Chapter 561 - 257: Witness the Miracle! (Part 1)

"The expression for the tangent is tanθ = c/b. If you establish a coordinate system, then c happens to be the projection of the straight line onto the y-axis, which is dy, and b is the projection onto the x-axis, which is dx." "Their ratio is exactly the derivative dy/dx; in other words, tanθ = dy/dx." Faraday listened attentively, spent two minutes performing calculations on paper, then suddenly slapped his forehead in realization: "So that’s it, I see now. Please go on, Luo Feng." Xu Yun nodded and continued explaining: "Because the wave function f(x, t) is a function of two variables, x and t, we can only take partial derivatives at a specific point." "So the tangent is equal to the partial derivative at that point, tanθ = ?f/?x, and the original wave equation can be written like this..." Then Xu Yun wrote a new equation on the paper: T(?f/?x|x+Δx ? ?f/?x|x) = μ·Δx·?2f/?t2. It looked a bit more complicated than before, but the eyes of the heavyweights present all immediately brightened. Reaching this step, the next line of thinking became clear. All you need to do is divide both sides of the equation by Δx; then the left side becomes the difference between ?f/?x at x+Δx and at x, divided by Δx. This is actually the derivative expression for the function ?f/?x itself. In other words, after dividing both sides by Δx, the left side becomes the 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, since we’ve already used ?2f/?t2 to represent the second-order partial derivative of the function with respect to t, here we can naturally use ?2f/?x2 to represent the second-order partial derivative with respect to x. Then, by dividing both sides by T, the resulting equation becomes much simpler: ?2f/?x = μ?2f/T?x2. And if your mind hasn’t gone fuzzy yet, you’ll notice... the unit of μ/T... happens to be the reciprocal of the square of velocity! In other words, if we define a quantity as the square root of T/μ, then its unit is exactly that of velocity. You can imagine, this velocity is naturally the propagation speed v of the wave: v2 = T/μ. So, after substituting this value in, a final formula emerges: ?2f/?x = ?2f/v2?x2. This formula is later referred to as... the classical wave equation. Of course, this equation doesn’t account for quantum effects.