Sketch the displacement of the string

[bon]

Homework Statement

2. At time t = 0, the displacement of an infinitely long string is:
y(x, t) = sin(pix/a) for −a ≤ x ≤ a

The string is initially at rest. Using d’Alembert’s solution for speed c, sketch the displacement of the string at t = 0, t = a/2c, and t = a/c.

Homework Equations

The Attempt at a Solution

So I see how to do this...my only problem is as follows:

D'Alembert solutions says y = f(x-ct) + g(x+ct)..

but we don't know if the wave goes left or right..so how do i know whether to sub in x-ct for x or x+ct for x? thanks!

 

[ehild]

Is the initial displacement zero outside the interval [-a,a]?

You had a very similar problem not long ago.

ehild

 

[bon]

Is the initial displacement zero outside the interval [-a,a]?

You had a very similar problem not long ago.

ehild

yes it is similar outside the interval..

I know that this is similar to the one before - my problem is just knowing what to substitute instead of x -- should it be x-ct or x+ct?...

Thanks!

 

[ehild]

Why do you think that the wave goes either left or right and not in both direction? Apply D'Alembert solution y = f(x-ct) + g(x+ct).

ehild

 

[rdl]

As a scientist, it is important to always consider all possibilities and analyze the given information before making any assumptions. In this case, we know that the string is initially at rest, and the given function is symmetric about the y-axis. This means that the wave could potentially travel in either direction with equal probability. Therefore, when using d’Alembert’s solution, we can substitute both x-ct and x+ct and sketch the resulting displacements for both cases. This will give us a more complete understanding of the possible movements of the string.