y(x,t) = Asin(kx - ωt + φ) represents a single harmonic wave that travels in the positive x-direction, for example a harmonic wave on a string.  The speed of the wave is v = ω/k = λf.

If two or more waves travel in the same medium along the positive or negative x-direction at the same time, then they interfere.  Interference can produce standing waves, beats, and wave packets.

When waves travel in two or three dimensions, we also observe diffraction.  Interference and diffraction are characteristic signatures of waves.

Some experiments suggest that light is an electromagnetic wave moving with speed λf = c.  We have observed diffraction and interference patterns produced by light.  The patterns we studied in detail are the single slit diffraction pattern and the double slit interference pattern.

Single Slit: destructive interference:  wsinθ = mλ,  m = 1, 2, ….
Double slit, grating: constructive interference:  dsinθ = mλ,  m = 0, 1, 2, ….

Other experiments suggest that light is a steam of particle, which we call photons, such as the photoelectric effect, Compton scattering, and single photon counting experiments.  The photons associated with light of a particular wavelength λ have energy E = hf and momentum p = h/λ.  Since for photons λf = c, we have E = hc/λ, λ= hc/E. 

But when counting single photons for a sufficiently long time using a double-slit experimental setup, we notice that the distribution of their arrival positions at the counter forms an interference pattern.

So is light a wave or a stream of particles.  Both models seem to have some merit.  In quantum mechanics we consider light to be a stream of particles.  But the behavior of these particles is not predicted by Newton's laws, but by a different equation, called a wave equation.

If a wave equation describes the behavior of photons, maybe a wave equation also describes the behavior of other microscopic particles.
Newton's laws cannot explain the results of the electron diffraction experiment, where electrons form an interference pattern.

The deBroglie relations give the wavelength and frequency of a particle with energy E and momentum p as E = hf = ћω and p = h/λ = ћk.  But for particle E = (1/2)mv² = p²/(2m), so E = ћ²k²/(2m), λ = h/√(2mE).  The relationship between λ and E is different for particles than for photons.

If Newton's laws do not describe the behavior of particles on a microscopic scale, then what does?  What does a wave equation look like?  We have studied the wave equation that describes the behavior of waves on a string.  This wave equation tells us how the displacement y of a string can possibly change as a function of position and time.  The key word is CHANGE.