Understanding Velocity in Standing Waves | Wave Motion Homework"

[joe:)]

Homework Statement

1)I'm analysing a standing wave formed by superposition of Asin(kx-wt) and Asin(kx+wt) so it becomes 2Asinkxcoswt

It asks me to comment on the velocity of this wave. But I thought it was a standing wave - so it has no velocty in the x direction..

So what is its velocity - what does it mean physically - is it just 0? Or w/k? thanks

2) I'm given that general solution of wave equation of string length L can be written as the sum from r=0 to r=infinity of Arsin(xrpi/L)sin(r pi c t /L) + Brsin(xrpi/L)cos(r pi c t /L)

Then I'm told that stationary waves y=f(x)g(t) exist on a string length L.

I'm told f(x) = Asin(kx) I'm told to find an expression for k and find an expression for g(t)

Homework Equations

The Attempt at a Solution

I'm not really sure how to solve 2) to be honest..

I thought the f(x) would have to be an infinite sum and k = r pi / L where r is any integer..

I'm a bit confused..any help? Thanks

 

[boardbox]

Problem 2 delves into a little Fourier analysis, so if you're familiar with that at all try thinking in that direction.

As for the expression, you can use separation of variable to arrive at forms for the solution.

The wave will be described by the infinite sum of f(x)g(t) so the f(x) is actually only a component.

Edit:
I've absent mindedly forgot to mention that you should look at the wave equation as well, should help you a bit.

 

[joe:)]

Thanks..

We haven't done Fourier yet..

I'm not sure how to find the expression by separation of variables..could you explain please? How will f(x) just be a constant A times sinkx? surely its an infitite sum? Confused :S

Also any ideas on Q1?

 

[boardbox]

Q1 I'm not really sure.

Fourier is actually pretty easy but I bet this one can be done without.

Separation of variables is a method of solving partial differential equations. The idea is that you have PDE that describes what you're looking at (in this case you're interested in the wave equation). You assume a solution f(x,t) = g(x)h(t) and plug that into your PDE. Then you separate terms so that you only have one variable of each type on one side, so all the x's on one side and all the t's on the other. You can then set each side of the PDE to a separation constant (this is your k) and solve the ordinary differential equation.Though to be honest, the above is probably far more than you're actually being asked. I think that if you look at the equation for number 2 a bit longer you'll see that you can actually just match terms and pull out a g(t). Just look at what you can factor out and remember that sums are valid solutions.

 

[joe:)]

sorry..i really can't see it at the moment...? I swear the expression for f(x) should be an infinite sum too?

thanks for your patience

 

[boardbox]

It's not. You have the right k.

Try factoring f(x) out the given equation. Remember the solution is f(x) times g(x).