Make a tunneling pipe (sometimes referred to as a boring rod) with a high-pressure nozzle (a nozzle made of metal with a small opening works best). These items are available in home improvement stores or irrigation supply stores.
A tunneling pipe can be made of polyvinyl chloride plastic pipe (PVC) with a PVC male hose end adapter glued on one end and a female hose end adapter glued on the other end. The PVC pipe should be either 1/2 inch, 3/4 inch or 1 inch in diameter.
Dig a trench up to both sides of the sidewalk to approximately a foot in depth. You will start boring at about 6 to 8 inches below the sidewalk.
Put on eye protection. Cut the PVC pipe to 2 feet longer than the sidewalk.
Glue the male and female hose end adapters to the PVC pipe. Screw the high-pressured nozzle to the male hose end adapter, and the garden hose to the female hose end adapter (this PVC pipe will now be referred to as the tunneling pipe).
Tape, with duct tape, whatever it is you want under the sidewalk--such as electrical wiring encased in PVC pipe (cap each end so you won't get dirt in it; remove caps later)--to the tunneling pipe. If you are planning on using the PVC pipe for a sprinkler system, you can just leave the tunneling pipe under the sidewalk for that purpose.
If you are boring for a larger pipe, such as a drainpipe, just make the tunnel large enough to push a pipe through.
Lay the tunneling pipe in the trench with the nozzle ready to spray under the sidewalk. Turn on your garden hose.
The force of the water will start to hydraulically bore a hole under the sidewalk.
Push the hose nozzle farther under the sidewalk as the hole gets bigger. You may have to use the hose end to bore out more dirt by thrusting the nozzle in and out under the sidewalk.
Keep spraying the water and boring under the sidewalk until the tunneling pipe breaks through on the other side.
Remove the garden hose. Cut off the nozzle and the male and female adapters with PVC cutters or a hacksaw.