The field you are describing is one possible solution to Maxwell's equations - a plane wave. It's a highly useful solution because all nonevanescent (propagating) fields in freespace / homogeneous mediums can be built from a superposition of plane waves.
But real solutions don't have to (and indeed never do) look like this particular solution. Real waves are superpositions of different plane waves; in particular, a superposition involving a spread of directions of constituent plane waves decreases swiftly with transverse distance from the center of the disturbance. Indeed, once we include a spread of directions and frequencies in the plane wave superposition, the disturbance at any time can have a truly compact support i.e. is precisely zero at any point that cannot be reached from the source travelling at less than light speed for the time since the disturbance began, and thus comply with special relativity.
Another, much more realistic propagating solution to Maxwell's equations is a spherical wave.
$$\mathbf{E}\left(r,\,\theta,\,\phi,\,t\right)=A\frac{\sin\theta}{r}\left[\cos\left(kr-\omega t\right)-\frac{\sin\left(kr-\omega t\right)}{kr}\right]\hat{\boldsymbol{\phi}}$$
$$$$
$$\mathbf{B}\left(r,\,\theta,\,\phi,\,t\right)=\frac{2A\cos\theta}{r^{2}\omega}\left[\sin\left(kr-\omega t\right)+\frac{\cos\left(kr-\omega t\right)}{kr}\right]\hat{\mathbf{r}}+\frac{A\sin\theta}{r^{3}\omega}\left[\left(\frac{1}{k}-kr^{2}\right)\cos\left(kr-\omega t\right)+r\sin\left(kr-\omega t\right)\right]\hat{\boldsymbol{\theta}} $$
where $(r,\,\theta,\,\phi)$ are the spherical polar co-ordinates.
As $r\to\infty$ such a wave looks exactly like a plane wave with amplitude varying like $1/r$ over regions subtending angles at the source.