Why do waves not change frequency but change velocity when going from light to heavy rope?
The velocity in a string is determined by the mechanical properties of the string, and is faster in the light string. At the junction between the strings, the wave in the light string excites a wave in the heavy string, by wiggling the heavy string end up and down. The number of times this happens per second – the frequency – is not affected by any string properties. The frequency of the wave in the heavy string is just dependent on how many times per second its end is wiggled up and down. So the frequency of the waves in the two strings is the same. But since velocity, frequency and wavelength are connected by , if is the same and goes down, then wavelength will shorten. The same idea holds for light, where a higher index of refraction is analogous to the heavier string.
