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Today we will talk about a short and straightforward problem.

Many people are puzzled by the energy of mechanical waves. Why is the elastic potential energy of the mass element in the equilibrium position the largest? Why does the kinetic energy of the mass element change synchronously with the potential energy?

Of course, you may use some special cases to help understand these problems. For example, when shaking a horizontal rope to produce shear waves, you notice that those are in equilibrium positions, because the pulling forces on both sides point in the opposite direction, so the deformation is the most severe; while in those positions at the highest or lowest points, the forces on both sides point in the same direction, although they have the maximum acceleration, but the deformation is zero. So the position with the maximum potential energy should be in the equilibrium position.

If you shake a soft spring to form a shear wave, you will see that at the equilibrium position, the spring deformation is the most obvious, while the lowest and highest points are almost the same, as shown in the following figure.

As for kinetic energy, it is also easy to think of a tried-and-tested rule: if the acceleration of the equilibrium position is zero, the velocity must be maximum, so kinetic energy is maximum.

In this way, the synchronization of kinetic energy and potential energy reaches the maximum and, of course, the synchronization reaches the minimum, so the law of their synchronous changes is established.

But if you want to be serious, how on earth do you get the law of the energy of mechanical waves? Then you have to use a little math. A process of reasoning and analysis is given in more detail below.

Considering that a long spring is stretched, what is its elastic potential energy?

If you have studied physics in high school, of course you know it is like this.

Among them is the coefficient of stubbornness. If the material and thickness of the spring are determined, the longer the spring, the smaller the stubbornness coefficient, this should be easy to understand, the longer the spring, of course, it will not be so hard to pull. According to this law, the stubbornness coefficient can be written as a constant, depending on the thickness of the spring and the material properties.

So the potential energy above can be written as a new variable deliberately cobbled together here-- the relative elongation. Because it is constant, the potential energy stored by a spring (determined by material, thickness and original length) is determined by its relative elongation.

If the spring is compressed, however, the energy expression is the same.

In short, the relative shape of the spring determines its potential energy!

Then, for a long medium, the potential energy it stores is similar: it depends on its relative shape.

You might think that the offset of the center of this medium will produce potential energy, which is actually an illusion. The absolute offset of each point in the medium does not necessarily form elastic potential energy!

Imagine a ruler in your hand, suppose it moves for a distance now, obviously, the gravitational potential energy of the whole ruler has changed, but there is no deformation, so there is no elastic potential energy! But obviously, aren't the points on the ruler offset?!

Therefore, fundamentally speaking, elastic potential energy will be formed only when the displacements of the internal points are different, which leads to the relative displacement between the points, that is, when the matter is deformed.

Obviously, the medium of mechanical waves is the best representative.

For mechanical waves, the variable of the wave function describes the offset of the midpoint of the medium. It is a function of position and time, in other words, the offset of different points is different at the same time. So it will naturally lead to relative displacement between the points, that is, the deformation of the wave medium, which can lead to elastic potential energy!

So, what is the law that the potential energy in the medium changes with the position?

Considering a certain moment, the wave behaves as a curve-waveform curve.

It describes the variation of the offset of each point in the medium with the coordinates, which is expressed as a function.

When the position produces an increment, the offset also produces an increment.

Now analyze a prime element of the original length in the medium.

Before deformation, the coordinates of its two ends are and, respectively.

After deformation, the longitudinal offsets of the two ends are and. The coordinates at the left end become + and the coordinates at the right end become +.

Therefore, the length of the deformed medium is +, and the absolute elongation is, so the relative deformation-relative elongation of the original length is

According to the law that the relative elongation determines the potential energy mentioned earlier, the potential energy stored by the mass element of the original length is

If you divide the one from the right to the left, the average potential energy density in the medium is obtained, so the transition from the average velocity to the instantaneous velocity is imitated-the time is obtained at any time, and now an infinitesimal section of the medium is considered. In the above formula, the infinitesimal becomes a differential component, and instead, the potential energy density of any point in the medium is

See?! The division of the two differentials leads to the derivative, which gives you the derivative.

