We have calculated the first and second sound modes of a dilute interacting Bose gas in a spherical trap for temperatures
(0.6<T/Tc<1.2) and for systems with
104 to
108 particles. The second sound modes (which exist only below
Tc) generally have a stronger temperature dependence than the first sound modes. The puzzling temperature variations of the sound modes near
Tc recently observed at JILA in systems with
103 particles match surprisingly well with those of the first and second sound modes of much larger systems.
This set of four lectures reviews various aspects of the theory of a dilute low-temperature trapped Bose gas, starting with (I) a review of the Bogoliubov description of the elementary excitations in a uniform system. The treatment is then generalized (II) to include the new physical effects of a confining harmonic trap potential on the condensate and its normal modes. An equivalent hydrodynamic description (III) focuses directly on the density and velocity fluctuations. The physics of vortices (IV) in an incompressible fluid is summarized and extended to the case of a trapped Bose condensate.