In this senior thesis, we explore a wave equation for the energy level spectrum of the electrons of hydrogen in terms of Webster's horn equation instead of the Bohr's radial orbiting model. We seek to derive the area function of Webster's horn equation by matching it to the Rydberg series for hydrogen's electrons. First we derive the dispersion relation for the Rydberg series. Then we utilize that dispersion relation to find a complex analytic impedance function. In our derivation we develop an inverse technique that takes the hydrogen spectrum and yields a closed-form area function of a horn. From this we find a fixed length horn with an closed-form area function to be incompatible with the Rydberg series. Then we start the investigation of a different inverse technique from D. C. Youla which states that given the reflectance and a few other parameters we can completely determine a horn an area function. This addition is still in progress.
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