The physical meaning of electromagnetic waves What, fundamentally, is an electromagnetic wave? As far as I know, all wave phenomena are derivations of an oscillating processes, e.g. particles vibrating in a medium. I can't imagine a wave process which couldn't be explained as moving particles: sound wave - movement of gas molecules, waves on the water - movement of water molecules, earthquake - movement of ground particles (and finally molecules), even phonons - vibrational motion of crystal lattice.  Phonons are virtual but the atoms arranged in lattice are real.
I tend to think that the reason has to do with the restless nature of the photons along with the uncertainty principle - the impossibility to localize a primary chunk of matter (with momentum) - when we catch the wave (stop the process) the frequency is gone. But again, it looks like it is a particle and we just can't observe it with our methods.
So, what is an electromagnetic wave? How can it move vibrating across the vacuum? How can we physically interpret phenomenon of electromagnetic waves?
 
Update
The concept of field is itself very complex phenomenon to understand its physical meaning by our "everyday" mind. The field looks like a math tool to describe and predict nature events. But its physical representation is unclear. I can't think of light as some 3-D helix shaped pulsing of continuous substance :) The world is quantized and it's obvious that the sine-shape like objects cannot be elementary in reality.
 A: An EMR is a term used to explain the effects "light" has on the electric and the magnetic field. We don't know what light looks like so the only way we can relate it to the physical world is by observing the effects it has on the surroundings. Instead of oscillating particles, it has oscillating electric and magnetic vectors which, again, can't be seen but they can be detected. A field can be imagined as a plane which can have excitations. These excitations mean different things in different fields, for example, in the electron field, excitations represent the chances of an electron being present at that particular place.
A: Let us start from the beginning, what does physical mean and what does "mean" mean:

The physical meaning of electromagnetic waves

The language of physics we learn not only at school but by everyday experience, is a language developed to describe the behavior of nature for dimensions larger than 10^-9 cm  , and times larger than 10^-9 seconds; in the present day physics language, dimensions large enough that the   Heisenberg Uncertainty Principle (HUP) is automatically fulfilled. There is also a mathematical description for students of physics.
So  we have given a " mathematial meaning" to a sea wave , by modelling its motion in transfer of energy with sinusoidal functions which are solutions of differential equations. 
Then we have instruments which discover physically that at the small dimensions of the  HUP there exists a substructure of atoms and molecules. The enlightenment coming from this experimental observation is what powers the "physical" in the title question. It considers "physical" something that can have an experimental under layer which layer can be modeled mathematically again. 
As a highschool teacher I had used to say: "problem number one has been reduced to problem number two". I.e., "what is the physical meaning of atoms and molecules" . Again an underlying level was revealed by experimental observations, the quantum mechanical models of atoms and molecules, and again the problem of what is "physical" has been pushed a layer below.
From the above, we can see that the term "physical meaning" is defined by the existence of an experimentally measurable under-layer that explains the behavior under study, pushing the "unknown" a layer below. We are making great efforts with our colliders etc to find lower and lower layers, if they exist. At present we are stopped with the standard model of physics which has pushed the under-layer to the posited existence of the particles in the table, the elementary particles, which mathematically are modeled as point and are the building blocks up to all layers I have been discussing above.
Notice the photon in this table. It is the building block of the electromagnetic wave, so it is the under-layer of the electromagnetic wave. That is the physical meaning of the electromagnetic wave, that it emerges, its mathematical models, from a confluence of photons. This needs quantum field theory to be understood, but it is just mathematics, as our experiments have not reached an under-layer which will be modeled mathematically so that the photon can be considered a measured confluence of the under-layer, the way the molecules are an under-layer for the water wave.

So, what is an electromagnetic wave? 

An over-layer emerging from the confluence of photons.

How can it move vibrating across the vacuum? How can we physically interpret phenomenon of electromagnetic waves?

It is riding on an enormous number of photons .  The mathematical description of this can be seen here.
We are stuck with the mathematical meaning because photons themselves are the bottom layer that can be measured in our experiments. That means we are stuck with quantum mechanical mathematical models describing the photons, which correctly gives the E and B fields of the classical wave and their energy transport in space. We have not reached experimentally the point of seeing whether there exists an under-layer to this or not, i.e. whether we have reached the end of the line with the standard model and its extensions as far as describing nature and giving meaning to "physical".
