Chapter 258 Witness the miracle! (Part 1)
Much, much later.
When Wheat mentioned today's experiment in his memoir "He Changed Cambridge", he once wrote a very affectionate sentence:
"常娨遁, Luo Feng!"
This sentence contains the extremely complex emotions of Xiaomai. In short, it means the embarrassment of social death to the point of picking one's feet.
After all, in addition to Xiaomai himself and Xu Yun, there were also a series of units from physics books such as Prince Albert, Faraday, and Joule.
Of course.
At this time, Xiaomai was still a very honest young man, and he had not yet realized what a middle-class thing he had done.
Although he blushed a little after reading this sentence, he still didn't want to kill Xu Yun with an ax as he did later.
Then he handed the piece of paper back to Xu Yun and asked:
"Mr. Luo Feng, what should we do next?"
Xu Yun glanced at him, patted his shoulder earnestly, and said:
"Didn't I say that I would unlock the seal of the electromagnetic world?"
Wheat:
"."
Then Xu Yun's expression became serious and he led him to Faraday and others:
"Mr. Faraday, according to the thinking of the Fat Fish ancestors, there are two things we have to do next."
Faraday and others acted like they were all ears.
Xu Yun raised a finger and explained:
"The first is derivation, the second is experiment."
"Derivation?"
Faraday adjusted his glasses, repeated the word, and asked Xu Yun:
"Derivating what?"
Xu Yun did not answer the question directly, but asked:
"Mr. Faraday, I heard that you once proposed a theory, that is, there must be an electric field around the charge, right?"
Faraday nodded.
Students who have studied physics should all know this.
Faraday was the first to introduce the concept of electric field and proposed the idea of using electric field lines to represent the electric field.
At the same time, the iron filings around the magnet were used to simulate the magnetic field lines.
Xu Yun smiled slightly when he saw that, suppressed the emotions in his heart, and said as calmly as possible:
"What we are going to deduce next is something that exists in the electric field."
Then he picked up paper and pen and drew a wave pattern on the paper.
That is the image of the sine function.
Then he drew a circle on the image and said to Faraday and others:
"Mr. Faraday, the purpose of our study of physics is to summarize some consistency from the ever-changing phenomena of nature."
"Then use mathematical language to quantitatively and accurately describe this consistent phenomenon."
"For example, F=ma proposed by Mr. Newton, △S>0 in thermodynamics in 1824, readers = handsome and beautiful women, 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 and slowly shook their heads.
Wave.
This is a very common word or phenomenon in life.
In addition to the ripples, when a stone falls into water, it produces waves.
What appears when the rope shakes is also a wave.
The wind blowing across the lake creates waves.
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.
It's just that the unit of description is still minus several cubic meters, unlike later generations where there are nanometers and micrometers.
in this case.
Naturally, many people have tried to study waves, including Mavericks in the distance and Euler in the near side.
But sadly.
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 laws of waves.
But now Xu Yun asked this question
Could it be
"Classmate Luo Feng, has Mr. Feiyu deduced the mathematical expression of wave motion?"
Xu Yun still did not answer this question directly, but continued to write on the paper.
He first made a basic coordinate system on the previously drawn function image.
Then draw a → in the X-axis direction and write a V.
This means that a wave moves in the positive direction of the x-axis at a certain speed v.
Then Xu Yun explained:
"First of all, we know that a wave is constantly moving."
"This image is just what the wave looks like at a certain moment, and it will move a little to the right the next moment."
Faraday and others nodded,
This is standard human speech and is not difficult to understand.
As for how much the wave has moved at the next moment, it is also easy to calculate:
Because the wave speed is v, after Δt time, the wave will move to the right by a distance of v·Δt.
Then Xu Yun drew a circle on one of the wave crests and said:
"Mathematically speaking, we can think of this wave as a series of points (x, y), so we can use a function y=f(x) to describe it, right?"
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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) forms a point in the coordinate system, and by connecting all such points, a curve is obtained - this is the real first concept.
Then Xu Yun wrote a t next to it, which means time.
Because simply 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.
In other words, the waveform changes with time, that is:
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 the author is lifted up and shaken by the reader, what is the essential difference between them and waves?
The answer is also very simple:
When waves propagate, 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. Although the wave is not in this place one second later, it is still in this shape.
This is a strong constraint.
Since f(x,t) is used to describe the wave, the initial shape of the wave (the shape when t=0) can be expressed as f(x,0).
After time t has passed, the wave speed is v.
Then this wave moves a distance of vt to the right, that is, the initial shape f(x, 0) moves to the right by vt.
So Xu Yun wrote another formula:
f(x,t)=f(x-vt,0).
Then he took his first look at Farah.
Most of the big guys present came from professional courses, and only Faraday was an apprentice.
Although he later gained a lot of knowledge, mathematics is still a weakness of this electromagnetic master.
But what made Xu Yun relax slightly was.
The electromagnetic expert's expression showed no fluctuation, and it seemed that he was not left behind for the time being.
