Why wavefunction becomes exponentially smaller during quantum tunneling?

I am interested in quantum tunneling and I am wondering why the wavefunction of a particle would becomes smaller so that there is a slight possibility of finding it at the other side of a big energy barrier? Is there any interaction otherwise how can the wavefunction knows there is a barrier?

Answer

Wavefunctions are solutions of quantum mechanical differential equations, with given boundary conditions for the problem at hand, i.e. tunneling:

Is there any interaction otherwise how can the wavefunction knows there is a barrier?

The boundary condition of a barrier defines the wavefunction by construction. It has been found that experiments validate this model and its predictions.