text: For an incident plane wave at a single frequency  (and the angular frequency ) on a crystal, the diffracted waves from the crystal can be thought as the sum of outgoing plane waves from the crystal. (In fact, any wave can be represented as the sum of plane waves, see Fourier Optics.) The incident wave and one of plane waves of the diffracted wave are represented as
  where  and  are wave vectors for the incident and outgoing plane waves,  is the position vector, and  is a scalar representing time, and  and  are initial phases for the waves. For simplicity we take waves as scalars here, even though the main case of interest is an electromagnetic field, which is a vector. We can think these scalar waves as components of vector waves along a certain axis (x, y, or z axis) of the Cartesian coordinate system.
  The incident and diffracted waves propagate through space independently, except at points of the lattice  of the crystal, where they resonate with the oscillators, so the phases of these waves must coincide. At each point  of the lattice , we have 
  or equivalently, we must have
  for some integer , that depends on the point . Since this equation holds at ,  at some integer . So
  (We still use  instead of  since both the notations essentially indicate some integer.) By rearranging terms, we get
  Now, it is enough to check that this condition is satisfied at the primitive vectors  (which is exactly what the Laue equations say), because, at any lattice point , we have 
  where  is the integer . The claim that each parenthesis, e.g. , is to be a multiple of  (that is each Laue equation) is justified since otherwise  does not hold for any arbitrary integers .
  This ensures that if the Laue equations are satisfied, then the incoming and outgoing (diffracted) wave have the same phase at each point of the crystal lattice, so the oscillations of atoms of the crystal, that follows the incoming wave, can at the same time generate the outgoing wave at the same phase of the incoming wave.