Assuming you've already setup WOL correctly on NIC, bios, router, given a static ip to the computer, tested it out and were initially successful doing WOL over WAN, but failed after a period of shut-down time, is because your router clears its arp table as stated above.
However, there is a KISS way to bypass that pesky cleared arp list that obviously doesn't contain your shut-down computer. No telnet, no hacking static entry in DHCP that will eventually lockup your router, no suspect firmware, no VPN, no dedicated computer required.
Although similar, this is not VPN. The caveat is you must have a printer, voip or any device with an IP/MAC address that is constantly on, then a simple inexpensive $15 hub after the router will work. NOTE, it must be a hub. A hub broadcasts to all devices connected to it, a switch will only send packets to the specific device its intended for and not the other connected devices. Set your router to port forward (Use port 9) to go to the constant active device (I have a cheap HP Combo Printer/Fax that is always on) and thus will always show up in the dynamic arp cache. Therefore the packet will not get dumped since it has someplace to go.
When you send a magic packet to the router's IP address, it will forward it through the hub to the printer, however the magic packet contains the MAC address of the computer I want woken up. According to the router, the target device for the magic packet is the printer, but the hub inside the LAN will also broadcast it to all devices, one being the hub-connected computer even though its not in the router's dynamic arp list.
Always a good idea to reserve the IP addresses for the constant on device and the computer to be woken up, if your router supports it.
