at the stated mode of vibration.
(b) Calculate the length Lp the pipe must have to resonate at 480Hz. Identify the harmonic number involved for the pipe.
(c) During a performance, the room temperature rises so that the speed of sound increases by 2.0%. Assuming the pipe length cannot be changed, compute the new resonance frequency of the pipe and the percentage by which it differs from 480Hz.
(d) To retune the guitar string to the new pitch found in part (c), by what percentage must the string tension be adjusted? Treat μ as constant and assume small changes so that linear approximations are valid.