# Why this equation called as a plane wave equation?

I'm confused why this equation is called as plane wave equation. Is that because the direction is constant and indicates it to propagate in only one direction?

Thanks for any comment

First of all, that is not an equation, but it is a way of expressing a plane wave in mathematical form.

In fact the equal sign there doesn't mean "equality", but "equal by definition". You don't solve it, it simply tells you that the lhs (a general expression for the electric field) is equal to the rhs in the case of a plane wave. In other words, that's the mathematical definition of a plane wave.

It is called "plane" because the value of $$\ |E| \$$ doesn't depend on R, but only on the phase, which is the same for all points in the same plane. That is the plane for which the scalar product $$\ \hat k \cdot R \$$ is constant, which is perpendicular to the vector $$\\hat k\$$ (which lies in the direction of propagation).

BTW, note that that expression technically is the phasor of the electric field, because there is no time dependency.

You may want to see, for example, this site.

• how is it not an equation? Commented May 5, 2023 at 21:07
• @user253751 An equation is something which you can solve because you have an unknown. That's not an equation, it's a definition. It defines the mathematical form of a plane wave (in phasor form). Surprisingly enough, not every mathematical writing where an equal sign appears is an equation. Commented May 5, 2023 at 21:57
• @user253751 In fact the equal sign may be ambiguous in math and most rigorous texts use a different symbol for definitions, which usually is an equal sign with a little "def" superscript, or an uppercase greek "delta" superscript, or even the "Pascal assignment" symbol := or the congruence symbol (those are the most common I've seen) Commented May 5, 2023 at 21:57
• You may have E0(k^,w) and k and k^ and R and want to know E(R,w). Commented May 5, 2023 at 21:58
• @user253751 Sorry, no. The LHS is just the general form of the phasor of electric field (in vector function notation), the RHS is the specific form for a general plane wave. It makes no sense to solve that. It's a definition. Commented May 5, 2023 at 22:00

It's called a "plane" wave because it's the equation for a wave that is, well, planar. It's planar in that the values of the wave only vary in time, and in the direction of propagation -- in the two directions perpendicular to the direction of propagation (the plane), all the values are the same. This also means that at any point in space, the direction of propagation is constant.

Contrast this to a spherical wave, where the energy propagates from a point, so the direction of propagation changes from point to point.

Note that when you're introduced to electrodynamics, they just teach you the plane wave. They save spherical propagation for later, after you thoroughly understand propagation in a plane.

• One way to imagine a plane wave is that it's an approximation of a wave far away from its source. For example, sunlight on earth, because earth is pretty far away from the sun. Commented May 5, 2023 at 21:08
• I was going to attempt to define that, but was caught up in a geeky urge to explain that my "almost a plane wave" might by your "way too curved for the approximation to work". So I gave up. But -- yes. Commented May 6, 2023 at 2:54