So Xu Yun continued the derivation.
"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 initial shape translated by a section, then it represents a wave."
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"This is a purely mathematical description, but it is not enough. We also need to conduct some analysis from a physical perspective."
"For example. Tension."
As everyone knows.
A rope is stationary when placed on the ground. When we shake it, a wave will appear.
So here comes the question:
How does this wave spread to such a distance?
Our hands only held one end of the rope and did not touch the middle of the rope, but when the wave reached the middle, the rope did move.
If the rope moves, it means that force is acting on it. So where does this force come from?
The answer is also very simple:
This force can only come from the interaction between adjacent points of the rope.
Each point "pulls" the point next to it, and the point next door moves - just like when we line up to count, we only notify the person next to you. The force inside the rope is called tension.
Another example is when we pull a rope hard. I obviously exert a force on the rope, but why doesn't the rope get stretched?
Why doesn't the point closest to my hand get pulled?
The answer is naturally that points near this point exert an opposite tension on the particle.
In this way, one side of this point is pulled, and the other side is pulled by its adjacent point, and the effects of the two forces cancel out.
But the effects of the forces are mutual. If a 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 is not moving, so it must also be subject to tension from the inner point.
This process can continue to propagate, and the end result is that there will be tension in all parts of the rope.
Through the above analysis, we can summarize a concept:
When a rope is at rest on the ground, it is in a relaxed state with no tension.
But when a wave passes here, the rope will change into the shape of a wave, and then there will be tension.
It is this tension that makes points on the rope vibrate up and down, so analyzing the effect of this tension on the rope becomes the key to analyzing the wave phenomenon.
Then Xu Yun wrote another formula on the paper:
F=ma.
That's right.
It is the second law of Niu that Mavericks summed up.
As everyone knows.
Mavericks' first law tells us that "an object will remain at rest or move in a straight line at a uniform speed when there is no force or the net external force is 0." So what if the net external force is not 0?
Maverick’s second law goes on to say:
If the net 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 to say.
If we know the mass m of an object, as long as you can analyze the net external force F it receives.
Then we can calculate its acceleration a according to Calf's second law F=ma.
If you know the acceleration, you know how it will move next.
Xu Yun then randomly picked two points on a certain segment of the function image.
Write A on one and B on the other, and the arcs of the two are marked △l.
After writing, push it in front of Xiaomai:
"Classmate Maxwell, would you like to analyze the total external force on this section? Ignore gravity."
Hearing this, Xiaomai was stunned, pointed at himself, and said in surprise:
"Me?"
Xu Yun nodded and sighed slightly in his heart.
What he is going to do today will have great significance in promoting Faraday, the electromagnetic field, or more broadly, the entire historical process of mankind.
But for Mai and Hertz, it may not be a good thing.
Because this means that some of their original contributions have been erased.
Just like one day, a migrant worker with a monthly salary of 4,000 suddenly learns that he could have become a billionaire. As a result, a reborn person took away your opportunity on the grounds of "common development of mankind". How would you feel?
To be fair, it's a bit unfair.
So deep down in Xu Yun's heart, he felt a little guilty towards Mai.
How to compensate for Wheat in the future is another matter. In short, in the current process, all he can do is to get Wheat into the sight of these big guys as much as possible.
Of course.
Xiaomai didn't know Xu Yun's inner thoughts. At this time, he was holding a pen and writing stress 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 from point A toward the lower left and the tension T from point B toward the upper right, which are equal to each other."
"But the band area is curved, so the directions of the two T's are not the same."
"Assuming that the angle between the direction of the tension at point A and the horizontal axis is θ, the angle between point B and the horizontal axis is obviously different, and is recorded as θ+Δθ."
"Because the points on the band move up and down when they fluctuate, only the up and down components of the tension T need to be considered."
“The upward tension at point B is T·sin (θ+Δθ), and the downward tension at point A is T·sinθ. Then, the resultant force on the entire segment AB in the vertical direction is equal to the subtraction of these two forces. ”
soon.
Wheat wrote a formula on the paper:
F=T·sin(θ+Δθ)-T·sinθ.
Xu Yun nodded with satisfaction and said:
"So what is the mass of the wave?"
"The mass of a wave?"
This time.
Mai's brows frowned slightly.
If we assume that the mass per unit length of the band is μ, then the mass of the band with length Δl is obviously μ·Δl.
However, because Xu Yun took a very small interval.
Assume that the abscissa of point A is x, and the abscissa of point B is x+Δx.
That is to say, the projected length of rope AB on the abscissa is Δx.
Then when the length of the rope is very short and the fluctuation is very small, Δx can be approximately used instead of Δl.
In this way, 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 somewhat dry English:
"Wait a minute. The net external force and mass have been determined. Now let's find the acceleration"
Hear what Kirchhoff said.
The classroom, which was not very noisy before, suddenly became a little quiet again.
yes.
Unknowingly, Xu Yun had deduced the resultant external force and mass!
