### A.37 Maxwell’s wave equa­tions

This note de­rives the wave equa­tions sat­is­fied by elec­tro­mag­netic fields. The de­riva­tion will use stan­dard for­mu­lae of vec­tor analy­sis, as found in, for ex­am­ple, [41, 20.35-45].

The start­ing point is Maxwell’s equa­tions for the elec­tro­mag­netic field in vac­uum:

Here is the elec­tric field, the mag­netic field, the charge den­sity, the cur­rent den­sity, the con­stant speed of light, and is a con­stant called the per­mit­tiv­ity of space. The charge and cur­rent den­si­ties are re­lated by the con­ti­nu­ity equa­tion

To get a wave equa­tion for the elec­tric field, take the curl, , of (3) and ap­ply the stan­dard vec­tor iden­tity (D.1), (1) and (4) to get

 (A.235)

Sim­i­larly, for the mag­netic field take the curl of (4) and use (2) and (3) to get
 (A.236)

These are un­cou­pled in­ho­mo­ge­neous wave equa­tions for the com­po­nents of and , for given charge and cur­rent den­si­ties. Ac­cord­ing to the the­ory of par­tial dif­fer­en­tial equa­tions, these equa­tions im­ply that ef­fects prop­a­gate no faster than the speed of light. You can also see the same thing pretty clearly from the fact that the ho­mo­ge­neous wave equa­tion has so­lu­tions like

which are waves that travel with speed in the -​di­rec­tion.

The wave equa­tions for the po­ten­tials and are next. First note from (2) that the di­ver­gence of is zero. Then vec­tor cal­cu­lus says that it can be writ­ten as the curl of some vec­tor. Call that vec­tor .

Next de­fine

Plug this into (3) to show that the curl of is zero. Then vec­tor cal­cu­lus says that it can be writ­ten as mi­nus the gra­di­ent of a scalar. Call this scalar . Plug that into the ex­pres­sion above to get

Next, note that if you de­fine mod­i­fied ver­sions and of and by set­ting

where is any ar­bi­trary func­tion of , , , and , then still
since the curl of a gra­di­ent is al­ways zero, and
be­cause the two terms drop out against each other.

The fact that and pro­duce the same phys­i­cal fields is the fa­mous gauge prop­erty of the elec­tro­mag­netic field.

Now you can se­lect so that

That is known as the “Lorenz con­di­tion.” A cor­re­spond­ing gauge func­tion is a “Lorenz gauge.”

To find the gauge func­tion that pro­duces this con­di­tion, plug the de­f­i­n­i­tions for and in terms of and into the left hand side of the Lorentz con­di­tion. That pro­duces, af­ter a change of sign,

That is a sec­ond or­der in­ho­mo­ge­neous wave equa­tion for .

Now plug the ex­pres­sions (6) and (7) for and in terms of and into the Maxwell’s equa­tions. Equa­tions (2) and (3) are sat­is­fied au­to­mat­i­cally. From (2), af­ter us­ing (8),

 (A.237)

From (4), af­ter us­ing (8),
 (A.238)

You can still se­lect the two ini­tial con­di­tions for . The smart thing to do is se­lect them so that and its time de­riv­a­tive are zero at time zero. In that case, if there is no charge den­sity, will stay zero for all time. That is be­cause its wave equa­tion is then ho­mo­ge­neous. The Lorenz con­di­tion will then en­sure that is zero too.

In­stead of the Lorenz con­di­tion, you could se­lect to make zero. That is called the “Coulomb gauge” or “trans­verse gauge” or “trans­verse gauge.” It re­quires that sat­is­fies the Pois­son equa­tion

Then the gov­ern­ing equa­tions be­come

Note that now sat­is­fies a purely spa­tial Pois­son equa­tion.