Into Unscientific

a long time later.

When Wheat mentioned today's experiment in his memoir "He Changed Cambridge", he once wrote a sentence very kindly:

"Hey, Luo Feng!"

This sentence contains Maimai's extremely complicated emotions, and the abbreviation is the embarrassment of being so dead that he picks his feet.

After all, apart from Wheat himself and Xu Yun, there were also a series of units in physics books such as Prince Albert, Faraday, and Joule.

Of course.

At this time, Mai Mai was still a very simple and honest young man, and he didn't realize what a second-year thing he had done.

Although he blushed a little after reading this sentence, he hadn't yet reached the point where he wanted to kill Xu Yun with an axe.

Then he handed the piece of paper back to Xu Yun and asked:

"Mr. Luo Feng, what are we going to do next?"

Xu Yun glanced at him, patted his shoulder earnestly, and said:

"Didn't I just say, let's break the seal of the electromagnetic world."

wheat:

"."

Then Xu Yun straightened his expression and brought him to Faraday and the others:

"Mr. Faraday, according to the idea of ​​Fat Fish's ancestor, there are two things we need to do next."

Faraday and others made an allusion to listening.

Xu Yun raised a finger and explained:

"Derivation first, experiment second."

"Derivation?"

Faraday adjusted his glasses, repeated the word, and asked Xu Yun:

"Deduce what?"

Xu Yun did not answer the question directly, but asked rhetorically:

"Mr. Faraday, I heard that you once proposed a theory that there must be an electric field around an electric charge, correct?"

Faraday nodded.

Anyone who has studied physics should know this.

Faraday first introduced the concept of electric field and proposed the idea of ​​using electric field lines to represent electric field.

At the same time, the iron filings around the magnet are used to simulate the situation of the magnetic field lines.

Xu Yunjian said with a slight smile, suppressed the emotions in his heart, and said as calmly as possible:

"What we are going to derive next is something that exists in the electric field."

Then he picked up the paper and pen, and drew a wave on the paper.

That is, the graph of the sine function.

Then he drew a circle on the image and said to Faraday and others:

"Mr. Faraday, we study physics for the purpose of summarizing some kind of consistency from various phenomena in the ever-changing nature."

"Then use the language of mathematics to describe this consistent phenomenon quantitatively and precisely."

"For example, F=ma proposed by Mr. Newton, △S\u003e0 in thermodynamics in 1824, reader = handsome beauty, etc."

"Then the question is, is there a mathematical equation that can describe waves in our current world?"

Faraday and the others were silent for a moment, then slowly shook their heads.

Wave.

This is a very common word in life, or a phenomenon.

Except for naizi, stones falling into water produce waves.

Waves appear when the rope is shaken.

The wind blowing across the lake creates waves.

It was introduced earlier.

The level of physics in 1850 was actually not low. At this time, the scientific community could already measure relatively fine values ​​such as frequency and wavelength of light.

Nothing more than the unit of description is still minus several square meters, unlike the later generations that have nanometers and micrometers.

in this case.

Naturally, many people have tried to study waves. Mavericks are far away, and Euler is close.

But unfortunately.

Due to the limitations of the thinking of the times, the scientific community has not been able to derive a standard mathematical equation that can describe the law of waves.

But now Xu Yun asked such words

Could it be

"Student Luo Feng, has Mr. Fat Yu already deduced the mathematical expression of wave motion?"

Xu Yun still didn't answer this question directly, but continued to write on paper.

He first made a basic coordinate system on the function image drawn before.

Draw a → in the direction of the X axis, and write a V character.

This represents a wave moving with a certain velocity v in the positive direction of the x-axis.

Then Xu Yun explained:

"First we know that a wave is constantly moving."

"The image is just what the wave looks like at one moment, and it moves a little bit to the right the next moment."

Faraday and the others nodded in unison,

This is standard human speech, not difficult to understand.

How much the wave moves at the next moment is easy to calculate:

Because the wave speed is v, the wave will move to the right by a distance of v·Δt after Δt time.

Then Xu Yun drew a circle on one of the crests, and said:

"Mathematically, we can think of this wave as a collection of points (x, y), so we can describe it with a function y=f(x), right?"

A function is a mapping relationship. In the function y=f(x), every time an x ​​is given, a y can be obtained through a certain operation f(x).

This pair (x, y) constitutes a point in the coordinate system, and connecting all such points gives a curve-this is a real first-year concept.

Then Xu Yun wrote a t next to it, which means time.

Because the simple y=f(x) only describes the shape of the wave at a certain moment.

