Error Propagation in Transcendental Equation

[jst6981]

Hey guys,

I'm in a class where we're learning about waveguides, and without going into too much depth, we often solve an equation

$$ \tan{(\kappa (\frac{a}{2}))} = \frac{\gamma}{\kappa} $$

for $\kappa$ numerically since there isn't an analytic solution for $\kappa$. I'm doing a project where $a$ has an error $\Delta a$, and I want to be able to solve this in a way so that I have a $\kappa$ written $\kappa \pm \Delta \kappa$ so that I can propagate the error in $\kappa$ due to the error in $a$ throughout the rest of my calculations for the project.

If anybody has a nice way I can do this, please let me know. I appreciate help in advance!

 

[mfb]

Vary a by Δa up and down, solve for $\kappa$ in both cases. If the resulting deviations of $\kappa$ are reasonably symmetric, use that as uncertainty. Otherwise it might need more sophisticated approaches.

 

[jst6981]

I had considered that, each function is roughly linear when they intersect as below.

I think it may work

 

[haruspex]

If you write that k is a function of a and differentiate your equation wrt a, you can eliminate the trig and get a differential equation for k. Manipulate that to get k' as a function of k and a.

 

[ChrisVer]

Do you want an analytical solution or you are fine with using numerical solutions?
In the latter you can generate some toys, let's say 10,000...
In each toy, you randomly sample $\alpha$ from a gaussian (for example) with mean $\alpha$ and standard deviation $\Delta \alpha$, so that you get a set $\{\alpha_i\} , i=1,~2,~...,~10000$ measurements...
You can get the $\{k_i\}$ set from your equation.
See how it's distributed and try to find the 68% central coverage (aka determine $\Delta k_{\pm}$ where plus/minus means the up/down uncertainty - that means the range [nominal-down, nominal] contains the 34% of your toys and the range [nominal,nominal+up] contains the other 34%)...