A wave in 3D is very hard to visualize. This diagram represents a part of a wave in 3D.

If you draw lines following the crests of these waves then we can represent the wave by lines.

This results in the following image. This actually shows us the level curves of the wave.

As seen in affine transformation this looks like the many lines of Ax+By=C. We can use this to describe the equation of the wave.


This diagram shows us the crests of the wave, the direction that the wave is travelling, the cosine max(which are the crests) and the wavelength which is one cycle.

Crests
The wave in 2D or 3D is very similiar to the waves in 1D.

In 1D the crests are the top of cosine wave or the cosine max.

Wavelength
The wavelength of a 2D and 3D wave is similiar to the wavelength of a 1D wave as shown below.
This is in 2D/3D.

This is in 1D.

Frequency
The definition for frequency is the same for both 1D, 2D and 3D waves. It is how many crests pass a point per second. ie how many crest pass the yellow line per second.
In 2D and 3D.

In 1D.

Velocity
Similiarly to the frequency the velocity of a wave in 2D and 3D is similiar to the velocity in 1D.
It is the speed at which a crest travels.
In 2D and 3D.

In 1D.

Amplitude and Wave Height
Amplitude and wave height are hard to visualize in 2D and 3D but we can picture it in 1D. Amplitude(A) is the distance from the xaxis to the crest while wave height is the distance from the trough to the crest.

The wave height is equal to Acos(ax+by+dt)=0 in 2D and Acos(ax+by+cz+dt)=0 in 3D.
Where t=0 and the angle is measured in radians.
This is our affine function, where Acos(ax+by) or Acos(ax+by+cz) is our linear function and the dt part is our translation.

Waves in 2D and 3D and the Wave Equation
The wavelength () is perpendicular to the level lines and we can consider this to be the normal to the level lines. The wavelength is also a multiple of [a,b] in 2D and [a,b,c] in 3D.

Using the signed distance (C ''C ')/(A^{2}+B^{2})^{1/2} and (C ''C ')/(A^{2}+B^{2}+C^{2})^{1/2} from one level to the next ie C '=0 to C ''=2 we get
=2/(a^{2}+b^{2})^{1/2} in 2D
=2/(a^{2}+b^{2}+c^{2})^{1/2} in 3D

Since we know that = k[a,b] in 2D and k[a,b,c] in 3D,
=2/(a^{2}+b^{2})^{1/2}=k(a^{2}+b^{2})^{1/2}
=> k=2/(a^{2}+b^{2}) in 2D
=2/(a^{2}+b^{2}+c^{2})^{1/2}=k(a^{2}+b^{2}+c^{2})^{1/2}
=> k=2/(a^{2}+b^{2}+c^{2}) in 3D
Therefore becomes =2/(a^{2}+b^{2})x[a,b] in 2D.
=2/(a^{2}+b^{2}+c^{2})x[a,b,c] in 3D.
This is an expression of in terms of a,b in 2D and a,b,c in 3D.

The constant term dt can be written as t, where is the radian frequency. As changes then this changes the translation of the curve.
The equation of the wave without horizontal scaling is
Acos(ax+byt) for 2D.
and Acos(ax+by+czt) for 3D.
This is the same as the translation of the 1D wave by cT.
