 Modulation sideband

Multiplying a carrier $\cos \omega t$ by $a+b\cos \Omega t$ produces a carrier and components at $\omega \pm \Omega$, each with amplitude $b/2$. For a nondispersive travelling signal all three depend on retarded time $t-x/c$, so their wavenumbers also differ: $k_{\pm }=(\omega \pm \Omega )/c$. A time-only amplitude multiplied into a spatial carrier is the same boundary signal at $x=0$, but is not generally a solution of the free wave equation throughout space.