  Use of a superconducting traveling wave accelerating (STWA) structure with a
small phase advance per cell rather than a standing wave structure may provide
a significant increase of the accelerating gradient in the ILC linac. For the
same surface electric and magnetic fields the STWA achieves an accelerating
gradient 1.2 larger than TESLA-like standing wave cavities. The STWA allows
also longer acceleration cavities, reducing the number of gaps between them.
However, the STWA structure requires a SC feedback waveguide to return the few
hundreds of MW of circulating RF power from the structure output to the
structure input. A test single-cell cavity with feedback was designed,
manufactured and successfully tested demonstrating the possibility of a proper
processing to achieve a high accelerating gradient. These results open way to
take the next step of the TW SC cavity development: to build and test a
traveling-wave three-cell cavity with a feedback waveguide. The latest results
of the single-cell cavity tests are discussed as well as the design of the test
3-cell TW cavity.

Accelerator Physics