Pipe Open or closed from a Standing Wave Equation

Question

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?

Answer 1

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.

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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.