Example of transcendental equation in Physics

Question

Hello, everyone, 

I was asked by my numerical calculus teacher (undergraduate course) to solve for roots of a transcendental or a non-linear equation using some numerical method. The problem is that this equation has to be related with physics, and I don't know any equation of this kind in physics (unfortunately, I don't have a deep knowlege in physics yet); actually, the ones I know involve differential and integral calculus, so they would require more advanced methods that I don't know yet. So, could you guys provide me with an equation of such kind? You don't have to explain the physics behind it if you don't want to; I just need the equation and its name (or something that specifies it) - the theory behind it I can search myself. 

Just some points concerning this problem: the equation should not be a differential or an integral equation; and it should be (please) an easy one (especially, an one variable one), because I will have to apply this numerical method both with and without a computer (to compare the results), which means that a hard equation could lead to a problem that I could not answer without a computer.

Thank you for your attention to this matter.

Answer 1

I can add the simplest example: $$\sin(kx_1)=\sin(k(L-x_1))\qquad (1).$$ The first $\sin$ is a solution turning into zero at $x=0$, the second one turns into zero at $x=L$. This is a solution for an elastic guitare string fixed at $x=0$ and $x=L$. At $x=x_1$ we require continuety of the total solution; thus the equation (1). You must find the possible values of $k_n$ to satisfy this equation.

Comments

Great! Thank you very much, @VladimirKalitvianski ! :) (I'm the question author) 

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Note, the possible discrete values of $k$ will not depend on $x_1$ since I proposed the case of a uniform string. Still, it is interesting to make sure that it is so and compare with the analytical solutions $\sin(k_nx)$ where $k_n=\pi\cdot n/L$, $n=1, 2, 3,...$.

In  fact, the continuety equation should read as follows: $$\sin(kx_1)=\pm\sin(k(L-x_1))\qquad (1).$$

Answer 2

The expected spin using mean field theory in the Ising model. For spin $\sigma$ between negative one and one, external magnetic field $B$, spin coupling $J$, and temperature $T$, we have:

$$\langle \sigma \rangle =\tanh \left(\frac{ J \langle \sigma \rangle+B}{k_B T} \right)$$

If you plot this for different temperatures, you'll see exactly how the phase transition happens. It has one solution for high temperatures but three (with one solution unstable) for low temperatures.
 

Answer 3

It happens quite often in quantum mechanics, but I guess you are not that advanced in physics.  :-)

OK, in electricity I can come up with a diode (Shockley diode equation) connected to a generator.

The diode lets current i = i_0 (exp (V / V0) - 1).  (Note: In introductory courses you probably see that a diode lets i=0 for V < V0 and whatever current for V = V0.)    We also know that the voltage is V = v - ri (the generator has internal resistance r, so that the voltage is a bit smaller than nominal voltage v).  Thus i = (v - V) / r so that you get the equation:

     (v - V) = r i_0 (exp(V/V0) - 1).

Comments

Perfect! Thank you ever so much! :) (I'm the question author)