wavefunction - Help me to visualize this wave equation in time, to which direction it moves?

The wave is

$\bar{E} = E_{0} sin(\frac{2\pi z}{\lambda} + wt) \bar{i} + E_{0} cos(\frac{2 \pi z}{\lambda}+wt) \bar{j}$

Let's simplify with $z = 1$. Now the xy-axis is defined by parametrization $(sin(\frac{2\pi }{\lambda}+wt), cos(\frac{2\pi }{\lambda} + wt)$ where $t$ is time and $\lambda$ is wavelength. This parametrization satisfy the equation $1^2=x^{2}+y^{2}$, a circle.

Now, let's variate the value of $z$. We know now that it cannot move into x or y coordinates or do we? Not really, the latter simplification is naive -- $x-y$ parametrization depends on the dimension $z$ -- but can we see something from it? If so, how to proceed now?

The solution is that the wave moves along the $z$ -axis to the negative direction as $t$ increases, a thing I cannot see.

The way I am trying to solve this kind of problems is:

1. Parametrize the equation


2. suppose other things constant and change one dimension, observe


3. check other variable

...now however I find it hard to parametrize the $z$ so a bit lost. So how can I visualize the wave with pen-and-paper?

Answer

Would you agree that $\vec{E}$ depends only on $\frac{2 \pi z}{\lambda} + \omega t$ (taking $E_0$ to be a constant)?

If so, we can imagine picking some spot it space and time, taking note of the value of $\vec{E}$ at that point and looking to see how we have to move to keep the value constant in time

$$ \frac{2 \pi z}{\lambda} + \omega t = C $$

where C is determined entirely by our initial choice of space--time location. So:

$$ z = z(t) = \frac{ \lambda }{2 \pi} \left( C - \omega t \right) $$

represents a locus of $z$-positions as a function of time where $\vec{E}$ continues to have the same value it had at our starting point. And those positions move in the negative $z$ direction as time increases.

Question for the studuent: how fast do they move?

You should be able to answer by inspection.