Chapter 560 257 Witness the Miracle! (Part 1)_4

This is the most primitive set of equations, its content is not very clear, the left side of the equation looks too troublesome.

Therefore it still needs some modification.

As for where the idea for modification comes from?

Of course, it is sinθ.

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

As everyone knows.

The sine value sinθ equals the opposite side c divided by the hypotenuse a, the tangent value tanθ equals the opposite side c divided by the adjacent side b.

Xu Yun drew another right triangle with a very small angle, estimating only a few degrees:

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

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

Subsequently, he wrote on the paper:

[Thus c/b≈c/a, that is, tanθ≈sinθ.]

[The previous formula can be written as F= T·tan(θ+Δθ) - T·tanθ = μ·Δxa∂²f/∂t².]

"One moment."

Upon seeing this sentence, Faraday suddenly frowned and interrupted Xu Yun.

It's obvious.

At this time, he was vaguely showing signs of being left behind:

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

Xu Yun glanced at Wheat, and Wheat immediately understood:

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

"The expression for the tangent value is tanθ=c/b, if a coordinate system is established, then c is just the projection of the line on the y-axis dy, and b is the projection on the x-axis dx."

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

Faraday listened carefully, spent two minutes calculating on the paper, and then suddenly hit his forehead in realization:

"I see, I understand, please continue, student Luo Feng."

Xu Yun nodded and continued to explain:

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

"Then the tangent value equals its partial derivative at this point tanθ=∂f/∂x, the original wave equation can be written like this..."

Then Xu Yun wrote down a new equation on the paper:

T(∂f/∂xlx+△x-∂f/∂xlx)=μ·Δxa∂²f/∂t².

It looks a bit more complex than before, but the eyes of these bosses on the scene brightened considerably.

At this step, the following ideas become much clearer.

Just divide both sides of the equation by Δx, then the left turns into the difference of function ∂f/∂x at x+Δx and x divided by Δx.

In essence, this is the derivative expression of the function ∂f/∂x.

That is to say.

When both sides are divided by Δx, the left becomes the second derivative of the partial derivative ∂f/∂x with respect to x, which is the second-order partial derivative of f(x, t) with respect to x.

At the same time, the function for the second-order partial derivative with respect to t has already been expressed as ∂²f/∂t² above, so here ∂²f/∂x² can naturally be used to express the second-order partial derivative with respect to x.

Then divide both sides by T again, and the equation becomes much simpler:

∂²f/∂x=μ∂²f/T∂x².

And if you are not dizzy yet, you will find.....

μ/T's unit.....

Is exactly the reciprocal of speed squared!

That is to say, if we define a quantity as the square root of T/μ, this quantity's unit is precisely the unit of speed.

Imagine, this speed is naturally the propagation speed of the wave v:

v²=T/μ.

Therefore, after substituting this value in, a final formula appears:

∂²f/∂x=∂²f/v²∂x².

This formula in later generations is also called...

The classical wave equation.

Of course.

This equation does not take quantum effects into account.

If quantum effects are considered, this classical wave equation becomes useless, one must turn to using the quantum wave equation, that is the famous Schrödinger equation.

It was from this classical wave equation that Schrödinger combined de Broglie's concept of matter waves to conjecture the Schrödinger equation.

Yes, it was conjectured.

The specifics won't be elaborated further for now, but in short, this equation freed physicists from the fear dominated by Heisenberg's matrix and returned to the beautiful world of differential equations.

For now, Xu Yun doesn't need to consider the quantum aspects, so the classical wave equation is sufficient.

He then wrote a new formula on the paper.

And as this new formula was written, Faraday suddenly realized...

The portion of nitroglycerin he had left seemed insufficient.

.......

Note:

Some say Volta is just a patch for a bug, speechless.... Would I commit such a basic mistake, I replaced Taylor expansion with Han Li expansion before, and yet have made no changes to the photovoltaic?..

I've written such foreshadowing before, and even mentioned in the "Lai Fu Sword Technique" chapter that this cultivation technique would be used in the next dungeon, and had already designed the Wheat dungeon then.

Being told that the remaining lines are a patch, is beyond words.jpg.

Upon seeing this sentence, Faraday suddenly frowned and interrupted Xu Yun. It was obvious. At this time, he was vaguely showing signs of being left behind: "Student Luo Feng, what is the significance of replacing sinθ with tanθ?" Xu Yun glanced at Wheat, and Wheat immediately understood: "Mr. Faraday, because the tangent value tanθ can also represent the slope of a line, which is the derivative of a curve at a certain point.