In fact, considering that at any time, it is a function of sum, so it is better to use partial derivative symbols instead of rewriting! Whose derivative? The partial derivative of the offset to the prime element coordinates, that is, the rate of change of the offset of the prime element to the coordinates! Isn't that the slope of the tangent at some point on the waveform curve?

Therefore, the potential energy of the mass element in the medium depends on the slope of the tangent on the waveform curve. The larger the absolute value of the slope is, the greater the potential energy is, and the smaller the absolute value is, the smaller the potential energy is.

Obviously, the absolute value of the slope at the equilibrium position is the largest on the waveform curve. So this position has the greatest potential energy! Where is the slope zero? The peaks and troughs, of course, so the potential energy of these positions is zero!

In fact, according to the mutual derivative relationship between sine and cosine, a more general rule can be seen in the picture above.

The above derivation is carried out according to the longitudinal wave, if it is a shear wave, the matter will undergo transverse deformation, the calculation is a little more complicated, but the final law is consistent.

Let's take a closer look at the expression of mechanical wave energy.

The first is the potential energy part.

From the above analysis, we can see that in order to obtain the exact expression of potential energy, it is necessary to determine the value in the formula, which reflects the influence of the stubbornness coefficient of this medium except the length. So what are these so-called "other effects"?

The elasticity of the material? Yes! It's called the modulus of elasticity. What else? What is the thickness of this medium? The thicker it must be, the harder it is to deal with, right? That's right!

The modulus of elasticity is for solids, and for liquids, there is a corresponding amount. In short, it is the elastic property of the object itself.

Assuming that the modulus of elasticity is expressed by the thickness of the medium, that is, its bottom area, then the potential energy above can be expressed as the volume of the medium. Generally speaking, the relationship between elastic modulus and wave velocity is density, according to which the above potential energy can be written as if we look at the microelement of a medium, then the above formula is to replace the partial derivative of the wave function pair, that is, the expression of the potential energy is the kinetic energy part.

This medium is vibrating, so it has kinetic energy. Kinetic energy is very simple, directly press the expression of kinetic energy, that is

Among them is the vibration velocity of the medium-notice that the medium does not move with the wave! So what's here is not the wave velocity, but the velocity of the medium offset-- the vibration velocity, which, according to the definition of velocity, is the kinetic energy.

It was found that the kinetic energy and potential energy of the mass element change exactly the same with position and time. In more physical terms, they both change in the same amplitude and in the same phase, and the values of the two are exactly the same at every moment for a certain point.

The energy of the mechanical wave is the sum of kinetic energy and potential energy, so naturally, the mechanical energy of the mass element can be seen thoroughly, and the part of energy related to coordinates and time has the standard form of simple harmonics. Therefore, the energy of the mechanical wave itself forms a simple harmonic, and its frequency is twice that of the wave. For simple harmonics, the propagation velocity of energy is the velocity of the wave itself, that is, the phase velocity.

Since the energy of the simple harmonic varies with the time period, it shows that the energy of the mass element of the simple harmonic medium is not conserved, which is different from the simple harmonic vibration. Many people find this counterintuitive. They think: isn't the mass element of the simple harmonic medium doing simple harmonic vibration? Such being the case, the energy of the element of matter should be conserved.

The crux of the problem is that the prime element of a simple harmonic is not a simple harmonic vibration, because of the powerful action between the adjacent elements, so the prime element does a forced vibration rather than a simple harmonic vibration.

When the wavefront has just reached a certain element of mass, it is still in the equilibrium position, and the energy of the element of mass is maximum. When a wave crest or trough reaches a proton, its energy flows out along the propagation direction of the wave-because it needs to instigate its neighbors to move along with it; when the crest or trough reaches the proton, it begins to absorb the energy coming from the direction of the wave source and returns to the equilibrium position again. Over and over again, the energy travels forward in the direction of the wave line.

This article is from the official account of Wechat: University Physics (ID:wuliboke), by Xue Debao.

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