[talanum1 (OP)]

How does this: "the energy of a wave is proportional to the amplitude of the wave" rhyme with this: "the energy of a wave's particle is proportional to the frequency of the particle"?

Which is it now?

[alancalverd]

Both!

The power associated with a continuous wave is proportional to the square of its amplitude.

The energy of a single photon is proportional to the frequency of the wave that models it.

[yor_on]

So Alan, can we from that find the amount of particles (photons) assumed to be equivalent to a defined magnitude of a wavefront? Or should that be a continuous wave?

[alancalverd]

Yes. If a single visible photon has energy of, say, 3 eV = 4.8 x 10^-19 joules, then a flux of 2.1 x 10²⁰ photons per second delivers 1 watt to the absorbing surface.

With higher energy photons, say 0.1 - 1 MeV x-rays, it's quite possible to measure either the photon flux or the beam power.  In principle this could be extended down to 10 kV x-rays where direct measurement of wavelength is possible too, but I'm getting too old to do the work!