Failure of the Wave Theory of Light

What the classical wave theory predicts. In the classical (Maxwell) picture, a light wave carries its energy CONTINUOUSLY, spread out over its entire wavefront, with the energy delivered per second to a given area being proportional to the INTENSITY of the wave (which is itself proportional to the square of the wave's amplitude). On this picture, an electron sitting in a metal surface simply absorbs energy gradually and continuously from the incoming wave for as long as the light keeps shining, until it has accumulated enough energy to break free.

Why this fails to explain the threshold frequency (Law 3). If energy is delivered continuously and can simply be accumulated over time, then ANY frequency of light, however low, ought EVENTUALLY to deliver enough total energy to free an electron, provided the light is left shining for long enough or is made intense enough -- there is no reason, in this picture, for a strict cut-off frequency below which emission can NEVER happen, however long one waits. Yet Lenard's experiments found exactly such a cut-off, sharp and absolute, for every metal tested.

Why this fails to explain the intensity-independence of $K_{max}$ (Law 2). A MORE INTENSE wave, in the classical picture, delivers energy to the surface at a FASTER rate -- so a brighter light of the same frequency should let an electron accumulate its escape energy faster and, having done so, should also be free to go on absorbing yet more energy from the same intense wave, leaving with a correspondingly LARGER kinetic energy. The classical picture therefore predicts that $K_{max}$ SHOULD increase with intensity -- directly contradicting Lenard's finding that $K_{max}$ depends only on frequency, never on intensity.

Why this fails to explain the instantaneous emission (Law 4). For very weak (low-intensity) light, the classical picture requires the electron to accumulate energy slowly, drop by drop, from the continuously-spread wave -- a calculation of how long this ought to take, using the classical rate of energy delivery per unit area and the known size of an atom, gives an estimated time lag of MINUTES, or longer, before an electron could plausibly gather enough energy to escape, even for the weakest measurable illumination. Experimentally, however, photoelectric emission begins effectively at once, with no detectable delay, at any intensity above zero (provided the frequency is above threshold) -- a flat contradiction of the classical estimate. …