If we deduce the acceleration
Then can’t we express the wave equation in the classical system in the form of Niu Er?
Think of this.
Several big guys took out pens and papers and tentatively calculated the final acceleration.
Speaking of acceleration, we must first talk about its concept:
This is a measure of how fast the speed changes.
As for acceleration, it must be that the faster the speed increases, the greater the acceleration value.
For example, we often hear "I want to speed up" and so on.
Suppose a car's speed in the first second is 2m/s and its speed in the second second is 4m/s.
Then its acceleration is the speed difference (4-2=2) divided 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, the value is calculated by dividing the distance difference by the time difference.
For example, a car is 20 meters away from the starting point at the first second and 50 meters away from the starting point at the second second.
Then its speed is the distance difference (50-20=30) divided 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!
Divide the distance difference by the time difference to get the speed, and then divide the speed difference by the time difference to get the acceleration. Both processes are divided by the time difference.
So
What if these two processes were combined?
Can we just 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 an idea. Strictly speaking, this expression is not very accurate, but it can be very convenient for everyone to understand this idea.
If distance is regarded as a function of time, then take the derivative of this function:
It is the distance difference above divided by the time difference, but it tends to be infinitesimal, and the function of speed is obtained.
Taking the derivative of the velocity function again gives us a representation of acceleration.
Few students knew whether they knew it or not, but the big guys present all thought of this very quickly.
Yes.
The function f(x, t) listed before describes the position of a certain point on the band at different times t!
So as long as you take the derivative of f(x, t) twice with respect to time, you will naturally get the acceleration a at this point.
Because function f is a function of two variables, x and t, it can only be partial derivative of time f/t. If you find the partial derivative again, add 2.
So it's quick.
Including Faraday, all the big guys wrote down a value:
Acceleration a=f/t.
Combining this value with the previous resultant force and mass, a new expression appears:
F=T·sin(θ+Δθ)-T·sinθ=μ·Δxf/t.
Then William Weber took a serious look at this expression and frowned slightly:
"Classmate Luo Feng, is this the final expression? I seem to feel that it can be simplified?"
Xu Yun nodded:
"Of course."
F=T·sin(θ+Δθ)-T·sinθ=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 are too troublesome to look at.
Therefore it still needs to be modified.
As for the idea of renovation?
Of course it’s sinθ.
I saw Xu Yun pick up the pen and draw a right triangle on the paper.
As everyone knows.
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 triangle with a very small 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 will almost coincide."
"At this time we can approximately think that a and b are equal, that is, a≈b."
Then he wrote 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 moment."
Seeing this sentence, Faraday suddenly frowned and interrupted Xu Yun.
It's obvious.
At this time, he was already showing signs of falling behind:
"Classmate Luo Feng, what is the significance of replacing sinθ with tanθ?"
Xu Yun looked at Wheat again, and Wheat immediately understood:
"Mr. Faraday, because the tangent value tanθ can also represent the slope of a straight line, that is, it represents the derivative of the curve at a certain point."
"The expression of tangent value is tanθ=c/b. If we build a coordinate system, then c is exactly 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, which means tanθ=dy/dx."
After listening carefully, Faraday spent two minutes doing some calculations on paper, and then suddenly slapped his forehead:
"I see, I understand, please continue, classmate Luo Feng."
Xu Yun nodded and continued to explain:
"Because the wave function f(x, t) is a binary function about x and t, we can only find the partial derivative at a certain point."
"Then the tangent value is equal to its partial derivative tanθ=f/x at this point, and the original wave equation can be written like this."
Xu Yun then wrote a new equation on the paper:
T（f/xlx+Δx-f/xlx）=μ·Δxaf/t.
It looked a little more complicated than before, but the eyes of the big guys at the scene were all 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 function f/x.
That is to say.
After dividing both sides by Δx at the same time, the left side becomes the partial derivative f/x and takes another derivative with respect to x, that is, f(x, t) takes the second-order partial derivative with respect to x.
At the same time, f/t has been used above to represent the second-order partial derivative of the function with respect to t, so naturally f/x can be used here to represent the second-order partial derivative of the function with respect 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 out
Unit of μ/T.
It happens to be the reciprocal of the square of speed!
That is to say, if we define a quantity as the square root of T/μ, then the unit of this quantity is exactly 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 appears:
f/x=f/vx.
This formula was also called
in later generations.
Classical wave equation.
Of course.
This equation does not take into account quantum effects.
If you want to consider quantum effects, this classical wave equation is useless, and you must turn to the quantum wave equation, which is the famous Schrödinger equation.
Schrödinger started from this classical wave equation and combined it with de Broglie's concept of matter waves to guess the Schrödinger equation.
That's right, it's just a guess.
I won’t go into details about the specific content. 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 quantum aspects, so the classical wave equation is enough.
Then he wrote a new formula on the paper.
As this new formula was written, Faraday suddenly discovered
The tablet of nitroglycerin I had left seemed not enough. (End of chapter)