Why must there be a reflected wave at a boundary?

[aaaa202]

When my book derives equations for reflection between two media it simply says something like this:
Suppose a plane electromagnetic wave meets the boundary between two media. This gives rise to an incident and a reflected wave. But I am always puzzled. What is that states that there MUST be reflection? I mean a transmitted wave is also a solution to the wave equation so why can't a transmitted wave without a reflected wave be just as good as a solution. To sum up, it just all seems like guesswork to me.

 

[haruspex]

it's reasonable to assume there is both a transmitted and a reflected wave because these include the case that there is not, i.e. when the amplitude turns out to be zero.
The change in medium leads to a difference between the incident and transmitted waves. On various grounds, such as conservation of energy, you may be able to deduce that the reflected wave has nonzero amplitude.

 

[andrien]

normal component of D and tangential component of E must be continuous across the boundary which imposes some restriction on reflection and transmission.