Wave and vector may not be strange words for one who deals in physics. Both the terms have very much to do in the world of physics. Actually these all terms come up in the region of optics. Well, let’s talk about the topic. Wave vector is the vector that describes a vector that defines a wave. You know that a vector has got magnitude as well as direction. Likewise, the wave vector has also got both of these. The magnitude of a wave vector is either a angular wave number or just wave number. Moreover it tells the direction of wave propagation.

In the context of special relativity, the wave vector is also defined as a four-vector. Actually, the wave vector has got two definitions. The difference occurs over a factor of 2π in the magnitude. Also these two definitions are in two different parts of science. One is in physics whereas the other is in crystallography.

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The definition in physics is as follows:

Where 'x' is defined to as the position, 't' denotes the time. 'Ψ' is the disturbance that describes the waves as a function of x and y.

'A' denotes the amplitude of the wave. 'ω' denotes the phase offset, 'ω' is the angular frequency and 'k' the wave number. This will happen in the case of a perfect 1-dimensional travelling wave. The same can be found in the case of 3-dimensional space as

Where the ‘.’ denotes the dot product and 'r', 'k' will be the position and wave vectors respectively. In the field of cryptography the same can be explained as:

In the case of 1-dimensional space, it will be as follows

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And in the case of 3-dimensional space, it will be as follows

The sole difference that we can notice here is the usage of v instead of ω.

Where, 2πv=ω