Backing up to an Airport Timecapsule over non-Airport WiFi I recently upgraded some gear and I am no longer using my Airport as a wireless host. Instead, I now have another device acting as the primary server for DHCP, Router, and WiFi. 

The old Airport is connected via Ethernet to the new device (plugged into the LAN port on the Airport). I have turned the Wireless feature off on the Airport, and set it in Bridge mode.

After making these network changes, my Time Machine is no longer able to find the Time Capsule.

My question is, can Time Machine backup over a Wireless network to a wired Airport device?

Simplified diagram of how I'm trying to connect: 
[Mac] -- Wireless Network -- [Router] -- [Airport]
The answer is Yes, it can do that but needs to be set up first.

Here are detailed instructions.

In summary:

Connect your Mac to TC via Ethernet to set it up.

Now Disable the wireless functions of the Time Capsule and, Set your Time Capsule to Bridge Mode.--Update the configuration.

Next: use an Ethernet cable to connect one of your router’s LAN ports to your Time Capsule’s WAN port.