# Homework Help: Wave equation ( partial differential equations)

1. Sep 1, 2011

### zonedestruct

Consider a string of length 5 which is fixed at its ends at x = 0 and x = 5. The speed of waves along the string is v = 2 and the displacement of points on a string is defined by the function f(x,t). At the initial time the string is pulled into the shape of a triangle, defined by

f(x,0) = x for 0 <= x < 1
f(x,0) = 5/4 - x/4 for 1 <= x <= 5

and then released from rest.

a. What are the 2 initial conditions for this problem?
b. What are the 2 boundary conditions for this problem?
c. Use the method of separation of variables and Fourier methods to find an equation for how the shape of the string changes with time.

How do I do this question, please somebody help me i'm terribly lost its depressing :(

2. Sep 1, 2011

### Herr Malus

I assume that your textbook (or resource or whatever) has the actual wave equation, and the procedure for solving i listed somewhere. I suggest you start there.

Now, as to some helpful hints if you don't happen to have that text available.
-In the problem statement, you noted that the string was clamped down at two ends. Further you mentioned a speed for the string. These are your initial and boundary conditions.
-I suggest you quickly go through this:
http://en.wikipedia.org/wiki/Wave_equation

Cheers,
-Malus

3. Sep 1, 2011

### zonedestruct

I think the boundary conditions are:
f(0,t) = 0 and f(5,t)=0 but im not sure about the initial conditions can you please tell me what they would be? would one of them possibly be df(x,t)/dt = 2?

4. Sep 1, 2011

### vela

Staff Emeritus
That's correct.
That's not correct. Where did you get df(x,t)/dt = 2 from?

What does the word initial imply?

5. Sep 1, 2011

### zonedestruct

isnt the velocity at all point on the string always going to be 2 do then partialf(x,t)/partial(t) = 2?? what would at least one of them be?

6. Sep 1, 2011

### vela

Staff Emeritus
Oh, OK. I see what you're thinking.

If you have a horizontal string, the wave moves down the string, that is, in the horizontal direction, and v=2 (units?) is the speed at which it propagates down the string.

If you look at a single point on the string, it moves vertically. The velocity of each point is called the transverse velocity, and it's given by ∂f/∂t, which, like f, is a function of x and t.

The one of the initial conditions has to do with ∂f/∂t. It has nothing to do with the speed at which the wave moves down the string.

7. Sep 1, 2011

### vela

Staff Emeritus
Here's a Flash animation of a wave on a string.