Difference in the direction of propagation of em wave [duplicate]

marked as duplicate by Rob Jeffries, John Rennie electromagnetism StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Apr 28 at 11:11
Generally, the phase of the eletromagnetic wave is going to be $$\phi = \vec k \cdot \vec r \pm wt$$; from this expression, we can interpret the direction of the wave. Take a fixed time, then look for direction of the $$\delta \vec r$$ that maximizes your $$\delta \phi$$. That's the direction of propagation of the wave. It will be parallel to $$\vec k$$.
The "sense" of propagation in the direction of $$\vec k$$ is determined by the sign on $$wt$$; If your $$\delta \vec r$$ has the same direction and sense of $$\vec k$$, then, to maintain the phase, your $$\pm w\delta t$$ should be negative. That means a wave going "forward" on the sense of the propagation has a phase $$\phi = \vec k \cdot \vec r - wt$$, and a wave going "backwards" has a phase $$\phi = \vec k \cdot \vec r + wt$$.