waves - What is the cause of reflection?

A wave that meets the boundary of two substances with different propagating velocities is partially reflected.

Does anybody know the reason behind reflection?

Answer

The question is general, so one can start with the simplest waves, the ones that are easiest to picture: transverse waves on a string.

First one needs to consider reflection at the ends of strings. There are two cases: where the end is fixed, or where the end is free. Those are the extreme ends of boundary conditions: where the string on the right is infinitely heavy or where the string on the right is massless.

It is often is a bit of math to solve differential equations with boundary conditions, but at these types of ends it can be done with picture-type arguments, and the principle of superposition. In both of these cases, one can add a mirror wave, a wave traveling in the opposite direction, like in the animation below:

The wave is on the blue string. In the first case, the string end is fixed at the black dot. One could imagine the string continuing behind this point (pictured in red), and a wave coming from the other side. When the distance is the same and the phase is inverted, the dot remains stationary in the superposition of the two waves traveling through eachother. So this satisfies the wave equation with the boundary condition of a fixed end.

Similar for a free end, but now the reflected wave has the same phase.

Now to the question. Imagine ropes with different masses connected at the dot. Then part of the wave continues, but there will also be reflection. The phase depends on whether the wave velocity to the right is higher or lower, the amplitude depends on how large the difference is. This is very similar to water waves (wave speed changes at a change of the depth of the water), or sound, or light. Or even the quantum mechanical problem of a particle transmitted at a potential step.

See also this page (with animations): http://www.acs.psu.edu/drussell/Demos/reflect/reflect.html

More mathematically: http://www.people.fas.harvard.edu/~djmorin/waves/transverse.pdf

In 4.2 there, also the boundary condition is explained. It is obvious that the displacement of the string must be continuous at the boundary. But also the slope must be continuous.