Mathematical Relationship to find number of antinodes/nodes in a standing wave

Question

So I have been learning about closed pipes (nodes at both ends), open pipes (antinodes at both ends) and open/closed pipes (node at one end and antinode on the other).

I have also learnt that for a closed pipe and an open pipe the length of the pipe is equivalent to half a wavelength ($L = \frac{\lambda}{2})$ and for a open/closed pipe the length is equivalent to a quarter wavelength ($L = \frac{\lambda}{4})$.

I was wondering if there is a mathematical relationship to calculate how many nodes and antinodes there are in each one of these 3 pipes (is there a similar mathematical formula for all 3 or is it the same), I tried doing some research and could not find any formula to calculate the number of nodes and antinodes in these 3 types of pipes (standing waves).

Answer 1

There is (in theory) no limit to the number of nodes $N$ or anti-nodes $A$, except that these numbers cannot differ by more than 1. Also, standing waves in the pipe of length $L$ can have other wavelengths $\lambda$ besides those you have stated :

• for a pipe closed at both ends $N=A+1$ and $\lambda=\frac{2L}{A}$

• for a pipe closed at one end $N=A$ and $\lambda=\frac{4L}{N}=\frac{4L}{A}$

• for a pipe open at both ends $A=N+1$ and $\lambda=\frac{2L}{N}$