This would really be more valid on Geekslutz, but we'll do what we can!
First of all - you need to find a driver that's suitable. It has to be high enough power to cause the physical thing to move. A speaker is a viable option but, as I found out with my own trials, the speaker is going to give you an awful lot of direct sound - which is a PITA to deal with.
A DC motor, however, or other mechanicle transducer will allow you to move the spring without moving air. There's a couple of youtube videos demonstrating construction and most stores should sell a DC motor - I would emphasis that this can NOT be a Stepper Motor as they take an input from a PWM source.
Once you know the load from the motor on either side, it's time to design an amplifier around that - though we can expect it to be about 30-100 ohms - a pair of LM4562NA in parallel will comfortably drive the motor - or headphone drivers down to about 50 ohms at around 3-4 volts (from what I've read, you'll need a supply rail of +/- 6Volts, though you can do a split supply design if needs be. If you want to drive it harder, or you have a lower resistance still, you'll have to think about increasing your power rails or setting up another pair of op amps - both can be messy, so breadboard the thing first.
On the output side you've a few options - another motor or speaker, a pickup or a piezo. A transducer will be the most efficent, though the LF content will be very quiet - and a simple High Z buffer and amp should do the trick - a TL072 will give you what you need to comfortably drive a Line In.
As an aside, plenty of Amp Companies sell replacement spring reverbs - so if you're willing to spend about 50 quid on one of those you simply have to find a way to drive it (which I've described above). If time is a factor I'd strongly recommend doing that.
Why don't you just knock it off with them negative waves?