If you want to describe a complete dynamic wave, you have to take time t into account.

That is to say, the waveform changes with time, namely:

The vertical coordinate y of a certain point in the image is not only related to the horizontal axis x, but also related to time t. In this case, a binary function y=f(x, t) must be used to describe a wave.

But this is not enough.

The world is full of things that change with time and space.

For example, when an apple falls, or when the author is shaken by the reader, what is the essential difference between them and waves?

The answer is equally simple:

When the waves are propagating, although the positions of the waves are different at different times, their shapes are always the same.

That is to say, the wave was in this shape one second ago, and although the wave is no longer in this place after one second, it still has this shape.

This is a strong constraint.

Since the wave is described by f(x, t), the initial shape of the wave (the shape at t=0) can be expressed as f(x, 0).

After the time t has elapsed, the wave speed is v.

Then this wave moves to the right by the distance of vt, that is, the initial shape f(x, 0) is moved to the right by vt.

So Xu Yun wrote down another formula:

f(x,t)=f(x-vt,0).

Then he took a first look at Farah.

Among the bigwigs present, most of them came from professional courses, and only Faraday was a "nine leaking fish" who was born as an apprentice.

Although he added a lot of knowledge later, mathematics is still a weakness of this electromagnetic tycoon.

But what made Xu Yun slightly relaxed.

The expression of this electromagnetism master did not fluctuate, and it seems that he has not been left behind for the time being.

So Xu Yun continued to deduce.

"That is to say, as long as there is a function that satisfies f(x, t)=f(x-vt, 0), and the shape at any time is equal to the translation of the initial shape, then it represents a wave."

"This is a purely mathematical description, but it's not enough. We also need some analysis from a physical point of view."

"Like. Tension."

well known.

When a rope is placed on the ground, it is stationary, and when we flick it, there will be a wave.

So here comes the question:

How did this wave travel to such a distance?

Our hand is only pulling one end of the rope, and does not touch the middle of the rope, but when the wave reaches the middle, the rope does move.

If the rope moves, it means that there is a force acting on it, so where does this force come from?

The answer is equally simple:

This force can only come from the interaction between adjacent points of the rope.

Each point "pulls" the point next to itself, and the point next door moves—just like we only notify the person next to you when we queue up to count. The force between the inside of this rope is called tension.

Another example is that we pull a rope hard. I obviously exert a force on the rope, but why is the rope not stretched?

Why won't the point closest to my hand be pulled?

The answer is naturally that the points near this point exert an opposite tension to this mass point.

In this way, one side of this point is pulled, and the other side is pulled by its neighboring points, and the effects of the two forces cancel out.

But the action of the force is mutual, and the nearby point exerts a tension on the end point, then the nearby point will also receive a pulling force from the end point.

However, this nearby point has not moved, so it must also be under the tension of the more inner point.

This process can be propagated all the time, and the final result is that all parts of the rope will be tense.

Through the above analysis, we can conclude a concept:

When a rope is at rest on the ground, it is in a state of slack and no tension.

But when a wave passes here, the rope will become a wave shape, and there is tension at this time.

It is this tension that makes the points on the rope vibrate up and down, so analyzing the effect of this tension on the rope is the key to analyzing the fluctuation phenomenon.

Then Xu Yun wrote down another formula on the paper:

F=ma.

That's right.

It is the second law of Niu summed up by Mavericks.

well known.

Mavericks' first law tells us that "an object will remain at rest or move in a straight line at a constant speed when there is no force or the resultant external force is 0", so what if the resultant external force is not 0?

Mavericks' second law goes on to say:

If the total external force F is not zero, then the object will have an acceleration a, and the relationship between them is quantitatively described by F=ma.

That is.

If we know the mass m of an object, as long as you can analyze the resultant external force F it receives.

Then we can calculate its acceleration a according to the calf's second law F=ma.

Knowing the acceleration, you know how it will move next.

Then Xu Yun randomly picked two points on a certain section of the function image.

Write A on one and B on the other, and the arc of the two is marked as △l.

After writing, push it in front of the wheat:

"Student Maxwell, let's try to analyze the combined external force in this section? Don't consider gravity."

Mai Mai was taken aback when she heard the words, pointed to herself, and said in surprise:

"I?"

Xu Yun nodded and sighed slightly in his heart.

What he is going to do today will be of great significance to Faraday, to the field of electromagnetism, or to a greater extent, to the entire historical process of mankind.

But only for Mai and Hertz, it may not be a good thing.

Because it means that some contributions that belonged to them have been erased.

