  Broadband microwave spectroscopy can probe material properties in wide
spectral and temperature ranges, including superconductors at cryogenic
temperatures. The quality of such measurements crucially depends on the
calibration, which also removes from the obtained spectra signatures of
standing waves. Here we consider low-temperature reflection measurements in
Corbino geometry, and we show that the non-linear response of superconducting
samples close to the critical temperature can lead to strong signatures of
standing waves even in a well-calibrated Corbino spectrometer. We demonstrate
our findings with microwave measurements as a function of frequency and
temperature for a variety of superconducting samples and for different length
of the microwave transmission line. Finally we show that such non-linear
effects extend beyond the case of superconductors by probing a VO$_2$ thin film
at the insulator-metal transition.
