# Electromagnetic waves and other waves

Hello everybody, im new here :)
what exactly is the energy of the wave? how to think about it generally, in electromagnetic waves and in relation to amplitude? I can think about the kinetic or potential energy that a particle can get from that kind of wave or the sum of all such energy in all particles under wave's influence but im not sure what's the intuition behind this term. also how can we derieve its proportion to the amplitude of the wave?

how can we derieve its proportion to the amplitude of the wave?
An intuitive idea, step by step?

Step # 1: Let's describe a spring using generic words, which can then describe other cases, not just the spring. Not to say stretching or compression, excursion is called, to call the fact of departing from the relaxed condition of the system. If the system is a spring, the excursion is a stretch or a compression. If the system is of another type, the excursion will have other characteristics, but it will retain the basic characteristic, which is to separate the system from the relaxed condition. The water of a pond, in a relaxed condition, exhibits a flat surface. When a wave propagates in water, the surface presents alternating low and high zones. You can propose more examples, to describe in them the excursion regarding a relaxed condition.

Step # 2: The spring energy is expressed as

[TEX] E_s = \ {1} {2} \ k \ x ^ 2 [/TEX]

[TEX] k \ \ \ \ rightarrow \ \ \ spring \ constant [/TEX]
[TEX] x \ \ \ rightarrow \ \ \ tour [/TEX]

The variation of energy for an infinitesimal excursion is

[TEX] dE_s = k \ x \ dx [/TEX]

Step # 3: The mechanical type harmonic oscillator is the mass / spring system. What would happen if you line up many mechanical oscillators, putting something in the space that separates two contiguous oscillators, to allow each oscillator to interact with its two neighbors? In other words, a system of coupled oscillators. If you separate spring 1 from the relaxed condition, the coupling will make spring 2 separate as well, then spring 3, so on, transferring energy along the row of springs. You will notice that when spring 1 excursion is maximum, spring 2 excursion does not. Spring 2 reaches the maximum excursion some time later. That is, both oscillators do not oscillate in phase. Each oscillator has an oscillation phase related to its position in space.

Step # 3: Instead of mechanical oscillators, think of a physical property that can fill a region of space continuously, without that kind of separation that we see in the mechanical oscillator system. Can that physical property have anything in common with the mechanical oscillator system? If it can vary around the relaxed condition without breaking down, being able to oppose the cause that separates it from the relaxation, the more the greater the separation, tending to return to the relaxed condition, although later it surpasses it and goes away from the relax in the inverse direction, then that continuous physical property is able to behave like a system of infinitesimal coupled oscillators. At each point in the space where that property is present, an infinitesimal portion of the property is oscillating, that is, exhibiting the periodic variation of some detectable physical property. Two adjacent infinitesimal oscillators will exhibit an infinitesimal phase difference. And if you compare two infinitesimal oscillators separated by a finite distance, the phase difference will be finite.

Step # 4: Try to read the equation of the spring energy in generic terms. The energy is formulated as a constant multiplied by the square of the excursion with respect to the relaxed condition. In the case of a wave, the energy is not located in a device. It is distributed throughout the region where the wave propagates. Then the energy density is interesting, which is expressed as a constant multiplied by the square of the amplitude. The amplitude is the excursion with respect to the state that would exist in the region if the wave did not exist. If you analyze an electromagnetic wave, you must take into account the excursions of two physical magnitudes, which are the electric field and the magnetic field. And you can formulate an energy density for each field, so that the sum of both gives the total energy density.
An intuitive idea, step by step?

Step # 1: Let's describe a spring using generic words, which can then describe other cases, not just the spring. Not to say stretching or compression, excursion is called, to call the fact of departing from the relaxed condition of the system. If the system is a spring, the excursion is a stretch or a compression. If the system is of another type, the excursion will have other characteristics, but it will retain the basic characteristic, which is to separate the system from the relaxed condition. The water of a pond, in a relaxed condition, exhibits a flat surface. When a wave propagates in water, the surface presents alternating low and high zones. You can propose more examples, to describe in them the excursion regarding a relaxed condition.

