Through experimentation, I've found that the audio works fine (i.e., no static) if I remove all USB connections from the Thunderbolt Display.
If I plug my Apple keyboard into a USB port on the Thunderbolt Display, and I let the audio play for a few minutes, the audio begins to degrade with static.
As a short-term workaround, I've removed all USB connections from the Thunderbolt Display. When I connect the MacBook Air to the display, I plug a USB hub into the MacBook Air (not into the Thunderbolt Display). I plug the Apple keyboard (etc.) into the USB hub. With this configuration, the audio works fine.
Of course, I do want to be able to use the USB ports on the Thunderbolt Display, so this is a short-term workaround at best.
Better (and less fiddly and time-consuming temporary fix) is to leave Settings open on the Sound widget and simply cmb-tab to it and quickly press up then down on the keyboard (thus selectling Internal Speakers then Display Audio in quick succession). Takes seconds and your hands don't even have to leave the keyboard.