Question

A particular organ pipe can resonate at 264 Hz, 440 Hz, and 616 Hz, but not at any other frequencies in between. (a) Show why this is an open or a closed pipe. (b) What is the fundamental frequency of this pipe

Answer 1

(a) it is a closed pipe

(b) the fundamental frequency,
`f_0` is 88 Hz.

Step-by-step explanation:

Given overtones;

`f_1 = 264 \ Hz\\\\f_2 = 440 \ Hz\\\\f_3 = 616 \ Hz`

The overtones for open pipes are given as;

`f_1 = 2f_0\\\\f_2 = 3f_0\\\\f_3 = 4f_0\\\\f_4 = 5f_0`

The overtones for closed pipes are given as;

`f_1 = 3f_0\\\\f_2 = 5f_0\\\\f_3 = 7f_0\\\\f_4 = 9f_0`

Where;

`f_0` is the fundamental frequency

(a) From the given overtones;

test for open pipe;

`f_1 = 2f_0\\\\264 = 2f_0\\\\f_0 = 132 \ Hz\\\\then, f_2 \ should \ be \ equal \ to\ 440 Hz\\\\f_2 = 3f_0\\\\f_2 = 3 \ * \ 132 \ Hz = 396 \ Hz\\\\`

396 Hz ≠ 440 Hz (it is not open pipe)

Test for closed pipe;

`f_1=3f_0\\\\264 \ Hz = 3f_0\\\\88 \ Hz = f_0\\\\then, f_2 \ should \ be \ equal \ to \ 440 \ Hz\\\\f_2 = 5f_0\\\\f_2 = 5 \ * \ 88 \ Hz = 440 \ Hz\\\\f_3= 7 \ * \ 88 \ Hz = 616 \ Hz\\\\Thus, \ the \ pipe \ is \ closed`

(b) the fundamental frequency,
`f_0` is 88 Hz.