Why is frequency a fundamental property of waves? We are taught that although wavelength can change from one medium to other, frequency of a wave doesn't whenever Velocity varies in different media. 
But why at a deep level, is frequency so fundamental? Why can't both frequency and wavelength change? Or why isn't wavelength fundamental? Or velocity fundamental?
 A: When modeling waves, we require that the amplitude of the wave is continuous across the boundary between media - in other words, we require that there is no infinitely-sharp change in amplitude at the boundary. This means that in a sufficiently small region around the boundary, the amplitude of all of the points must be similar at all times, or else there would be a discontinuity.
Since the rate of change of the amplitude at any given point is determined by the frequency, this therefore means that in a small region around the boundary, the frequency of oscillations must be uniform. Therefore, frequency doesn't change across the boundary.
A: If a chunk of matter absorbs 100 wave crests and emits 101 wave crests, then we would say that the chunk of matter created some wave crests by itself, at least one. Right?
If 100 crests go in and then 100 crests come out at slower rate, then it is possible that the waves that came out are the same as the waves that went in.
Like if we send a continuous sound wave through an air mass whose temperature is dropping, then 100 waves that went in during 1 seconds time, those same waves may come out during 2 seconds, so this changing medium halves the frequency, and doubles the duration. (And contains at least 50000 wave crests after 1000 seconds)
