Question

The fundamental frequency of a pipe that is open at both ends is 594 Hz .
How long is this pipe?

If one end is now closed, find the wavelength of the newfundamental.

If one end is now closed, find the frequency of the newfundamental.

Answer 1

Concepts and reason

The concept used to solve this problem is open and closed end pipe harmonics.

Use the fundamental frequency of a pipe that is open at both ends to find the length of the pipe.

Then use the length of pipe to find the wavelength of fundamental and the fundamental frequency of a one end closed pipe.

Fundamentals

Expression for the fundamental frequency of a pipe that is open at both ends is,

Here, is the fundamental frequency of a pipe that is open at both ends, is the velocity of sound, and is the length of the pipe.

Expression for the fundamental frequency of a pipe that is closed at one end is,

Here, is the fundamental frequency of a pipe that is open at both ends, is the velocity of sound, and is the length of the pipe.

Expression for the wavelength of fundamental of a one end closed pipe is,

Here, is the wavelength of fundamental.

(1)

Expression for the fundamental frequency of a pipe that is open at both ends is,

Rearrange above expression for the length of the pipe.

Substitute for and for .

Therefore, the length of the pipe is .

(2)

Expression for the wavelength of fundamental of a one end closed pipe is,

Rearrange the above expression for .

$1=4L $

Substitute for .

Therefore, the wavelength of fundamental of a one end closed pipe is .

(3)

Expression for the fundamental frequency of a pipe that is closed at one end is,

Substitute for and for and for .

Therefore, the fundamental frequency of a pipe that is closed at one end is .

Ans: Part 1

The length of the pipe is .

Part 2

The wavelength of fundamental of a one end closed pipe is .

Part 3

The fundamental frequency of a pipe that is closed at one end is .