I run this way in my Living Room where I dont have a wired connection. It takes a bit of wifi fu to get it working is my experience. I have set it up both with the wire being dhcp and static and my wifi is static wpa-psk. So it will work either way. When I first started testing this with both volumio and rune, I couldnt get it working without using an ethernet loopback, but it may work without one.
There are various utilities to use....I see one called wifi-menu that may be useful to you and walk you through the setup.....the main one is netctl. The config files for netctl are in /etc/netctl/
I had to use something like this IIRC:
netctl disable wlan0
netctl enable wlan0
netctl start wlan0
at which point it gave me an error some of the time....

however doing this and rebooting took care of it