Step # 2: The spring energy is expressed as
$$E_s = \dfrac {1} {2} \ k \ x^2$$
## k \ \ \ \rightarrow \ \ \ spring \ constant##
##x \ \ \rightarrow \ \ \ excursion##
The variation of energy for an infinitesimal excursion is
$$dE_s = k \ x \ dx$$
Step # 3: The mechanical type harmonic oscillator is the mass / spring system. What would happen if you line up many mechanical oscillators, putting something in the space that separates two contiguous oscillators, to allow each oscillator to interact with its two neighbors? In other words, a system of coupled oscillators. If you separate spring 1 from the relaxed condition, the coupling will make spring 2 separate as well, then spring 3, so on, transferring energy along the row of springs. You will notice that when spring 1 excursion is maximum, spring 2 excursion does not. Spring 2 reaches the maximum excursion some time later. That is, both oscillators do not oscillate in phase. Each oscillator has an oscillation phase related to its position in space and the time when you see it.

Step # 3: Instead of mechanical oscillators, think of a physical property that can fill a region of space continuously, without that kind of separation that we see in the mechanical oscillator system. Can that physical property have anything in common with the mechanical oscillator system? If it can vary around the relaxed condition without breaking down, being able to oppose the cause that separates it from the relaxation, the more the greater the separation, tending to return to the relaxed condition, although later it surpasses it and goes away from the relax in the inverse direction, then that continuous physical property is able to behave like a system of infinitesimal coupled oscillators. At each point in the space where that property is present, an infinitesimal portion of the property is oscillating, that is, exhibiting the periodic variation of some detectable physical property. Two adjacent infinitesimal oscillators will exhibit an infinitesimal phase difference. And if you compare two infinitesimal oscillators separated by a finite distance, the phase difference will be finite.

Step # 4: Try to read the equation of the spring energy in generic terms. The energy is formulated as a constant multiplied by the square of the excursion with respect to the relaxed condition. In the case of a wave, the energy is not located in a device. It is distributed throughout the region where the wave propagates. Then the energy density is interesting, which is expressed as a constant multiplied by the square of the amplitude. The amplitude is the excursion with respect to the state that would exist in the region if the wave did not exist. If you analyze an electromagnetic wave, you must take into account the excursions of two physical magnitudes, which are the electric field and the magnetic field. And you can formulate an energy density for each field, so that the sum of both gives the total energy density.

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An intuitive idea, step by step?

Step # 1: Let's describe a spring using generic words, which can then describe other cases, not just the spring. Not to say stretching or compression, excursion is called, to call the fact of departing from the relaxed condition of the system. If the system is a spring, the excursion is a stretch or a compression. If the system is of another type, the excursion will have other characteristics, but it will retain the basic characteristic, which is to separate the system from the relaxed condition. The water of a pond, in a relaxed condition, exhibits a flat surface. When a wave propagates in water, the surface presents alternating low and high zones. You can propose more examples, to describe in them the excursion regarding a relaxed condition.

Step # 2: The spring energy is expressed as

[TEX] E_s = \ {1} {2} \ k \ x ^ 2 [/TEX]

[TEX] k \ \ \ \ rightarrow \ \ \ spring \ constant [/TEX]
[TEX] x \ \ \ rightarrow \ \ \ tour [/TEX]

The variation of energy for an infinitesimal excursion is

[TEX] dE_s = k \ x \ dx [/TEX]

Step # 3: The mechanical type harmonic oscillator is the mass / spring system. What would happen if you line up many mechanical oscillators, putting something in the space that separates two contiguous oscillators, to allow each oscillator to interact with its two neighbors? In other words, a system of coupled oscillators. If you separate spring 1 from the relaxed condition, the coupling will make spring 2 separate as well, then spring 3, so on, transferring energy along the row of springs. You will notice that when spring 1 excursion is maximum, spring 2 excursion does not. Spring 2 reaches the maximum excursion some time later. That is, both oscillators do not oscillate in phase. Each oscillator has an oscillation phase related to its position in space.

Step # 3: Instead of mechanical oscillators, think of a physical property that can fill a region of space continuously, without that kind of separation that we see in the mechanical oscillator system. Can that physical property have anything in common with the mechanical oscillator system? If it can vary around the relaxed condition without breaking down, being able to oppose the cause that separates it from the relaxation, the more the greater the separation, tending to return to the relaxed condition, although later it surpasses it and goes away from the relax in the inverse direction, then that continuous physical property is able to behave like a system of infinitesimal coupled oscillators. At each point in the space where that property is present, an infinitesimal portion of the property is oscillating, that is, exhibiting the periodic variation of some detectable physical property. Two adjacent infinitesimal oscillators will exhibit an infinitesimal phase difference. And if you compare two infinitesimal oscillators separated by a finite distance, the phase difference will be finite.

