Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/173

where λ, is determined by the equation

$\lambda ^{2}+m^{2}={\frac {\rho _{1}}{k_{1}+{\tfrac {4}{3}}n}}$;

and a longitudinal refracted wave,

$e_{y}=E{\frac {\partial }{\partial y}}f(t+\lambda _{1}z+my)$; $e_{z}=E{\frac {\partial }{\partial z}}f(t+\lambda _{1}z+my)$,

where λ₁, is determined by

$\lambda _{1}^{2}+m^{2}={\frac {\rho _{2}}{k_{2}+{\tfrac {4}{3}}n}}$.

Substituting these values for the displacement in the boundary-conditions which have been already formulated, we obtain the equations which determine the intensities of the reflected and refracted waves; in particular, it appears that the amplitude of the reflected transverse wave is given by the equation

${\frac {A-B}{A+B}}={\frac {l_{1}\rho _{1}}{l\rho _{2}}}+{\frac {m^{2}}{l}}{\frac {(\rho _{1}-\rho _{2})^{2}}{\rho _{2}(\lambda \rho _{2}+\lambda _{1}\rho _{1})}}$.

Now if the elastic constants of the media are such that the velocities of propagation of the longitudinal waves are of the same order of magnitude as those of the transverse waves, the direction-cosines of the longitudinal reflected and refracted rays will in general have real values, and these rays will carry away some of the energy which is brought to the interface by the incident wave. Green avoided this difficulty by adopting Fresnel's suggestion that the resistance of the aether to compression may be very large in comparison with the resistance to distortion, as is actually the case with such substances as jelly and caoutchouc: in this case the longitudinal waves are degraded in much the same way as the transverse refracted ray is degraded when there is total reflexion, and so do not carry away energy. Making this supposition, so that k₁ and k₂ are very large, the quantities λ and λ₁, have the values m$\textstyle {\sqrt {-1}}$, and we have

${\frac {A-B}{A+B}}={\frac {l_{1}\rho _{1}}{l\rho _{2}}}+{\frac {m^{2}}{l}}{\frac {(\rho _{1}-\rho _{2})^{2}}{\rho _{2}(\rho _{1}+\rho _{2})}}{\sqrt {-1}}$.