Harnessing the power of high pressure water washer, a PVC pipe longer than the sidewalk and a shovel to do the job makes it go by quickly. Water under pressure cuts through soil very easily and the pipe directs it where to go. The job is messy and the water may kick back a stone or other hard object. Always wear eye protection when operating a high pressure water hose.
Mark the PVC pipe with a series of large dots 1 inch apart using a marking pen. The dots should run the length of the pipe, and be in a straight line from start to finish. These allow you to aim the pipe at the flag. Shovel under the sidewalk a few inches at the entry point. Place the pipe in the outside trench and push the front of it in the hole under the sidewalk. Run string through the two dots just before the sidewalk on the PVC pipe and pull it across the walk to the flag. The pipe should now align with the flag.
Set the nozzle at the smallest diameter and push the pressure hose and the nozzle into the PVC pipe until it reaches the end under the sidewalk. Turn the water on and let it blast a hole in the soil under the walk. Advance the pipe and the hose as the soil comes loose and flows back out. Stop ever six inches and double check the dots to ensure that they are still lined up with the opposite flag. Occasionally you may have to pull the hose and nozzle out along with the pipe if the sidewalk is very wide. Before pulling the pipe, mark it so that you know how much of it is under the walk.
Wash the inside of the pipe out and place it back into the trench with the dots lined up again. Push the pipe into the hole until it won't go any further. If it is at the mark where you pulled it out, then continue with water as before. If it doesn't go in quite all the way, hammer the end of the pipe with a maul to force it more into the hole. Continue to wash away the dirt until the pipe is able to fit completely under the side walk and sticking out of both ends by several inches.
Force the pipe as low in the hole as it will go, and shovel as much sand into each side of the hold as you can. Adjust the nozzle to a much reduced pressure and wash the sand along the top of the pipe into the under-sidewalk trench. Continue to do this to both sides until no more sand can fit into the trench.