Now let's see what is going on, when a particle passes inside the box WALL. Then E is less than U and k being square root from now negative number becomes IMAGINARY, thus making ikx REAL number, and we have pure exponential dectrease (or increase depending where particle originally was - inside or outside)of probability inside wall. Because exponent never drops to zero, thus there is always a chance to find a particle outside regardless how thick or how strong are the walls. That is the nature and explanation of quantum tunnelling - electrons, protons, alpha particles, atoms and molecules can penetrate via variety of microscopic walls and result in radioactivity, various decay processes, in diffusion, in chemical reactions, etc.

Now, outside of the box particle energy E is greater than potential energy U again, so k is real, ikx is imaginary and sum of exponents results in real harmonic function (sin(kx) or cos (kx)), thus making particle a wave again - as imaginary numbers require.