Chapter 560 - 257: Witness the Miracle! (Part 1)

As everyone knows.

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

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

"But once the angle θ gets extremely small, the adjacent side b and the hypotenuse a become almost identical."

"At this point, we can approximate a and b as equal, that is, a≈b."

Then he wrote on the paper:

[So we have c/b≈c/a, that is, tanθ≈sinθ.]

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

"Wait a minute."

Seeing this sentence, Faraday suddenly furrowed his brow and interrupted Xu Yun.

Obviously.

He was already showing signs of lagging behind:

"Luo Feng, what’s the point of substituting tanθ for sinθ?"

Xu Yun glanced at Wheat, and Wheat immediately caught on:

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

"The expression for tangent is tanθ = c/b. If we set up a coordinate system, then c is the projection on the y-axis, dy, and b is the projection on the x-axis, dx."

"Their ratio is exactly the derivative dy/dx, so tanθ = dy/dx."

Faraday listened carefully, took two minutes to work it out on paper, then suddenly slapped his forehead in realization:

"I see, now I understand. Please go on, Luo Feng."

Xu Yun nodded and continued explaining:

"Since the wave function f(x, t) depends on both x and t, we can only calculate the partial derivative at a specific point."

"So the tangent value equals the partial derivative at this point: tanθ = ?f/?x. The original wave equation can now be written like this..."

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

T(?f/?xlx+△x - ?f/?xlx) = μ·Δxa?2f/?t2.

It looked a bit more complicated than before, but the eyes of the big shots present all brightened noticeably.

At this stage, the next steps become quite clear.

As long as we divide both sides of the equation by Δx, the left side becomes the difference between the values of the function ?f/?x at x+Δx and at x, divided by Δx.

This is actually the derivative expression for the function ?f/?x.

That is to say.

After dividing both sides by Δx, the left side becomes the derivative with respect to x of the partial derivative ?f/?x, which means it’s the second partial derivative of f(x, t) with respect to x.

We’ve already used ?2f/?t2 above to represent the second partial derivative of the function with respect to t, so naturally, we can use ?2f/?x2 to denote the second partial derivative of the function with respect to x.

Then, dividing both sides by T, the equation becomes much simpler:

?2f/?x = μ?2f/(T?x2).

And if your mind isn’t already fried, you’ll notice.....

The units of μ/T.....

Are exactly the reciprocal of velocity squared!

So if we define a quantity as the square root of T/μ, then its unit matches that of velocity.

It’s easy to imagine that this velocity is naturally the wave propagation speed v:

v2 = T/μ.

Therefore, substituting this value in, an ultimate formula emerges:

?2f/?x = ?2f/(v2?x2).

This equation is later known as...

The classical wave equation.

Of course.

This equation does not take quantum effects into account.

If we want to take quantum effects into consideration, this classical wave equation won’t work; we have to switch to the quantum wave equation—that’s the famous Schr?dinger equation.

Schr?dinger started from this classical wave equation and, by combining de Broglie’s concept of matter waves, managed to guess the Schr?dinger equation.

That’s right: he just guessed it.

No need to dwell on the details here; anyway, this equation freed physicists from the terror of Heisenberg’s matrices and led them back into the wonderful world of differential equations.

Now, since Xu Yun doesn’t need to consider quantum stuff, the classical wave equation will do just fine.

He then wrote a new formula on paper.

And as this new formula appeared, Faraday suddenly realized...

that he might not have enough nitroglycerin left.

.......