Brian Bi

What is light or an electromagnetic wave a wave in?

So, when you ask what an electromagnetic wave is a wave in, you're coming at this thinking that a wave should involve a material medium physically moving in an oscillatory fashion. That is what's happening in an ocean wave or a sound wave.

But it's possible to generalize this idea. Electromagnetic waves belong to this more generalized definition of waves.

This is how you can look at it. When there's a sound wave, it's true that air molecules are moving. But mathematically, we model this by considering the pressure in the air as a function of space and time. Over time, the pressure in the air oscillates, like a sine function, or something like that.. It also oscillates in space in a similar way. When a transverse wave travels down a string---or say a jump rope---we model it in terms of the displacement of each small segment of the string. Over time, the vertical displacement of a small segment of the string oscillates, as it moves up and down. It also oscillates with space, as some segments at a given time are at crests and others are at troughs, while still others are at nodes.

So a wave occurs when some physical variable that depends on both space and time---like say pressure or displacement---oscillates in space and time. It doesn't matter whether or not there is a material medium that's moving. In the case of electromagnetic waves, that variable is either the electric field or the magnetic field (depending on which one you care about---usually the electric field).

Another way of saying this is that a wave is a physical phenomenon that can be described by a function that satisfies a wave equation. For example, the electromagnetic wave equation is
$\frac{\partial^2 \mathbf{E}}{\partial t^2} - c^2 \nabla^2 \mathbf{E} = 0$
where $c$ is the speed of light. What wave equations have in common is that they are second order partial differential equations in both time and space, with the second derivative w.r.t. time having one sign and the second derivatives w.r.t. space having the opposite sign. (Note: There can be exceptions when complex numbers are involved; for example, the Schrödinger equation.) When a function satisfies a wave equation, its behaviour in space and time will be wavelike. If you have a physical observable that satisfies a wave equation, then the oscillation of that observable is referred to as a wave. That's how waves are defined and understood in modern physics.