Calculating the components of an electromagnetic wave

[Luck0]

Homework Statement

An electromagnetic planewave (non-monochromatic) propagates in vacuum along the positive x axis. The electric field vector is parallel to the y axis. We know the dependence of the component $E_y$ on the variable $x$ at the moment $t = 0$:

$E_y(x) = E_0\ \text{if}\ |x + a| < b$
$E_y(x) = 0\ \text{if}\ |x + a| > b$

$a/2 > b > 0$

An ideal plane mirror is placed at $x = 0$. Find the components of the electric and magnetic field as functions of the variable at the following time instants: $t_1 = a/2c,\ t_2 = a/c,\ t_3 = 2a/c$.

Homework Equations

One dimensional electromagnetic planewave propagating in the positive x direction:

$E = E(x - ct)$
$B= (1/c)E$

The Attempt at a Solution

As the wave propagates in the x direction, and the electric field is in the y direction, the magnetic field only has a nonzero component in the z direction. So all I have to do is find the $E_y$ behavior at the given times and multiply it by $1/c$.

Let $a = ct$. Then, for any time greater than zero, the electric field is null, because $a$ is always greater than $b$, and $x$ is always positive, so $|x + a|$ has to be greater than $b$. So it is a wave that exists only when $t = 0$ for certain values of $x$, and vanishes for any $t > 0$ or any $x > b$. But what is the purpose of a exercise like this if the wave does not exist at the given times?

Am I wrong? How can I use the information about the mirror?

 

[Luck0]

Anyone? I asked my professor about this question, and he said to me that $|x + a|$ is the distance between the mirror and the wave, centered at $a$, and $2b$ is the length of the wave. But now I am even more confused.