\(\cfrac{\psi_{n\,i}}{\psi_{n\,r}}=\cfrac{sin\,\theta}{sin\,\alpha}=n\)
that this component of \(\psi\), \(\psi_n\) has been reduced by a factor that is the refractive index of the material, \(n\). A simultaneous increase in the slide component \(\psi_s=\psi cos\,\alpha\) along the surface of the intercepting medium, ensure that the total \(\psi\) is conserved (ie. a constant) pass after the inter-surface.
\(sin\theta =nsin\alpha \)
\( cos^{ 2 }\theta =1-n^{ 2 }sin^{ 2 }\alpha \)
\( sin^{ 2 }\alpha =\cfrac { sin^{ 2 }\theta }{ n^{ 2 } } \)
\( cos^{ 2 }\alpha =1-\cfrac { sin^{ 2 }\theta }{ n^{ 2 } }\)
\(\cfrac { cos^{ 2 }\theta }{ cos^{ 2 }\alpha } =\cfrac { n^{ 2 }\left( 1-sin^{ 2 }\theta \right) }{ n^{ 2 }-sin^{ 2 }\theta } \)
\(\cfrac{\psi_{s\,i}}{\psi_{s\,r}}=\sqrt{\cfrac { 1-sin^{ 2 }\theta }{ 1-\cfrac { sin^{ 2 }\theta }{ n^{ 2 } } } }\)
The increase in the slide component \(\psi_s\), depends on the incident angle \(\theta\) and \(n\).
Graphically, \(\psi\) is squashed on impact. \(\psi\) is shorten perpendicular to the surface of the medium and is lengthened along the surface of the medium.
Total \(\psi\) remains constant.
A changing refractive index \(n\) with different wavelength of light suggests that different colored \(\psi\) are deformed differently. \(\psi\) is a standing wave centered at the particle. It has a wavelength, \(\lambda_\psi\) associated with it. At the fundamental oscillation, where there is one complete wave around a radius of \(a_\psi\),
\(2\pi a_\psi=\lambda_\psi\)
As such, the physical size of \(\psi\),
\(a_\psi=\cfrac{1}{2\pi}\lambda_\psi\)
is related to the wavelength of \(\psi\) and consequently the frequency of \(\psi\).
Putting these facts together, they suggest that the deformation in \(\psi\) depends on the initial size of \(\psi\) before the impact. So much so, refractive index \(n\), depends on the size of \(\psi\) and so the wavelength of \(\psi\). This way, we relate the wavelength of \(\psi\) with the wavelength of light as perceived, directly.
Considering both Cauchy's and Sellmeier equations with the experimental plots of refractive index vs wavelength in general, such deformations in \(\psi\) decrease with increasing \(a_\psi\).
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