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# Stationary waves in a pipe

## Stationary waves in a pipe

Sound waves which are projected into one end of a hollow pipe are reflected off the other end, producing stationary sound waves between the two ends.

For a closed pipe (i.e. a pipe closed at one end), the closed end is a node and the open end an antinode.

For an open pipe (i.e. both ends are open), both ends are anti-nodes.

Stationary waves are produced in open pipes (even though there is no hard surface for the sound to reflect off of) as the incident sound wave gets reflected off the stationary air molecules at the other end of the pipe.

However, in this case there is no change in phase of the reflected wave (resulting in an anti-node created at the other end of the pipe).

Harmonics of closed and open pipes

## Characteristics of stationary waves in a pipe

When the air within a closed pipe is vibrating at its fundamental frequency, there is only a quarter wavelength within the pipe (formed from one node at the closed end and an anti-node at the open end).

For an open pipe, there would be half of a wavelength between the ends of the pipe (two anti-nodes at either end with a node in between).

There will be additional anti-nodes between the ends for the upper harmonics of the pipe (similar to stationary waves on a string). This restricts the number of harmonics for a closed pipe (i.e. no even numbered harmonics are allowed).

Harmonics of closed and open pipes

## Equations for stationary waves in a pipe

The length of the pipe $L$ and the wavelength of the stationary wave $\lambda$ for closed and open pipes are related by: \begin{gather*} L_{\text{closed}}=\frac{n\lambda}{4}\\ L_{\text{open}}=\frac{n\lambda}{2} \end{gather*} For a closed pipe, $n$ is an odd integer, while $n$ can take any integer value for an open pipe. This arises because of the limits (i.e. fixed node and anti-node) imposed on closed pipes, which does not allow certain harmonics to exist.

Harmonics of closed and open pipes