The quality of that wav file is good enough that you could just stick it in one of those little analog record/playback chips, put a small amplifier on it's output, then play it back through a small horn speaker.
Hello, I'm new and this is my first reply on this board. I have been a member of another board on www.airraidsirens.com (Q&A Forums), and I built a small indoor model tornado siren, inspired by American Signal's T-128. I wanted a genuine electromechanical sound, so I used a 2x20 sec Voice Recording Module (electronics123.com). I replaced the microphone with a 100 Ohm resistor and soldered a 10 K resistor leg to one side. This formed a 40 dB pad to directly feed a .wav file from my PC sound card without overdriving it. (I had to play with PC's volume a lil' bit). I just used mini-grabbers on one of the RCA's I pulled from behind one of the PC speakers. I loaded 2 different siren .wav files from the same siren site. My radio had an ALARM OUT that eased triggering issues. I have yet to install a motor to rotate the speaker for realism, but I used a 1/4" headphone jack as a slipring when I am ready to. I have pics/video if anybody is interested.
I have never heard the siren on a police car go, "wha, wha, wha" like it was playing though a rotating Leslie speaker. Maybe you heard a police car that went too fast around a corner and was rolling over and over.