Chapter 559 - 257: Witness the Miracle! (Part 1)

Of course.

Wheat had no idea what Xu Yun was thinking. At this moment, he was holding a fountain pen, quickly sketching out a force analysis on the paper:

"Mr. Luo Feng says to ignore gravity, so we only need to analyze the tension T at both ends of segment AB."

"Segment AB experiences a left-downward tension T at point A and a right-upward tension T at point B, both equal in magnitude."

"But this segment is curved, so the directions of the two T’s are not the same."

"Suppose the angle between the tension at point A and the horizontal axis is θ, then it’s obvious that the angle at point B is different—let’s call it θ+Δθ."

"Because the points on the segment move up and down during oscillation, we only need to consider the vertical component of tension T."

"At point B, the upward tension is T·sin(θ+Δθ); at point A, the downward tension is T·sinθ. So, the net force in the vertical direction for the whole AB segment is the difference between these two forces..."

Soon enough,

Wheat wrote down a formula on the paper:

F = T·sin(θ+Δθ) - T·sinθ.

Xu Yun nodded with satisfaction and asked:

"So, what’s the mass of the wave?"

"The mass of the wave?"

This time,

Wheat’s brows furrowed slightly.

If we assume the mass per unit length of the segment is μ, then the mass of a segment of length Δl is clearly μ·Δl.

However, since Xu Yun chose an extremely small interval,

suppose the horizontal coordinate of point A is x, and for point B, it’s x+Δx.

That is, the projection length of rope AB onto the x-axis is Δx.

When the segment is very short and the oscillations are tiny, we can approximate Δl by Δx.

So, the mass of this segment can be represented as...

μ·Δx

Meanwhile,

Kirchhoff, off to the side, suddenly seemed to realize something, his pupils contracting slightly as he spoke in somewhat dry English:

"Wait... both the net external force and the mass have been determined. If we can find the acceleration..."

Upon hearing Kirchhoff’s words,

the classroom—which was never very noisy—suddenly grew even quieter.

That’s right.

Without even noticing, Xu Yun had already derived the net external force and the mass!

If the acceleration is derived next...

then couldn’t we write the equation for the wave in the classical framework in Newton’s second law form?

Thinking this,

several big names quickly pulled out paper and pen, attempting to calculate the final acceleration.

Speaking of acceleration, let’s first talk about its concept:

It’s a quantity used to measure how fast velocity changes.

As for acceleration, the faster the speed increases, the greater the value of acceleration.

For example, we often hear things like "I’m gonna speed up!" and so forth.

If a car’s speed in the first second is 2m/s, and in the second second it’s 4m/s,

then its acceleration is the difference in speed (4-2=2) divided by the difference in time (2-1=1), which results in 2m/s2.

Now, let’s recall: how do we find a car’s speed?

Of course, by dividing the difference in distance by the difference in time.

For example, if a car at the first second is 20 meters from the starting point, and at the second second it’s 50 meters away,

then its speed is the difference in distance (50-20=30) divided by the difference in time (2-1=1), resulting in 30m/s.

Did anyone notice anything from these two examples?

Exactly!

Dividing the difference in distance by the difference in time gives velocity, and dividing the difference in velocity by the difference in time gives acceleration—both processes involve dividing by the time difference.

So...

what if we combine these two processes?

Does that mean we could say:

Dividing the difference in distance by one time difference, then dividing by another time difference gives acceleration?

Of course.

It’s just a way of thinking; strictly speaking, this isn’t very accurate, but it’s a convenient way for everyone to understand the idea.

If we treat distance as a function of time, then taking the first derivative:

It’s like using the difference in distance over the difference in time, only tending toward zero, which gives the velocity function.

Then, taking the derivative of the velocity function gives the expression for acceleration.

Whether the students present understand or not is unclear, but all the experts in the room quickly grasped this point.

Yes.

The function f(x, t) listed earlier describes the position of a specific point on the wave segment at different times t!

So, we just need to take the second derivative of f(x, t) with respect to time to get the acceleration a at that point.

Because the function f has two variables, x and t, we can only take the partial derivative with respect to t, ?f/?t, and then once more, add a 2 for the second derivative.

So, very quickly,

including Faraday, all the big names successively wrote down a value:

Acceleration a = ?2f/?t2.

And combining this value with the net force and mass derived earlier, a new expression emerges:

F = T·sin(θ+Δθ) - T·sinθ = μ·Δx?2f/?t2.

William Weber then looked carefully at this expression, his brow slightly furrowing:

"Luo Feng, is this the final expression? I feel like we could simplify it further."

Xu Yun nodded:

"Certainly."

F = T·sin(θ+Δθ) - T·sinθ = μ·Δx?2f/?t2.

This is the most primitive form of the equation; its content isn’t very clear, and the left side of the equation looks rather messy.

So it still requires some modification.

As for the direction of the modification?

Of course, it’s about sinθ.

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