- #1
Decimal
- 75
- 7
Homework Statement
Given the following expression $$ \Psi(x,t) = A cosh(36 x^2 - 12 x t + t^2)$$ Determine whether this is a traveling wave and if so what is its propagation velocity and propagation direction?
Homework Equations
Wave equation $$ \frac {\delta^2 \Psi(x,t)} {\delta x^2} = \frac {1} {v^2} \frac {\delta^2 \Psi(x,t)} {\delta t^2}$$
The Attempt at a Solution
To figure out whether this expression solves the wave equation I decided to write the expression as a general function ## f(u(x)) ## and then apply the chain rule. In this way I was able to prove this expression describes a traveling wave without having to explicitly differentiate the hyperbolic cosine. This was still quite difficult and a rather lengthy calculation so if someone maybe knows a quicker way I would be eager to learn it.
My question is how to now determine the propagation direction. I figured out the velocity from the wave equation ## \frac {1} {v^2} = 6^2 ## but this gives me a negative and positive value for v. Which one is correct? I don't really know how to tell this from the expression.
Thank you!