In a cyclotron there is an alternating voltage $V(t) = V_0 \sin(\omega t)$ applied between the Ds so the polarity looks like this:

If we are accelerating (for example) electrons then we need the electron to be in the correct position as the voltage alternates:

So the time taken for the electron to do one complete circuit must be the same as the time taken for the applied alternating voltage to complete one cycle. This is the resonance condition referred to in the book.

The reason an alternating voltage is used instead of a static is purely practical. It is very hard to create a very intense static electric field without getting breakdowns due to arcing. An intense alternating field is much easier to generate safely.

In a cyclotron there is an alternating voltage $V(t) = V_0 \sin(\omega t)$ applied between the Ds so the polarity looks like this:

If we are accelerating (for example) electrons then we need the electron to be in the correct position as the voltage alternates:

So the time taken for the electron to do one complete circuit must be the same as the time taken for the applied alternating voltage to complete one cycle. This is the resonance condition referred to in the book.