Question

Question

Answer 1

Why do we use this red part? Why this differential equation is improtant.
I know the proof, but why do we need this thing.

Answer 2

@ganeshie8

Answer 3

@hartnn

Answer 4

@Farcher

Answer 5

@ParthKohli

Answer 6

The teacher just tells LHS=RHS

Answer 7

but does not explain why he did this!

Answer 8

If you write down the mechanical equations between stress and velocity in an elastic and massive medium, then you will be able to prove that those quantities obey this partial differential equation.

This equation has an infinite number of solutions, including all derivable functions of the combined variable \(kx-\omega t\). The simplest one being \(A \cos(kx-\omega t)\) or\(A \sin(kx-\omega t)\), it is usually the one used to proceed further.

Answer 9

i am a little confused

Answer 10

can you please explain in detail @Vincent-Lyon.Fr

Answer 11

@IrishBoy123

Answer 12

The differential equation you're referring to is generally referred to as the wave equation. As mentioned above, it can be derived from a number of different physical models, and solutions of that equation propagate in a wave-like fashion.

What exactly is your question? Are you asking how to derive that equation from a physical model? You mention knowing the "proof", and I'm not sure what you mean by that.

Answer 13

What exactly is your question? I have the same problem figuring out what you exactly need as Jemurray3 above.

Answer 14

It appears to me that the instructor has just started a discussion of "the standard wave equation in one dimension" and is now discussing the general form of solutions of that equation. It'd be nice to know how to derive that equation, but for now I'd suggest you concentrate on how solutions to it are found.

Answer 15

it is the wave equation, and the professor, is writing the corresponding solution, by substitution of \(y=A \cos(kx-\omega t)\), into such PDE

Answer 16

No no,
my question is why did he proof LHS=RHS? if he knew the equation?
How did he come across that equation?

Answer 17

@Michele_Laino @mathmale @Vincent-Lyon.Fr

Answer 18

@Jemurray3

Answer 19

The professor is proving LHS=RHS. I want to know why? I mean how did he come across that equation.

Answer 20

Sorry for late reply, I was out of station

Answer 21

He is not proving the wave equation itself; the wave equation is there beforehand.

What he is actually demonstrating is what you must do when you have a candidate wave function (the cosine function on the right).
Your intuition can lead you to think: Wouldn't this \(y(x,t)=...\) function be a wave equation?

Then you put it to the test by plugging it into the PDE on the left.
If LHS=RHS for all values of x and t, then the function IS a wave equation.
If LHS differs from RHS, the the function cannot be a wave equation, ie cannot represent a 1-dimensional wave.

The result (at about 6:00) shows that \(y(x,t)=A \cos(kx-\omega t)\) is a wave equation only if \(\omega\) and \(k\) are related by \(k=\pm \dfrac\omega v\)

Answer 22

ok ok @Vincent-Lyon.Fr
I got it.
So, this is a primary wave function which may be I will have to study in higher lectures. The professor is trying to proof LHS=RHS so that it satisfies that primary wave equation.

Am I going in the right direction? @Vincent-Lyon.Fr