# Pipe Open or closed from a Standing Wave Equation

Here is the full question:

The part in the brackets are the answers. I am still terribly confused.

Lets say we are given an equation for a standing wave in a pipe: $$y(x,t) = A\sin\left(\frac{3\pi x}{L}\right)\sin(\omega t).$$ Is this enough information to know whether it is an open-open or open-closed pipe system? Obviously at $$x = 0$$, we have a node, so at a minimum there is one closed end. $$k = \frac{2\pi}{\lambda} = \frac{3\pi}{L} \rightarrow \lambda = \frac{2}{3}L$$.

I know that a one-sided open system $$\lambda_n = \frac{2L}{n}$$ which would work if $$n = 3$$. Hence it is an open-closed system? I am super confused about what $$L$$ is here. Is this the length of the tube?

• Where did you get that equation? Oct 16, 2021 at 18:47
• What is the displacement at x=L?
– nasu
Oct 16, 2021 at 20:28
• Why do you assume that $\frac{2\pi}{\lambda}=\frac{3\pi}{L}$? Oct 16, 2021 at 21:38
• As x has not been defined and assuming that the length of the pipe is L then it is either an open/open (antinode to antinode) or closed/closed (node to node) pipe system, Oct 17, 2021 at 8:42
• Yes this is what I thought. I will attached the full question Oct 17, 2021 at 14:30

The question in isolation is not well posed for a single answer; in fact, there are an infinite number of "correct" answers that could be given. Further information could be provided (or assumed by the questioner) that would lead to a particular solution.

Analysis

First, I will assume that $$y$$ denotes the particle displacement within the tube (and not the pressure). Then, as you comment, we may say that the tube is closed at $$x=0$$. However, the length may be any value of $$x$$ such that $$\sin(3\pi x/L)=0$$ if the tube is closed at the far end, or any value of $$x$$ such that $$\cos(3\pi x/L)=0$$ (anti-node) if the tube is open at the far end. Why don't we calculate all of these distances, just for fun!

Closed Far End

We are looking for $$\sin\left( \frac{3\pi x}{L} \right) = 0 \hspace{15mm}\Rightarrow\hspace{15mm} \frac{3\pi x}{L} = n\pi,$$ where $$n$$ is any integer. Thus, we may conclude that the tube may be any of the following lengths: $$\left\{ \frac{nL}{3}: n\in\mathbb{Z} \right\}.$$ We are only interested in positive values of the length, and so the lowest possible length of a closed-closed tube would be $$n=1$$, and so the tube length is $$L/3$$.

Open Far End

In this case we are looking for $$\cos\left( \frac{3\pi x}{L} \right) = 0 \hspace{15mm}\Rightarrow\hspace{15mm} \frac{3\pi x}{L} = \frac{2n-1}{2}\pi,$$ where $$n$$ is again any integer. Thus, we may conclude that the tube may be any of the following lengths: $$\left\{ \frac{2n-1}{6}L: n\in\mathbb{Z} \right\}.$$ Again, we are only interested in positive lengths, and so the smallest tube would be one where $$n=1$$, which yields a tube length of $$L/6$$. The answer they provided is the case where $$n=3$$.

One set of information they could have provided to force a specific answer is how many nodes and anti-nodes are present in the tube (3 and 3 in the case given by the answer). I am sure there are other options as well.