How to find the direction of wave propagation

[Decimal]

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!

 

[Orodruin]

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.

Please show your actual computation. Your result seems correct, but a priori we cannot provide you with alternative solutions if you do not show that you have solved the problem.

However, let me give this general hint: A solution to the one-dimensional wave equation can always be written on the form ##\Psi(x,t) = f(x-vt) + g(x+vt)##, where the first term is a wave traveling to the right and the second a wave traveling to the left. Such a function will always satisfy the wave equation, as you can easily show by insertion.

 

[Decimal]

Here is what I did $$ \Psi(x,t) = f(u)$$ where $$ u = 36 x^2 - 12 x t + t^2$$ Then I started applying the chain rule to compute the partial derivatives. First for the partial derivative with respect to x: $$ \frac {\delta^2 f(u(x))} {\delta x^2} = \frac {\delta^2 f} {\delta u^2} (\frac {\delta u} {\delta x})^2 + \frac {\delta^2 u} {\delta x^2} \frac {\delta f} {\delta u} $$

Now I can compute the derivatives but leave the derivatives with respect to u, since they will be on both sides of the equation. If I do the same operation for right side of the wave equation I arrive at the following result: $$\frac {\delta^2 f} {\delta u^2} (72x-12t)^2 +72 \frac {\delta f} {\delta u} = \frac {1} {v^2} (\frac {\delta^2 f} {\delta u^2} (2t-12x)^2 + 2 \frac {\delta f} {\delta u}) $$

Solving this for v gives me my results. I hope this is enough info on my method. This gives me two possible values for v, so I am not quite sure how to now determine the propagation direction

 

[Orodruin]

Looking at your ##u##, can you rewrite it as the square of something?

 

[Decimal]

Ah yeah I can! ## u = (6x-t)^2 ## which I can rewrite to ## u = 36*(x- \frac {1} {6} t)^2 ## which means I can now write my function like $$ Ψ(x,t)=f(x- \frac {1} {6} t) $$ From this expression I can immediately see the velocity and propogation direction. That is indeed much easier than what I did! Thanks a lot!