The type of wave, its properties such as wavelength, and the difference between the two media will determine what happens to most of the wave energy.
Common examples of absorption of wave energy:
Common examples of transmission of wave energy:
Wavelength has a big effect on how much of a wave's energy is reflected, absorbed or transmitted at an interface.
For example, a greenhouse uses these differences to raise the temperature inside.
Ultraviolet light has a very short wavelength, and this is mostly transmitted by the glass. The plants, soil and floor in the greenhouse mostly absorb ultraviolet, and their temperature rises.
The plants, soil and floor emit infrared light which has a longer wavelength that is mostly reflected by glass. The infrared cannot escape the greenhouse and reflects until it is re-absorbed by objects inside the greenhouse. The amount of energy in the greenhouse rises so its temperature increases.