Physicist: Stripped down to it’s most essential parts, a Tesla coil is a wire sticking out of the ground. The electricity in the primary coil is what’s doing the pushing, and the electricity in the secondary coil is what’s being pushed. Answer gravy: To get sparks to really fly you need very high voltage (up to several million volts) at a fairly exact frequency. The AC mains have a low frequency (60 Hz) while the secondary coil needs to be driven at a high frequency (~1,000,000 Hz). the fast part of the circuit takes over, and the slow part is essentially ignored until all the energy is exhausted by exciting the secondary coil (string vibrates and slows).