Modulation sideband
= 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.