How do Xover works?

One capacitor, one inductor and one resistor indicates a first order crossover. The resistor is probably a 4-ohm because (maybe) one driver is probably 4-ohm and the other 8-ohm. The resistor is in series with the 4-ohm driver to give a nominal 8-ohm speaker impedance.

For first order filters, the 4.7uf cap lets through frequencies above 4kHz. The formula is

F = 1/{2*pi*C*R) where R=8 ohms

The inductor will let through frequencies under 4kHz

L = F/(2*pi*R) = 0.3mh at F=4kHz, R=8ohms

All of the parts here are critical. The function of the crossover is to roll off the frequencies, not to abruptly cut them. The order (capacitor quantity) determines how fast the frequencies roll off.

The way they work is: the capacitor is charged at a certain rate from the signal. If the rate of charge from the signal is greater than the capacitor's ability to absorb the charge, the excess frequencies are passed through the capacitor. As the frequency of charge decreases, the capacitor's ability to store it is greater than the charge rate - so these frequencies do not pass. This frequency is the F in the above equation.

The inductor, which is a coil, creates a magnetic field when a frequency charge is passed through. As the voltage rate of change (frequency) is increased, the inductor creates a proportionately stronger magnetic field which increases the resistance. As the frequency decreases, the magnetic field is created less frequently and the inductor settles down and lets current flow easier.

And the math is very simple especially if you know algebra:

The capacitive reactance Xr = 1/2*pi*f*C [Ohms] where pi is circular constant f is freequency(Hz) and C is capacitance (Farad)

The inductive reactance Xl = 2*pi*f*L [Ohms] where L is inductance H.

The crossover freequency can be found identically to the resonance from equation when you equalize both inductive and capacitive reactances.

After some algebra manipulation you get f = 1/2*pi*sqrt(LC) [Hz] where sqrt() is a square root function.

You should not mistake this formula from the parallel connection of capacitor and inductor which is the basic circuit of radio that oscilates at particular resonance freequency.

Normally the basic crossover consists of:
the tweeter shunt inductor and
the woofer shunt capacitor
which values are calculated to define a crossover freequency.

A value of a resistor is matched to define a total speaker DC resistance. Please note that both of reactive components should not create a resonance at audiable freequencies or low harmonics with drivers(now use the resonance formula to define an approximate valuable crossover point).

This formula is valid when you have an ideal resistor(i.e. non-reactive) along in the circuit and does not include the reactance of wires as well.

Now, after calculations the next step is tuuuning! Oscilloscope is used to set up the minimal harmonic interfearance and the maximum close-up to the theoretical calculations(yeah there might be adjustable elements used that than would be replaced with fixed ones).

Ever though of electronic crossovers:-)?

Not delving too deep into electronic theory... The white ceramic block is a resistor. The capacitors and inductors are reactive to alternating voltage/current. The resistor is resistive to voltage. Reactive inductors / capacitors resist AC nonlinearly. I.E. whereas a resistor resists voltages equally no matter what the frequency, a reactive element (capictor, inductor) resists altenating current at selective frequencies based on it's value, and how it's used in a circuit. Combinations of reactive and resistive elements in circuits can allow certain frequencies to pass through to the speaker, while blocking others. This allows each driver to function within it's chosen bandwith, and integrate with the other drivers in the speaker system.

Specifics:
A Resistor in series is linearly resistive to DC and AC

A capacitor in series is an open circuit to DC, and reactive to low frequency AC.

An Inductor in series is conductive to DC, but reactive to high frequency AC

Within your ProAc's,they are all critical components. Each as critical as the next.