How to calculate the amplitude increase by resonance?

In general you need to know both the strength of the driving force, its frequency, as well as the loss coefficient of the equations. A driven, damped simple harmonic oscillator in one dimension satisfies: $$frac{d^{2}z}{dt^{2}} + gamma frac{dz}{dt} + omega_{0}^{2} z = frac{F_{0}}{m}e^{jomega t} $$ Note that this describes the response of a simple system to a specific frequency of driving force. It makes intuitive sense that the solution would have a frequency equal to that of the driving force. However, it might have a phase factor. Thus, we assume a solution of the form $ z = A e^{j(omega t - phi)} $, and then we can solve for A and see what the response will be as a function of the driving force's frequency. Plugging in, we find: $$ (-omega^{2} + omega_{0}^{2} + jgamma omega)A e^{jomega t} = frac{F_{0}}{m} e^{jomega t} e^{jphi} $$ Since both sides are complex, we get two equations in two unknowns, $ A $ and $ phi $. After some algebra, we get that $$ A(omega) = frac{F_{0}/m}{((omega_{0}^{2} - omega^{2})^{2} + (gamma omega)^{2})^{1/2}} $$. That's the result you want. You asked about its dependence on the medium. The medium will determine the loss coefficient $ gamma $. As $ gamma $ gets larger, the denominator increases and the amplitude of the oscillation decreases.

This derivation is in Vibrations and Waves by A.P. French, along with a great deal more of information.