Explain open and closed end pipe with derivation
Answers
Answer:
Open and Closed Pipes
We have already covered the natural frequencies of strings and therefore tones that string instruments emit. What kinds of tones can be generated by wind instruments? Wind instruments consist of pipes, which are either open at both ends or closed at one end, in which standing waves can form. The frequencies of these standing waves will then be the frequencies of the instrument.
For open pipes, we can form standing waves that have antinodes at both ends. For closed pipes, we have to have a node at the closed end. As the picture shows, this has the consequence that the length, L, of the pipe determines the fundamental frequency and harmonics:
General solution (open pipe): The length of the pipe has to be an integer multiple, n, of half of the the wavelength, l:
L = n$\lambda$/2 ; n = 1, 2, 3, ....
The possible values for the wavelength, $\lambda$ and for the frequency, f, are then:
(The subscript n indicates that there are only certain values of the frequency and wavelength possible, one for each value of n) v is again the speed of sound in air.
General solution (closed pipe): Here the relationship between length of the pipe and wavelength is:
L = m$\lambda$/ 4 ; m = 1, 3, 5, ....
(We used a different letter, m, for the integer counting index. Please note that only odd-numbered values of m are possible!)
The possible values for the wavelength, $\lambda$ and for the frequency, f, are then: