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I´ve searched about this but I cant really find an answer that satisfies me.

What happens that determines " This wave has frequency value equal to x Hz and a value of wavelength equal to y "

I cant seem to understand this... Does the wavelength has anything to do with length is there something moving up and down and each cycle is a complete wave ?

I´m very confused especially when I know that the wave analysis that we see in textbooks is a mathematical analysis and then I hear stuff like " The frequency of a wave is related to it´s energy"

And i get even more confused when I hear that "The wavelength of a wave can stop a wave from passing trough a hole with a diameter equal to x"

I can´t connect the mathematical side to physical side.

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Does the wavelength has anything to do with length is there something moving up and down and each cycle is a complete wave

In classical EM, the answer is that the wavelength is the distance between the peak voltages in the electric field.

When you think of a water wave, it's easy to see the peak because its sticking up out of the water.

But consider a sound wave in metal... there's no physical peak in the same terms. In this case, the wavelength is the distance between the peak tension in the metal as the sound squeezes and stretches it. There's physical motion at the microscopic level, but you don't see it.

Classical EM works in the same fashion, it's the internal "tension" of the electric (or magnetic) field that you're measuring. Of course that begs the question what this field is that you're working on, but that's another question (or non-question, I guess)...

When this field passes a conductive material, the basic pattern of the field can be seen in the motion of the electrons in the conductor. That's how antennas create a signal that electronics can amplify. You won't see that either, but this is more directly like the sound case.

The frequency of a wave is related to it´s energy

Now that's something different. This statement is not true in classical EM, this is something that arises in QM. If the book does not make that clear, the book is crap.

The wavelength of a wave can stop a wave from passing trough a hole with a diameter equal to x

You're lucky this is so, otherwise you'd fry yourself looking through the holes in the metal plate in the door of a microwave!

So the reason for this is very complex, but it all boils down to the hole having to be in a material that interacts with the EM. So it's a metal plate in the door, not a plastic one.

Think about the old-school TV antennas you see, sadly hanging on rusting towers. Ever notice they always consist of a series of horizontal metal poles, with a clearly defined distance between them?

What happens is that any time you have a current in a conductor it radiates EM. So when a radio (TV) signal goes past the antenna it picks up that pattern and then rebroadcasts it. The strength of that signal is maximized if the rods, or elements, are a resonant length - which is why you have different lengths in a TV antenna to pick up different channels.

If you space the metal rods properly, that secondary signal will arrive at a given location exactly one wavelength after the "original" signal, and the two will constructively interfere. Put a bunch in a row and you amplify the signal. If you look closely at one of these antennas, you'll note the wires going to the TV actually only connect to one of the sets of rods, the others are passively adding to the signal through this interference process.

I hope that makes sense...

The same thing is happening in the plate, sort of. In this case the responding signal interferes destructively with the original signal on the back, and constructively on the front. The result is basically a mirror. But this only works when the holes are smaller than the wavelength and the plate has the right thickness. Microwaves are around 2.4 cm, so a hole that's a couple of mm is effectively like no hole at all. Light from the bulb inside is millions of times smaller, so it gets through no problem.

Handy, no?

Another example: have you noticed modern TV antennas don't look like the old-school ones? They have an X shaped active element at the front, and then a bunch of rods behind them. The rods are spaced close together, compared to the ~2m long signals they may as well be a solid sheet of metal. Yet the rods have a crapload less wind drag than a solid sheet.

Handy, no?

It's also much more complex than that, and you need to use all sorts of expansions and such to calculate it, but this is the basic idea.

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    $\begingroup$ "peak voltages in the electric field." incommensurate units. May I suggest "volts/meter"? $\endgroup$ Mar 22, 2018 at 20:41
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    $\begingroup$ You may indeed! $\endgroup$ Mar 22, 2018 at 20:43
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My understanding is that what is moving up and down is the strength of the electric field at a particular point in space. EM radiation is generated by the mechanical vibration of charged particles. Remember a charged particle casts an electric field that decays as 1/r^2. When a charged particle moves, say in the +x direction, all of the points in space have to acquire new field values to reflect the new distance from the charged particle, but it takes time for the "information" that the particle has moved to reach these points in space. Along the x axis, as the charged particle vibrates back and forth, the electric field at points on the x axis is alternately increased and decreased as electromagnetic pulses propagating at the speed of light pass through points on the axis. At any particular point, the field will be alternately high and low. This is the EM wave and the frequency is based on the vibrational frequency of the charged particle. It is -somewhat- similar to splashing your hands in a pool and the frequency of the water waves correspond to the frequency of your splashing. For me it seems intuitive that the faster you splash or vibrate the particle, the more energy is being used to reorganize the electric field. Einstein also predicted a similar effect with gravity waves which have now been measured with the motion of black holes.

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My impression is that your question is about waves in general.

A common simple wave would be a sine wave (google 'sine wave' for images). The distance between its positive peaks (say, in meters) would be its wavelength. (Wavelength is the spatial length of one cycle of the sine wave.)

One item missing from your question is the propagation speed of the wave--lets call it $v$.

Then using your notation there is a formula relating the three variables:

$$yx=v$$

So the wavelength times the frequency (in cycles per second) equals the wave speed. You can solve for any one variable using the other two.

This answers your: What happens that determines " This wave has frequency value equal to x Hz and a value of wavelength equal to y "

See

http://hubblesite.org/reference_desk/faq/answer.php.id=72&cat=light

Note that the individual points on the wave can move up and down (transverse) or back and forth (longitudinal). The math side of waves (particularly wave propagation) can become very involved.

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