I am looking for an outdoor motion detector that when an object about 4 feet away will sound if it does not move within 4 seconds. Does one exist?
Video motion detection is often used for applications like this. Are you specifically looking for an infrared detector or are you considering cameras? Commercial surveillance systems often have a loitering detection feature.
Our Alarm.com cameras with video analytics can do this. You can draw an area of interest on the picture and create a rule that if a person walks into the area of interest and remains there for 4 seconds you would get alerted with a video clip. It can also trigger Z-Wave lights to come on. But you could plug a buzzer or sounder into a Z-Wave switch and it would turn that on just like a light.