It’s like one day when a worker with a monthly salary of 4,000 suddenly realized that he could become a billionaire, but a reborn person took away the opportunity that belonged to him on the grounds of ‘common development of mankind’. How would you feel?

In all fairness, a little unfair.

So deep down in Xu Yun's heart, he felt a little guilty about Mai Mai.

How to compensate Wheat in the future is another matter. In short, in the current process, all he can do is to let Wheat come into the sight of these big guys as much as possible.

Of course.

Mai Mai didn't know what was going on in Xu Yun's heart. At this moment, he was holding a pen and writing the force analysis on the paper:

"Mr. Luo Feng said that gravity is not considered, so we only need to analyze the tension T at both ends of the band AB."

"Band AB is subject to the tension T of point A towards the lower left and the tension T of point B towards the upper right, which are equal to each other."

"But the region of the band is curved, so the directions of the two T's are not the same."

"Assuming that the angle between the direction of tension at point A and the horizontal axis is θ, the angle between point B and the horizontal axis is obviously different, and it is recorded as θ+Δθ."

"Because the points on the wave band move up and down when fluctuating, only the component of the tension T in the up and down direction needs to be considered."

"The upward tension at point B is T sin (θ+Δθ), and the downward tension at point A is T sin θ, then the resultant force on the entire AB segment in the vertical direction is equal to the subtraction of these two forces. "

soon.

Wheat wrote down a formula on paper:

F= T sin(θ+Δθ)-T sinθ.

Xu Yun nodded in satisfaction, and said:

"So what's the mass of the wave?"

"The mass of the wave?"

this time.

Mai Mai frowned slightly.

If it is assumed that the mass per unit length of a band is μ, then the mass of a band of length Δl is obviously μ·Δl.

However, because Xu Yun took a very small interval.

Suppose the abscissa of point A is x, and the abscissa of point B is x+Δx.

That is to say, the projection length of the rope AB on the abscissa is Δx.

Then when the length of the rope is very short and the fluctuation is very small, then Δx can be used approximately instead of Δl.

Then the mass of the rope can be expressed as.

μ·Δx

at the same time.

Kirchhoff on the side suddenly thought of something, his pupils shrank slightly, and he said in dry English:

"Wait. The resultant external force and mass have been determined. How to calculate the acceleration"

Hear what Kirchhoff said.

The classroom, which was not very noisy, suddenly became a little quieter.

yes.

Unknowingly, Xu Yun has deduced the combined external force and mass!

If we then deduce the acceleration

Then can't you express the equation of the wave in the classical system in the form of Niu Er?

Think here.

Several bigwigs took out pens and paper one after another, and tentatively calculated the final acceleration.

Speaking of acceleration, we must first talk about its concept:

This is used to measure the amount of speed change.

Acceleration, it must be that the faster the speed is added, the greater the value of acceleration.

For example, we often hear "I want to speed up" and so on.

If the speed of a car in the first second is 2m/s, the speed in the second second is 4m/s.

Then its acceleration is divided by the speed difference (4-2=2) by the time difference (2-1=1), and the result is 2m/s.

Think about it again, how is the speed of a car calculated?

Of course, it is the value obtained by dividing the distance difference by the time difference.

For example, a car is 20 meters away from the starting point in the first second, and 50 meters away from the starting point in the second second.

Then its speed is divided by the distance difference (50-20=30) by the time difference (2-1=1), and the result is 30m/s.

I wonder if you have discovered anything from these two examples?

That's right!

The speed is obtained by dividing the distance difference by the time difference, and the acceleration is obtained by dividing the speed difference by the time difference. These two processes are divided by the time difference.

So

What if these two processes are combined?

Would it be possible to say:

Divide the distance difference by a time difference, and then divide it by a time difference to get the acceleration?

Of course.

This is just a way of thinking. Strictly speaking, this expression is not very accurate, but it is very convenient for everyone to understand this idea.

If distance is viewed as a function of time, then take a derivative of this function:

It is the distance difference above divided by the time difference, but it tends to be infinitely small, and the function of speed is obtained,

Taking the derivative again as a function of velocity yields an expression for acceleration.

It is rare for the students to understand or not. Anyway, these bigwigs present quickly thought of this.

Yes.

The content described by the function f(x, t) listed before is the position of a certain point on the band at different times t!

So as long as the derivative of f(x, t) is calculated twice with respect to time, the acceleration a at this point is naturally obtained.

Because the function f is a function of the two variables x and t, it can only take the partial derivative of time f/t, and add 2 to the partial derivative again.

So soon.

