Would a water wave leak just like an electron with a barrier?
Assuming a watertight barrier, no not at all.
I'm not saying water is leaking through.
Say a 'water tight' barrier is surrounded by water on both sides. With a wave hitting one side.
Would there be a wave on the other side like in quantum tunneling.
If the barrier is perfectly rigid, then no water (pressure) waves will pass through no matter the thickness. If the barrier is not rigid, say it's made out of a thin film of flexible plastic, then of course the water wave can pass through. But this has nothing to do with quantum mechanics.
A better analogy for quantum-mechanical tunneling comes from classical E+M. When an EM wave reflects off of the boundary between two media at an angle greater than the critical angle, an evanescent wave forms in the second medium, and if the second medium is thin enough, the evanescent wave can actually transmit some energy from the EM wave through the second medium. The phenomenon is called "evanescent wave coupling." See http://en.wikipedia.org/wiki/Total_internal_reflection and http://en.wikipedia.org/wiki/Evanescent_wave
FOr a large object like a water wave it would require an impossible coincidence of every single particle in it tunneling at eactly the right time and then the ones behind them doing the same ... and doing do with all the protons and electrons remaining together. So improbably that it is impossible. It is just as impossible for even one atom with an electron AND a proton to do the same even if possible for indepenedent electrons to do it.
Superfluids (like liquid helium) can leak through a barrier. Just put a wall in a vessel with a superfluid then push it onto it. Some fluid will "climb" on the wall and fall on the other side, no matter how high the wall is, however this effect will vanish exponentialy with the height of the wall. This is an interesting phenomenon when a quantum-mechanical effect displays directly in a macroscopic world.
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