I am familiar with non-dimensionalizing a differential equation; however I am not familiar with a method to non-dimensionalize a non-differential equation. I just need to arrive to a non-dimensionalized formula. I have attempted it as shown here.
The equation i am trying to non-dimensionalize is as follows.
$$U_{orb} = \frac{\pi H_{rms}}{T_{p}sinh(2kh))}$$
Thank you for the response.
What I am looking for is U_{orb} to be completely dimensionless. so initially what I did to non dimensionalize it was the following:
$$U_{orb} \times\dfrac{T_p}{H_{mo}}$$
the unit of Uorb is $$\dfrac{m}{s }$$
the unit of T_p is $$s$$
the unit of H_rms is $$m$$
the unit of h is $$m$$
the unit of L is $$m$$
$$ U_{orb} = \frac{\pi H_{rms}}{T_{p}sinh(2(\frac{2\pi}{L}))h))}\\$$
$$ H_{rms} = m \;\;[meters]\\$$ $$ T_{p} = s \;\; [seconds]\\$$ $$ h = m \;\;[meters]\\$$ $$ L = m \;\;[meters]\\$$
$$ U_{orb} = \frac{\pi [m]}{[s]sinh(2(\frac{2\pi}{\cancel{[m]}}))\cancel{[m]}))}\\$$ $$ U_{orb} = \frac{\pi [m]}{[s]sinh(4\pi)}\\$$
$$ U_{orb} \times \frac{T_{p}}{H}= \frac{\pi H_{rms}}{T_{p}sinh(4\pi)}\times \frac{T_{p}}{H}\\$$
$$ U_{orb} \times \frac{T_{p}}{H}= \frac{\pi \cancel{[m]}}{\cancel{[s]}sinh(4\pi)}\times \frac{\cancel{[s]}}{\cancel{[m]}}\\$$
$$ U_{orb} \times \frac{T_{p}}{H}= \frac{\pi}{sinh(4\pi)}\\ $$
This is how I was thinking of doing it.