Partial derivatives and change of variables

[miew]

Homework Statement

Sorry I tried to use Latex but it didn't work out :/

Make the change of variables r = x + vt and s = x vt in the wave equation
partial^2y/partialx^2-(1/v^2)(partial^2y/partialt^2)=0

Homework Equations

partialy/patialx=(partialy/partialr)(partialr/partialx)+(partialy/partials)(partials/partialx)
partialy/partialy=(partialy/partialr)(partialr/partialt)+(partialy/partials)(partials/partialy)

partialr/partialx=1
partials/partialx=1
partialr/partialt=v
partials/partialt=-v

The Attempt at a Solution

I am a little bit confused about what to do. This is what I did:

Rearranging:
partial^2y/partialr^2(1-1/v)+partial^2y/partials^2(1+1/v)=0

I feel this is completely wrong, but I can't think of another way of doing it.

 

[hunt_mat]

Start from the basics:

$$\begin{array}{ccc} \frac{\partial}{\partial x} & = & \frac{\partial r}{\partial x} \frac{\partial}{\partial r}+\frac{\partial s}{\partial x} \frac{\partial}{\partial s} \\ \frac{\partial}{\partial t} & = & \frac{\partial r}{\partial t} \frac{\partial}{\partial r}+\frac{\partial s}{\partial t} \frac{\partial}{\partial s} \end{array}$$

Computing these yields:

$$\begin{array}{ccc} \frac{\partial}{\partial x} & = & \frac{\partial}{\partial r}+\frac{\partial}{\partial s} \\ \frac{\partial}{\partial t} & = & v\frac{\partial}{\partial r}-v\frac{\partial}{\partial s} \end{array}$$

Then just apply these again to get the second order derivatives.

 

[miew]

Yes, that is what i did and I got:

partial^2y/partialx^2=partial^2y/partialr^2+partial^2y/partials^2
partial^2y/partialt^2=vpartial^2y/partialr^2-v(partial^2y/partials^2)

Is that right? And then I just plugged that in the initial equation.

 

[hunt_mat]

Just use:
$$\frac{\partial^{2}}{\partial x^{2}}=\left(\frac{\partial}{\partial r}+\frac{\partial }{\partial s}\right)\left(\frac{\partial}{\partial r}+\frac{\partial }{\partial s}\right)$$

 

[miew]

Could you explain that ? And why what I did is wrong ? Can't I just take the partial derivative of partialy/partialx ?

 

[hunt_mat]

Where dies y come into it, I thought the original variables were x and t which you were transforming into r and s?
Okay, you should know that:
$$\frac{\partial}{\partial x}=\left(\frac{\partial}{\partial r}+\frac{\partial }{\partial s}\right)$$

So just apply this operator twice.

 

[miew]

Y comes from the original equation,

partial^2y/partialx^2-(1/v^2)(partial^2y/partialt^2)=0

I did what you said and I got

4partial^2y/(partialr * partials)

Is that right ?

 

[hunt_mat]

That looks very good. set that equal to zero and you have your original equation which you have now reduced to:
$$\frac{\partial^{2}y}{\partial r\partial s}=0$$

 

[miew]

Thank you very much !

How can I solve that equation ?

 

[hunt_mat]

integrate once and what do you get?

 

[miew]

do i get y=rs? I am confused...

 

[hunt_mat]

Integrate once w.r.t. r say and you obtain:
$$\frac{\partial y}{\partial s}=f(s)$$
Now what would you do?

 

[miew]

now Integrate with respect to s and I get: f(s)f(r)??

 

[hunt_mat]

No, you're forgetting your basic integration:
$$y=\int f(s)ds+g(r)$$
The integral of f(s) w.r.t. s is just another function, call it h(s) amd trhat is your solution.

 

[miew]

Oh okay ! thank you so much ! you were really helpful :)