Bash cannot kill itself?

[Clark J. Wang]

I have a bash script like this:

#!/bin/bash

trap 'echo killed by SIGALRM; exit 1' ALRM

function wait_kill()
{
    sleep 5
    kill -ALRM $$
}

wait_kill &

sleep 3600

### END OF THE SCRIPT ###

It does not work as I expected. The running script was not terminated after
5 seconds. So what's wrong here?

-Clark