Including Faraday, all bigwigs have written down a value one after another:

Acceleration a=f/t.

And combining this value with the previous resultant force and mass, a new expression appears:

F= T sin (θ+Δθ)-T sin θ=μ Δxf/t.

Then William Weber took a serious look at this expression, and frowned slightly:

"Student Luo Feng, is this the final expression? I seem to feel that it can be simplified?"

Xu Yun nodded:

"sure."

F= T sin (θ+Δθ)-T sin θ=μ Δxaf/t.

This is the most primitive system of equations, the content is not very clear, and the things on the left side of the equation look too troublesome.

So it needs to be reworked.

Where is the idea of ​​the transformation?

Of course it is sinθ.

Xu Yun picked up a pen and drew a right triangle on the paper.

well known.

The sine value sinθ is equal to the opposite side c divided by the hypotenuse a, and the tangent value tanθ is equal to the opposite side c divided by the adjacent side b.

Xu Yun drew another right-angled triangle with a very small included angle, estimated to be only a few degrees:

"But once the angle θ is very, very small, then the adjacent side b and the hypotenuse a are about to coincide."

"At this time, we can approximate that a and b are equal, that is, a≈b."

Then write on the paper:

[So there is c/b≈c/a, that is, tanθ≈sinθ. 】

[The previous formula can be written as F= T tan (θ+Δθ)-T tanθ=μ Δxaf/t. 】

"Wait a mininute."

Seeing this sentence, Faraday suddenly frowned and interrupted Xu Yun.

It is clear.

At this time, he has faintly shown signs of falling behind:

"Student Luo Feng, what is the meaning of replacing sinθ with tanθ?"

Xu Yun looked at Mai Mai again, and Mai Mai immediately understood:

"Mr. Faraday, because the tangent value tanθ can also represent the slope of a straight line, that is, the derivative of the curve at a certain point."

"The expression of the tangent value is tanθ=c/b. If a coordinate system is established, then this c just happens to be the projection dy of the straight line on the y-axis, and b is the projection dx on the x-axis."

"Their ratio is exactly the derivative dy/dx, that is to say tanθ=dy/dx."

After Faraday listened carefully, he spent two minutes doing calculations on the paper, and then suddenly slapped his forehead:

"I see. I understand. Please continue, Luo Feng."

Xu Yun nodded and continued to explain:

"Because the wave function f(x, t) is a binary function with respect to x and t, we can only find the partial derivative at a certain point."

"Then the tangent is equal to its partial derivative tanθ=f/x at this point, and the original wave equation can be written like this."

Then Xu Yun wrote down a new equation on paper:

T(f/xlx+△x-f/xlx)=μ·Δxaf/t.

It looked a little more complicated than before, but the eyes of these bigwigs on the scene were much brighter.

At this point, the next step is very clear.

As long as both sides of the equation are divided by Δx at the same time, the left side becomes the difference between the values ​​of the function f/x at x+Δx and x divided by Δx.

This is actually the derivative expression of the f/x function.

That is.

After both sides are divided by a Δx at the same time, the left side becomes the partial derivative f/x and the derivative of x is calculated again, that is, f(x, t) calculates the second-order partial derivative of x.

At the same time, f/t has been used above to represent the second-order partial derivative of the function to t, so f/x can naturally be used to represent the second-order partial derivative of the function to x.

Then divide both sides by T at the same time, and the equation is much simpler:

f/x=μf/Tx.

At the same time, if your brain is not dizzy, you will find that

The unit of μ/T.

It just happens to be the reciprocal of the square of the velocity!

That is to say, if we define a quantity as the square root of T/μ, then the unit of this quantity is just the unit of velocity.

It can be imagined that this speed is naturally the propagation speed v of this wave:

v=T/μ.

So after substituting this value, a final formula emerges:

f/x=f/vx.

This formula is later called

Classical wave equation.

Of course.

This equation does not take quantum effects into account.

If quantum effects are to be considered, this classical wave equation is useless, and the quantum wave equation must be used instead, which is the famous Schrödinger equation.

Starting from this classic wave equation, Schrödinger combined de Broglie's concept of matter waves to hard guess the Schrödinger equation.

Yes, guesswork.

I won’t go into details here. In short, this equation frees physicists from the fear of being dominated by Heisenberg’s matrix, and returns to the wonderful world of differential equations.

Now Xu Yun does not need to consider the quantum aspect, so the classical wave equation is enough.

Then he wrote down a new formula on paper.

With the writing of this new formula, Faraday suddenly discovered that

The piece of nitroglycerin I had left seemed not enough.