Heat capacity ratio and sound velocities for a Bose gas in multi-slabs

The heat capacity ratio ($c_p/c_V$) and adiabatic sound velocity ($c_S$) are reported for an inhomogeneous Bose gas with mass distribution arranged in multi-slabs. The mass distribution is generated by applying a Kronig-Penney potential in one direction on a 3D Bose gas while the bosons are free to move in the other two directions. Since the isobaric specific heat is indeterminate for $T \leq T_c$, we show the heat capacity ratio as a function of $T$ only for $T > T_c$, for six values of potential spatial period $a+b$: it diverges at $T_c$, after a threshold temperature we recover the 3D classical value 5/3, and for intermediate temperatures its behavior shows a dimensional ``crossover" from 3D to 2D. The average adiabatic sound velocity has two main features: above $T_c$ it is proportional to $T^{1/2}$, as a classical gas, while for temperatures below $T_c$ it goes as $T^{5/4}$ with a small deviation from this value which depends on the lattice parameter.