Codec Output has "Wavy" Amplitude on Chirp Input
These could be ripples in the frequency response of an (ill-configured) anti-aliasing filter involved in the acquisition or restitution of the signal. It could be that the code running on your MCU and initializing the codec sends invalid configuration information to the codec; causing the codec to work with filters coefficients defined for a wrong sample rate. One thing worth noting is that the STM32F4 I2S master can only output an MCK at 256x FS. Check that your codec can work at 48kHz with such an MCK frequency.

Another thing I can think of is your software "pass-through" routine on the MCU dropping samples. Depending on the ratio between the block size and the period of the waveform, an amplitude modulation effect could appear.