# Is the composition of two plane waves also a plane wave?

I'm really interesting to know more about Plasma physics for waves , it comes in my mind to ask this question related to dispertion relation topic exactly plane waves , It is well known that the plane waves defined by the following mathematical formula : $$A(x,t)= A_0 \exp (2i\pi (\frac{x-vt}{\lambda}))$$

$$A$$ is the amplitude of the wave, $$A_0 = A(0,0)$$

$$x$$ is a position along the wave's direction of travel, and $$t$$ is the time at which the wave is described. Now my question here is : if we take other plane wave:$$A'(x,t)= A_0 \exp (2i\pi (\frac{x-vt}{\lambda}))$$ $$A'(x,t)= A'_0 \exp (2i\pi (\frac{x-vt}{\lambda}))$$ and we look to do mathematical composition of $$A'$$ and $$A$$ like this : $$(A'\circ A)(x,t)$$, Now does this composition give us a plane wave ? and if it is What about its new parameters ?

• Composition is meaningless here. Composition means taking the output of one function and using it as the input for another function. That doesn’t work here because the output is not $x$ and $t$. Perhaps you are confusing composition with superposition, which means simply adding the two waves together? – G. Smith Dec 18 '19 at 21:53
• @G.Smith, I do not mean superposition, But I meant what composition operation give us between two plane waves and pleas could u explaine me more why the output is not x and y ? – zeraoulia rafik Dec 18 '19 at 22:05
• The output is the intensity of the wave at a point in space and time. That intensity is neither $x$ nor $t$ and cannot be fed into the other wave formula. – G. Smith Dec 18 '19 at 22:07
• If you think composition has meaning here, try writing down $(A'\circ A)(x,t)$ as an explicit function of $x$ and $t$. – G. Smith Dec 18 '19 at 22:13
• With all due respect, your understanding is wrong. Talking about the composition of these two functions is literally nonsense. See en.wikipedia.org/wiki/Function_composition. – G. Smith Dec 18 '19 at 22:24