Let's consider a transversal wave on a string, which is reflected on a wall. I understand that the velocity of the incident and reflected wave are equal. However, I don't understand why the frequencies of both waves are the same. Can anyone please explain this fact?
The spatial boundary conditions on the fields must hold for all times, something not possible unless the incident, reflected and transmitted waves have the same temporal part that “cancels out” for all times. You have a spatial boundary (or obstacle) so this changes the spatial part of the wave, but why should it change the temporal part?
In physics we like to apply the harmonic oscillator model, because this is the only model we actually understand. Hence, this is what I'd like to do:
- The wall is the "origin" of the reflected wave. Hence, we should consider the wall as an harmonic oscillator. The reflected wave is just depicting the chronological sequence of the harmonic oscillator.
- Also, the incident wave acts as an external periodic force, acting upon the harmonic oscillator.
Once we understand this analogy the fact that the incident and the reflected wave must possess the same frequency follows directly: An harmonic oscillator always oscillates with the frequency, with which it is excited. So, even if we consider a hypothetical setup, where the two waves have different velocities, the frequency of the two wave would remain the same. This also explains why the frequency of light (=transversal wave) does not change, if it enters a medium.
... the velocity of the incident and reflected wave are equal... why the frequencies of both waves are the same.
Take it as a common process. The transported energy in the wave is a function of its amplitude and forward motion (velocity). As long as there is no loss of energy, the relation of both parameters must remain constant after reflection for reasons of energy conservation.