Question

Problem 4 [8 pts] A long pipe, length L, is closed at both ends, and filled with a gas with speed of sound v. The pipe is excited in some fashion in order to produce standing waves. (a) Sketch the standing wave pattern for the four lowest frequencies supported by this pipe. Label the nodes and antinodes. (b) Make a table of the wavelengths and frequencies of the sound waves that are formed by these four excitations, in terms of L and v. (c) Find a general expression for the wavelength and frequency of the n-th harmonic (in terms of 14 and f1, the wavelength and frequency of the fundamental excitation). (d) Now consider the pipe to be closed at one end and open at the other. How do the above results change, for frequencies in particular? What is the new general expression relating frequency of the n-th harmonic to the fundamental frequency fi?

Answer 1

Length of the pipe - L. Velocity of sound = v. L=/2. fundamental. - L=d. 2ndet harmonie. L- L-21. and harmonies 4th 3and Ind hasmonic. Here - nodes. Antinodes , o 165 Fandamentali Warelength. frequency . Fundamental Si= 2L. 2nd of hasmonies, and hasmonies. I ha 3rd and h alo ? 12=La . Br - - = 24 andando harmonies. hg 7 2 2 2 2 2 2 2 3 1 for x = 3f, . = 31 Althood harmonies t = = = À FA = 48 of so, in genesal for noth hasmonics - & in 1 But, relocity is fixed - and, for nf. va di Ri - Anfr

Lax - L = 2 . Ist harmonies fundamental So, there, the fundamental frequency becomes half of the previous fundamental frequency And, the genesal expression of noth oudes frequency is - e (2n+ 1) Å