Energies and numbers of bound states in finite potential well

[71GA]

Hello I understand how to approach finite potential well. However i am disturbed by equation which describes number of states $N$ for a finite potential well ($d$ is a width of a well and $W_p$ is potential):
$$
N \approx \dfrac{\sqrt{2m W_p}d}{\hbar \pi}
$$
I am sure it has something to do with one of the constants $\mathcal L$ or $\mathcal K$ defined this way:
\begin{align}
\mathcal L &\equiv \sqrt{\tfrac{2mW}{\hbar^2}} & \mathcal{K}&\equiv \sqrt{ \tfrac{ 2m(W_p-W) }{ \hbar^2 }}
\end{align}
and the transcendental equations for ODD and EVEN solutions:
\begin{align}
&\frac{\mathcal K}{\mathcal L} = \tan \left(\mathcal L \tfrac{d}{2}\right) &&-\frac{\mathcal L}{\mathcal K} = \tan \left(\mathcal L \tfrac{d}{2}\right)\\
&\scriptsize{\text{transc. eq. - EVEN}} &&\scriptsize{\text{transc. eq. - ODD}}
\end{align}

QUESTION: Could anyoe tell me where does 1st equation come from? I mean $\tan(W)$ repeats every $\pi$, but if i insert $\mathcal L$ in transcendental equation i have $\tan(\sqrt{W})$! On what intervals does the latter repeat itself? Does this has something to do with it? It sure looks like it... Please help me to synthisize all this in my head.

 

[lightarrow]

QUESTION: Could anyoe tell me where does 1st equation come from? I mean $\tan(W)$ repeats every $\pi$, but if i insert $\mathcal L$ in transcendental equation i have $\tan(\sqrt{W})$! On what intervals does the latter repeat itself?

It's a strange question, I'm not sure to have understood it.
tan(x) is periodic with period = π; it means, for example, that tan(W) = 0 for: W = kπ where k is an integer; if it's tan(√(W)) = 0 it means that √(W) = kπ → W = k²π².

 

[71GA]

It's a strange question, I'm not sure to have understood it.
tan(x) is periodic with period = π; it means, for example, that tan(W) = 0 for: W = kπ where k is an integer; if it's tan(√(W)) = 0 it means that √(W) = kπ → W = k²π².

This did help. Thank you.