Calculating the components of an electromagnetic wave

1. The problem statement, all variables and given/known data

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 [itex]E_y[/itex] on the variable [itex]x[/itex] at the moment [itex]t = 0[/itex]:

[itex]E_y(x) = E_0\ \text{if}\ |x + a| < b[/itex]
[itex]E_y(x) = 0\ \text{if}\ |x + a| > b[/itex]

[itex]a/2 > b > 0[/itex]

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

2. Relevant equations

One dimensional electromagnetic planewave propagating in the positive x direction:

[itex]E = E(x – ct)[/itex]
[itex]B= (1/c)E[/itex]

3. 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 [itex]E_y[/itex] behavior at the given times and multiply it by [itex]1/c[/itex].

Let [itex]a = ct[/itex]. Then, for any time greater than zero, the electric field is null, because [itex]a[/itex] is always greater than [itex]b[/itex], and [itex]x[/itex] is always positive, so [itex]|x + a|[/itex] has to be greater than [itex]b[/itex]. So it is a wave that exists only when [itex]t = 0[/itex] for certain values of [itex]x[/itex], and vanishes for any [itex]t > 0[/itex] or any [itex]x > b[/itex]. 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?

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