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 = \ {1} {2} \ k \ x ^ 2
k \ \ \ \ rightarrow \ \ \ spring \ constant
x \ \ \ rightarrow \ \ \ tour
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.
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.
