It is the 1929 Auburn Cabin Speedster, a boat-tail coupe. For a detailed report, please link to "The Sensational Auburn 1929 Cabin Speedster" by Michael Lamm.
Boat-tail roadsters, where in plan view a car's sides converged to a point at the rear, thrived in the late 1920s into the mid-1930s. "Thrived" is actually likely too strong a word. Not many were built, but that was the era when they were built.
Regular roadsters -- a two-seater form of convertible coupé -- did not have tapered bodies. And their conventionally shaped rear ends could feature a rumble seat, an impossibility for a boat-tail.
What makes the Cabin Speedster interesting and unusual is that while its body had a boat-tail taper, the body was an enclosed coupé -- not any kind of convertible or drop-head.
The original car was destroyed along with many others when a huge tent for a Los Angeles car show caught fire. But the car was remembered, and eventually two replicas were built, as mentioned by Lamb. I came across one of these recently while visiting the excellent Auburn Cord Duesenberg Museum in Auburn, Indiana.
Below are some photos of the original car along with a few of mine taken at the museum.
Side view photo taken when the car was new.
Front quarter view showing that the headlights had not yet been installed. This image and the next two are from Lamm's article.
Rear quarter view. Note the tiny, impractical aft window.
This photo was taken at the Chicago car show. Woodlites are installed, but the car never received bumpers.
Now for my photos. Here is the wood body framing of the kind used before all-metal bodies became common in mid-1930s USA.
Front quarter view of the replica car built on Auburn components.
The Cabin Speedster was quite narrow, yielding comparatively small frontal area -- a theoretical aerodynamic advantage. So far as I know, the original car was never tested at high speed.
Interior view. The cane seats are an interesting, light-weight touch.
This shows the sculpting around the drastically raked dual-pane windshield.