Methods of calculating the roots of the dispersion equation for free oscillations of a gas bubble in a liquid

The problem of free radial oscillations of gas bubbles in a liquid is considered. The structure of the roots of the dispersion equation in the presence of heat transfer between the phases is studied in detail. It is shown that this equation has two complex-conjugate roots and an infinite number of real roots; all of the roots lie in the left complex half-plane, providing damping of radial oscillations. Approximate expressions for these roots are obtained.