If the energy of the wave becomes part of the internal energy of the object, then the wave's energy is 

The energy of the wave will be absorbed.  

Explanation:

The energy of the wave will be absorbed.   This is achieved by phonon exchange in mechanical waves.  For light, it's an association between photons.   if the energy of the wave is transmitted/reflected, it would not become part of the existing internal energy.

When it did, the energy will be doubled: it is transmitted back to original transmitting point and observed as internal energy within the object. This violet the basic principle of  energy conservation.

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