Question

Wave equation question.

Answer 1

\[\frac{ d^2u }{ dt^2 } =\left( \frac{ dx }{ dt } \right)^2 \frac{ d^2u }{ dx^2 } \]

Why isn't it written like this? It seems more memorable this way.

Answer 2

Consider the function: \[
f(x) = x^2
\]

Answer 3

That's not a wave, yeah?

Answer 4

What is the "wave" in question?

Answer 5

How is it written and how do you want it to be written?

Answer 6

None in particular, that's just something that's called the "wave equation". Usually I see it written as:

Utt=c^2 * Uxx

Answer 7

Actually I see it quite often like this:

Uxx=1/c^2*Utt

Answer 8

It just looks weird to me for no reason.

Answer 9

And \(c\) is \(\frac{dx}{dt}\)?

Answer 10

Yes

Answer 11

I don't get it either.... it looks like velocity squared...

Answer 12

If \(c\) is a velocity it makes me think of the speed of light.

Answer 13

I dunno why they did it that way. Maybe it's just your book in particular.

Answer 14

I've seen it multiple places, it's either c or v for velocity or speed of light.

Actually, when I took electricity and magnetism they showed that a changing electric field made magnetic fields and magnetic fields made electric fields. Then if you plug these electricity and magnetism constants into the wave equation to see how fast they make each other, it turned out that the speed of an "electromagnetic" wave travels at about what they measured the speed of light to be at the time and that's how we know. So yeah, c is speed of light.

Answer 15

\[
\frac{d^2u}{dt^2} = \frac{d}{dt}\frac{du}{dt} = \frac{d}{dt}\left(\frac{du}{dx}\frac{dx}{dt}\right) = \frac{d^2u}{dx^2}\left(\frac{dx}{dt}\right)^2 +\frac{du}{dx}\frac{d^2x}{dt^2}
\]So another way to think about it is just: \[
\frac{du}{dx}\frac{d^2x}{dt^2} = 0
\]So \[
\frac{d^2u}{dt^2}= \frac{d^2u}{dx^2}\left(\frac{dx}{dt}\right)^2 +\frac{du}{dx}\frac{d^2x}{dt^2} =\frac{d^2u}{dx^2}\left(\frac{dx}{dt}\right)^2 +0
\]