Progressive wave, wavelength moving in the opposite direction

[Redwaves]

Homework Statement

Finding the wavelength. if the wave is moving in the +z direction and -z direction.

Relevant Equations

$\Psi(z=15cm,t) = \hat{x} 6 cos (\frac{\pi}{3}t)$
$\Psi(z=12cm,t + 2s) = \Psi(z=18cm,t)$

I'm trying to find the wavelength. However, I don't understand why the wavelength is different if the wave is moving in the +z direction.

I have
$\Psi(z=15cm,t) = \hat{x} 6 cos (\frac{\pi}{3}t)$
$\Psi(z=12cm,t + 2s) = \Psi(z=18cm,t)$

For a wave moving on the -z direction

I know that the wavelength = $\frac{2\pi}{\kappa}$ and the shape of the wave is describe by this function $x(z,t) = A cos(\omega t +\kappa z + \alpha_0)$

$\kappa = \frac{\omega}{v}, \omega = \frac{\pi}{3}$ and $v = 12-18/(t+2)-t = -3$

thus, the wavelength = $\frac{2\pi}{\pi/9} = 18$, which is the right answer.

However, for a wave moving in the +z direction the wavelength is 9cm. Why is this different ?The velocity isn't the same? how can I find it.

 

[PeroK]

Homework Statement:: Finding the wavelength. if the wave is moving in the +z direction and -z direction.
Relevant Equations:: $\Psi(z=15cm,t) = \hat{x} 6 cos (\frac{\pi}{3}t)$
$\Psi(z=12cm,t + 2s) = \Psi(z=18cm,t)$

I'm trying to find the wavelength. However, I don't understand why the wavelength is different if the wave is moving in the +z direction.

I have
$\Psi(z=15cm,t) = \hat{x} 6 cos (\frac{\pi}{3}t)$
$\Psi(z=12cm,t + 2s) = \Psi(z=18cm,t)$

For a wave moving on the -z direction

I know that the wavelength = $\frac{2\pi}{\kappa}$ and the shape of the wave is describe by this function $x(z,t) = A cos(\omega t +\kappa z + \alpha_0)$

$\kappa = \frac{\omega}{v}, \omega = \frac{\pi}{3}$ and $v = 12-18/(t+2)-t = -3$

thus, the wavelength = $\frac{2\pi}{\pi/9} = 18$, which is the right answer.

However, for a wave moving in the +z direction the wavelength is 9cm. Why is this different ?The velocity isn't the same? how can I find it.

In general, we are given: $$\Psi(z_1, t+ t_1) = \Psi(z_2, t)$$Which implies that$$wt_1 = k(z_2 - z_1) \pm 2\pi n$$This gives us an infinite number of solutions, depending on the direction of motion and how many wavelengths there are between the points $z_1$ and $z_2$.

If we take $n = 0$, then we have $$\lambda = \frac{2\pi}{k} = 2\pi\frac{z_2 - z_1}{wt_1}$$And, in this case we have $$\lambda = 18cm$$. But, as above, there are infinitely many other solutions - one for ecah value of $n$.

 

[Redwaves]

How we know which value for n we choose? In my example., how can I know that for a wave moving in the +z direction n is -1?

 

[PeroK]

How we know which value for n we choose? In my example., how can I know that for a wave moving in the +z direction n is -1?

You don't know. Unless you have additional information, then there are multiple solutions.

 

[Redwaves]

In my case it should have a way to find wavelength since, that what I have to find.

Edit: from the information above, I have to find the wavelength.
Tell me if I'm not clear.