Reflection and refraction of Bose-Einstein-condensate excitations

We investigate transmission and reflection of Bose-Einstein-condensate excitations in the low-energy limit across a potential barrier separating two condensates with different densities. The Bogoliubov excitation in the low-energy limit has an incident angle where the perfect transmission occurs. This condition corresponds to Brewster's law for the electromagnetic wave. The total internal reflection of the Bogoliubov excitation is found to occur at a large incident angle in the low-energy limit. The anomalous tunneling named by Kagan can be understood in terms of the impedance matching. In the case of the normal incidence, comparison with the results in Tomonaga-Luttinger liquids is made.