An experiment has been performed to measure the quadratic Doppler shift, ~(1 - ν2/c2)-½, predicted by the special theory of relativity. A moving beam of radiating hydrogen atoms with velocities ranging up to 2.8×108cm/sec has been viewed from the incoming and outgoing directions simultaneously. Averaging wavelength measurements of a particular spectral line for the two observations gives a measurement of the quadratic shift. The number (½) in the theoretically predicted exponent can be compared to the experimental results. The latter gives the value of this exponent to be 0.498±0.025. The predominant factor leading to the experimental uncertainty is the width of the spectral lines involved in the measurement.
HIRSCH I. MANDELBERG and LOUIS WITTEN, "Experimental Verification of the Relativistic Doppler Effect," J. Opt. Soc. Am. 52, 529-535 (1962)