What property of a wave remains unchanged when a wave enters a different medium?

Frequency.

Explanation:  

A wave has the  basic properties like wavelength, frequency and velocity.

A wave that bounces back in the same plane and in the same medium is said to be reflected. No change happens in its velocity or wavelength or frequency.

If the wave enters a different medium it is said to have refracted ( bending of the wave). The interaction of the wave with the particles of the medium will cause the wavelength and speed to change,  but frequency  remains unchanged.

This is because frequency is the number of vibrations or cycles per second, undergone by the wave. So it is independent of the medium changes. It only depends on the source that is generating the waves.

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explanation: