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One failure of wave nature of light in photoelectric effect was that increasing the intensity of light did not increase the kinetic energy of electrons. I don't understand how this is true.
We know that an increase of intensity means an increase of energy $E$. Again, $E=hf$, where $f$ is frequency, so increasing intensity increases energy, which in turn increases frequency, since $E$ is proportional to $f$. Then, why is wave nature of light failing to explain photoelectric effect?
You are confusing the energy of a single photon with the energy of the light beam.
Increasing the intensity of light (while keeping the frequency $f$ constant) does not increase the energy per photon, Instead it increases the number of photons per second,
For the most part the photoelectric effect happens one photon/electron interaction at a time. So the effect relies solely on the energy of each photon and the binding strength or energy required to free the electron.
To say "wave nature of light" means to deal with it as EM wave. The EM energy is $ \propto (E^2 + B^2)$. For sinusoidal wave $E = E_0sin(kx-\omega t)$, and $B = B_0sin(kx-\omega t)$, that means proportional to the amplitude squared.
As can be seen, in the wave model there is no dependency on the frequency for the energy. It was in contradiction with the experience in the case of the photoelectric effect.
The photoelectric effect shows that the transition that frees an electron is a resonant transition, happening at a single frequency. Increasing the energy at the resonant frequency increases the number of electrons emitted, at a rate of 1 electron per quantum hf of energy. If the frequency is increased, but the number of quanta hf is kept constant, the number of electrons emitted remains the same and the extra energy goes into electron kinetic energy.
