# Solution of wave equation not obeying maxwell equation

I was reading ch. 9 of classical electrodynamics of dr. griffith's. there he wrote that if a electric/magnetic field satisfy maxwell's equation then they must solve the wave equation which is $$\frac{1}{c^2} \frac{\partial^2 \textbf{E}}{\partial t^2}-\nabla^2\textbf{E}=0$$ $$\frac{1}{c^2} \frac{\partial^2 \textbf{B}}{\partial t^2}-\nabla^2\textbf{B}=0$$ Thats all right.but the reverse is not true.cause maxwell's equations impose constrain. well i thought that every em wave must obey maxwell equation,after all the wave equation in vacuum are derived using maxwell equation.isn't it contradictory? again when one minimizes the action of em field which is $$A=-\sum\int md\tau +\int A_iJ^id^4x-\frac{1}{16\pi}\int F_{\mu\nu}F^{\mu\nu} d^4x$$ i take this action from the book Gravitation:Foundation and Frontier by prof.Thanu Padmanabhan eq 2.111 where the first two terms are free particle and field lagrangian and last term is the interacting part. when one minimizes the action one gets $$\partial_{\mu}F^{\nu\mu}=4\pi J^i$$ which is nothing but the maxwell's equation. but what this equation says is the dynamics of E and B field which are Maxwell's equation. But from the wave equation one can get solution which not necessarily obey maxwell equation.isn't it contradictory? i am confused here.cause the minimizing action gives the Euler-Lagrange equation.which are in this case maxwell's equation.so any E and B field must obey them.But from inhomogeneous wave equation one can get solution which dont obey maxwell.please help. $$\frac{1}{c^2} \frac{\partial^2 \textbf{E}}{\partial t^2}-\nabla^2\textbf{E}=-\Big(\frac{1}{\epsilon_0}\nabla\rho+\mu_0\frac{\partial j}{\partial t}\Big)$$ $$\frac{1}{c^2} \frac{\partial^2 \textbf{B}}{\partial t^2}-\nabla^2\textbf{B}=\mu_0 \nabla \times\ j$$ which are taken from modern electrodynamics by prof. zangwill.eq 20.4 and 20.5 Thank you.

• An example : Here $E$ and $B$ solve the wave equation independantly, but there's nothing forcing them to be orthogonal in an EM wave. – Slereah Jun 30 '18 at 14:41
• ok.but i am bothered by the fact that minimizing action gives dynamical equation for fields.which are maxwell.so why we are getting solution whi ch do not obey maxwell? the action tells it what the dynamics.why there is other solution besides maxwell? – jowadul kader Jun 30 '18 at 14:53