There he received bachelor's and master's degrees in mathematics from the State College of Washington, now Washington State University, and in 1928 received his Ph.D. from Columbia University for his dissertation On Certain Properties of the Metrical and Generalized Metrical Groups in Linear Spaces of n Dimension.
In 1933 he and Sally P. Mead began analysis of waveguide propagation discovered analytically by their colleague George C. Southworth.
Schelkunoff appears to have been the first to notice the important practical consequences of the fact that attenuation in the TE01 mode decays inversely with the 3⁄2 power of the frequency.
During his 35 year career at Bell Labs, Schelkunoff's research included radar, electromagnetic wave propagation in the atmosphere and in microwave guides, short-wave radio, broad-band antennas, and grounding.
He retired from Columbia U. in 1965, and served as a consultant on magnetrons for the United States Naval Station at San Diego.