To calculate the capacitor voltage requires the solution of a second order differential equation.
The "Ohm's Law" of inductance is V = L di/dt where V = the voltage across the inductance, i the current through it and t = time.
For capacitance, it is i = C dV/dt where i = current through capacitor, V = voltage across the capacitor and t = time.
The answer also depends upon whether the circuit is over damped, under damped or critically damped. This depends upon the L, C and R values.
It also assumes that there is enough charge in C1 to keep the voltage approximately constant for the time of interest.
If not, then the solution becomes even more complex.
The charge on C2 won't leak back into the buzzer coils as the diode is reverse biassed.
Len