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As far as I know (correct me if I’m wrong), for a mechanical wave (e.g. sound), the frequency determines the pitch, the amplitude determines the loudness, and the amplitude is proportional to the intensity, which represents the energy the wave carries.

In an EM wave like light, the frequency determines color and energy, the amplitude is simply the amplitude of the E and B fields, while it is the intensity that determines brightness.

Why are the roles of the same factors different in these two waves? Why does the frequency in light influence energy but not in mechanical waves? Does frequency or intensity determine energy in a light wave?

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You're mixing up two separate and unrelated meanings for the energy of the light.

  • In a sound wave the energy transmitted by the sound wave is proportional to the square of the amplitude.

  • In a light wave the energy transmitted by the light wave is proportional to the square of the amplitude, just like the sound wave.

However in the light wave the energy is quantised in units of $h\nu$ i.e. the energy can't take just any value, it has to be a multiple (normally a very large multiple) of $h\nu$. The frequency, $\nu$, determines the energy of a single photon, but not of the overall light wave.

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  • $\begingroup$ Can’t energy be quantized for a mechanical wave as well? (I.e. wave-particle duality.) $\endgroup$
    – lightweaver
    Jul 10, 2015 at 13:38
  • $\begingroup$ Lattice vibrations in solids can be sort of quantised, and the quanta are quasi-particles called phonons. However I don't think propagation of a sound wave can usefully be described using phonons. In liquids and gases I'm not sure whether phonons exist. $\endgroup$
    – John Rennie
    Jul 10, 2015 at 15:14

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