Questions on Waves: Answers to 3 Questions

[fluidistic]

My first question is: When a wave gets reflected over a surface, its phase gets delayed by $$\frac{\pi}{2}$$ rad or $$\pi$$ rad, I do not remember. How can I show this? I've access to Hecht's book on Optics but didn't find anything with the change of phase for wave's reflection. If someone could point me the exact page explaining this, I'd be glad. Any website explaining mathematically this is also welcome.

My second question is, if I send a plane wave over a perfect mirror. If the phase of the wave changes by $$-\pi rad$$ after 1 reflection, the wave will cancel itself?!
I'll have the sum of $$\vec E (x,t)=\vec E _0 (\omega t - \vec k \vec x)+ \vec E _0 (\omega t -\vec k \vec x - \pi)=\vec E _0 [\cos (\theta )+ \cos (\theta - \pi ) ]=0$$. It seems obviously wrong, what do I do wrong?

My third question is, how can I show that a spherical wave became a plane wave when $$r \to \infty$$? According to http://scienceworld.wolfram.com/physics/SphericalWave.html, a spherical wave can be written under the form $$\frac{\psi _0}{r} \cos (\omega t -kr + \phi)$$ while a plane wave under the form $$\psi _0 \cos (\omega t -\vec k \cdot \vec r + \phi)$$. So it seems they took the limit when $$r \to 1$$?!

 

[jtbell]

For the phase change (among other things):

Reflection at a dielectric boundary

 

[fluidistic]

For the phase change (among other things):

Reflection at a dielectric boundary

Thanks a lot, it really helped.
I've learned from it that there's a pi rad difference for the reflected wave if it reflexes over a medium with a greater refractive index than the incident one. Also, the refracted wave have no angle delay compared to the incident one. But the surprising thing to me is that the reflected wave doesn't have any phase change if it reflexes over a surface with a lower refractive index than the incident one.
Does that mean that internal reflexion occurring in water doesn't change the phase of the wave? The wave reflects over the surface water/air, but still in water? I'm a bit confused here. Does the wave traveling in water reflexes over water or over air, in case of an internal reflexion?