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

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).

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 $\lambda=\frac{2L}{N}$
• So there is no relationship that exists between (for example) the wavelength $\lambda = \frac{4L}{2n}$ and $N=A+1$ for a pipe closed at both ends that we can relate to find the number of nodes/antinodes. For example if L=1.00 and n=1 so the resulting wavelength is $2.00m$ , we can't use this value to find the number of nodes or antinodes? Jul 19, 2016 at 20:52