Question

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

Answer 1

The wave's energy is absorbed. In mechanical waves (sound, etc) this is done through phonon exchange. With light it's a photon/phonon interaction. Think about it like this:
If the wave's energy is reflected then it does not become part of the internal energy also. If it did, the energy would be doubled: it is reflected back into where it came from and is also now inside the object as internal energy. This violates the conservation of energy.

Answer 2

The energy of the wave will be absorbed.

Step-by-step 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.