I am attempting to build a trumpet, and need to calculate the lengths of the piping for it. I have attempted to find information on this online, but have been unable to find something that can help me to understand it. I am asking this question on this platform because the results are often more personalised to the specifi question being asked. My question is as such:
When x is the frequency that must be produced, what would its relationship to the length l of a double open ended tube be?
Thanks in advance for the help! (ps - if my question is flawed please tell me)
Although your question amounts more to physics, the main point of your interest within indeed is the geometrical aspect thereof.
Acoustics deals with longitudinal waves, i.e. the direction of amplitude is along the tubes axis. The ground state of a stationare longitudinal wave within a two-sided open tube has an oppositionally oriented amplitudinal maximum at either end and exactly one amplitudinal node (fixed point) at the center. (Higher states will have for the end conditions likewise maxima, but would add further nodes in between.)
Thus if $L$ is the length of your tube and $\lambda_0$ describes the wave length of the ground state, then you will have $L=\lambda_0/2$. More generally you get for the wave lengths $\lambda_n$ of the higher states accordingly $L=\frac{\lambda_n(n+1)}2$.
Further you'll have $c=\lambda f$ for the relation between the wave length $\lambda$, the frequency $f$ and the velocity (here: the velocity of sound within the according medium, i.e. air of according temperature) $c$.
Thence you'll simply get $$L=\frac{c(n+1)}{2f_n}$$ for the various harmonics $(n>0)$ and the fundamental frequency $(n=0)$ respectively.
--- rk