Displacement of Transverse Waves HELP

[grapejellypie]

Displacement of Transverse Waves HELP!

Homework Statement

At time t=0, the displacement of a transverse wave pulse is described by y=2/(x^(4) +1), with both x and y in cm. Write an expression for the wavefunction as a function of position x and time t if it is propagating in the positive x direction at 3.0 cm/s

Homework Equations

I'm not sure if this has to do with partial derivatives...and I don't quite understand partial derivatives.

The Attempt at a Solution

I know that v= 3.0 cm/s...

 

[gabbagabbahey]

Well your given a stationary wave pulse f(x,0)=y(x)=\frac{2}{x^4+1} and you want to find f(x,t) that satisfies the wave equation f_{xx}(x,t)-\frac{1}{v^2}f_{tt}(x,t)=0. You probably know that any function of the form f(x \pm vt) will satisfy the wave equation, and that if you want just the solution that travels forward at speed v, you choose the negative sign (i.e. f(x - vt)). You are given y(x), so what is y(x-vt)?

 

[grapejellypie]

what do you mean by fxx? is that the second derivative of x?

 

[gabbagabbahey]

Yes,

f_{xx}(x,t)=\frac{\partial ^2 f(x,t)}{\partial x^2}

 

[grapejellypie]

would y(x-vt) be [2/(x^(4) +1)] - 3.0 cm?

 

[gabbagabbahey]

No, just substitute x-vt everywhere you see an x.

 

[grapejellypie]

thank you so much for all of your help! i really, really appreciate it!

I'm sorry, but I have another question:

why do you set f{xx}(x,t)-1/v^(2) * f{tt}(x,t) = 0?

 

[gabbagabbahey]

You mean f_{xx}(x,t)-\frac{1}{v^2}f_{tt}(x,t)=0?

That's the one-dimensional wave equation; have you not seen it before?

Would it help if I wrote it like this:
\frac{\partial ^2 f(x,t)}{\partial x^2}-\frac{1}{v^2} \frac{\partial ^2 f(x,t)}{\partial t^2}=0

 

[grapejellypie]

i've seen the two second derivatives equal to each other, but i never thought of manipulating the equation to move the variables to one side.

thank you again for all of your help. i really appreciate it!

so just to double-check...the answer would be y(x,t) =2/[(x-3t)^(4)+1] ?

 

[gabbagabbahey]

Yes, you can check your answer yourself too by seeing what happens at t=0, you should get y(x) back. Also, you can take the partial second derivatives and verify that f_{xx}(x,t)-\frac{1}{v^2}f_{tt}(x,t)=0. You also know that the pulse should be traveling at 3cm/s to the right; which means that since the pulse is centered at x=0 for t=0, you should have a pulse that is centered at x=3cm for t=1s. These are checks that you should do to convince yourself that you have the correct answer.