Step # 4: Try to read the equation of the spring energy in generic terms. The energy is formulated as a constant multiplied by the square of the excursion with respect to the relaxed condition. In the case of a wave, the energy is not located in a device. It is distributed throughout the region where the wave propagates. Then the energy density is interesting, which is expressed as a constant multiplied by the square of the amplitude. The amplitude is the excursion with respect to the state that would exist in the region if the wave did not exist. If you analyze an electromagnetic wave, you must take into account the excursions of two physical magnitudes, which are the electric field and the magnetic field. And you can formulate an energy density for each field, so that the sum of both gives the total energy density.
An intuitive idea, step by step?
View attachment 225959
Step # 1: Let's describe a spring using generic words, which can then describe other cases, not just the spring. Not to say stretching or compression, excursion is called, to call the fact of departing from the relaxed condition of the system. If the system is a spring, the excursion is a stretch or a compression. If the system is of another type, the excursion will have other characteristics, but it will retain the basic characteristic, which is to separate the system from the relaxed condition. The water of a pond, in a relaxed condition, exhibits a flat surface. When a wave propagates in water, the surface presents alternating low and high zones. You can propose more examples, to describe in them the excursion regarding a relaxed condition.

Step # 2: The spring energy is expressed as
$$E_s = \dfrac {1} {2} \ k \ x^2$$
## k \ \ \ \rightarrow \ \ \ spring \ constant##
##x \ \ \rightarrow \ \ \ excursion##
The variation of energy for an infinitesimal excursion is
$$dE_s = k \ x \ dx$$
Step # 3: The mechanical type harmonic oscillator is the mass / spring system. What would happen if you line up many mechanical oscillators, putting something in the space that separates two contiguous oscillators, to allow each oscillator to interact with its two neighbors? In other words, a system of coupled oscillators. If you separate spring 1 from the relaxed condition, the coupling will make spring 2 separate as well, then spring 3, so on, transferring energy along the row of springs. You will notice that when spring 1 excursion is maximum, spring 2 excursion does not. Spring 2 reaches the maximum excursion some time later. That is, both oscillators do not oscillate in phase. Each oscillator has an oscillation phase related to its position in space and the time when you see it.

Step # 3: Instead of mechanical oscillators, think of a physical property that can fill a region of space continuously, without that kind of separation that we see in the mechanical oscillator system. Can that physical property have anything in common with the mechanical oscillator system? If it can vary around the relaxed condition without breaking down, being able to oppose the cause that separates it from the relaxation, the more the greater the separation, tending to return to the relaxed condition, although later it surpasses it and goes away from the relax in the inverse direction, then that continuous physical property is able to behave like a system of infinitesimal coupled oscillators. At each point in the space where that property is present, an infinitesimal portion of the property is oscillating, that is, exhibiting the periodic variation of some detectable physical property. Two adjacent infinitesimal oscillators will exhibit an infinitesimal phase difference. And if you compare two infinitesimal oscillators separated by a finite distance, the phase difference will be finite.

Step # 4: Try to read the equation of the spring energy in generic terms. The energy is formulated as a constant multiplied by the square of the excursion with respect to the relaxed condition. In the case of a wave, the energy is not located in a device. It is distributed throughout the region where the wave propagates. Then the energy density is interesting, which is expressed as a constant multiplied by the square of the amplitude. The amplitude is the excursion with respect to the state that would exist in the region if the wave did not exist. If you analyze an electromagnetic wave, you must take into account the excursions of two physical magnitudes, which are the electric field and the magnetic field. And you can formulate an energy density for each field, so that the sum of both gives the total energy density.

Hi thank you ! great explanation :)
so the fact that the oscilating element in electromagnetic wave is the value of the field and not a particle moving in a spring-like behaviour is irrelevant?
i will be grateful if you could show that a little more formally too (as long as it wont consume too much of your time)

also- could you give an electromagnetic field's oscilation intuition too in relation to the movement of charged particles? are the particles themselves oscilating as in mechanical waves? how to envision their actual movement pattern?

thanks

could you give an electromagnetic field's oscilation intuition too in relation to the movement of charged particles? are the particles themselves oscilating as in mechanical waves? how to envision their actual movement pattern?
What you ask and what you suppose matches something I've seen elsewhere. But I can not reproduce it here, because they are not mainstream sites. I can only remind you of the divergence theorem, the differential version of Gauss's theorem. In an electromagnetic wave propagating in a vacuum, ##\vec{E}## have divergence equal to zero. And it can not have divergence other than zero without violating Maxwell's equations. ##\vec{E}## corresponds to free charge. That means that no free charge can be involved in the propagation. You look for reasons to admit that there is some charge involved in the propagation. Analyzing only ##\vec{E}## you will never find the reasons. It is all I can say.