Um... sort of?
More to the point, if the antenna were in a reflective box, the standing waves would be trapped, and your transmitter throws a fit. It's equivalent to a coaxial resonator, or helical resonator or what have you.
Standing waves are bad. Unless you need them specifically for filtering out undesired frequencies, but in that case, you only need them inside the filter, and not outside the filter at the frequency(ies) of interest. That is, you might have a transmitter, feeding a filter of some sort, but the transmitter always sees a good match at the driven frequency, while the elements inside the filter handle additional reactive energy.
So, qualified in this way, as measured at a matched amplifier feedpoint, say, you don't want SWR.
So the antenna, then. The sheer fact that it's open to free space, reduces its SWR, both within the elements, and at the feedline. Less SWR on the elements means lower Q, wider bandwidth. Bandwidth is generally considered A Good Thing in antenna design; but, see the above mention of filtering and desirability of frequencies.
So, qualified again, when you don't have a special interest in some frequency band: you should desire a wideband antenna, i.e., one with low SWR on its elements.
Still following?
So then, a wideband antenna. This is a structure which has similar geometry over a wide scale. A dipole doesn't: it's a fixed length, and the element diameter doesn't scale along the length. A conical dipole does, however. Or a bowtie antenna. Or a horn, or some fractals (but not all of them, indeed I might even say, not most of them, because there are a lot of curve and plane fractals that people seem to draw because they look cool, but which aren't chosen because of any electromagnetic intuition, or simulation results!).
These examples share a common trait: the free-space EM wave interacts almost seamlessly with the antenna structure. It doesn't get trapped by the structure. The waves don't, well, stand around.
I prefer to think of simpler antennas (like the dipole) as a worse version of a wideband antenna. By removing material, you cause resonance (and anti-resonance) to occur at special frequencies, thus cutting off reception inbetween. Instead of a wide open window, you have an opaque mask with holes punched in it.
Going back to material properties: very little of the wave enters the material. After all, it would be a poor conductor if that were the case, and you'd have an absorber, not an antenna.
The key is to direct the waves into your transmission line, from whatever directions and phases it's set up to do. From the transmission line out to free space, the waves are shaped, guided by the conductors, residing in the space between them. Currents aren't carried
in wires, they are carried
on wires, and the voltage is carried between them.
It's not that an antenna element has an impedance -- though it's still kind of fair* to say it does have one, given a size, length, shape and frequency -- it's that the materials have an impedance different enough from the space around them, to serve this wave-guiding function without excessive losses.
*Trouble is, impedance is a scalar -- just a number. It's supposed to be measured at a point. A resistor at DC, for example. You measure the current into, and voltage across, a pointlike element. At AC, we can still work in terms of impedance (complex), where the elements are much less than a wavelength -- still pointlike. Or where the elements are comparable to a wavelength, but we measure at an RF "port" -- a pointlike transmission line connection. If we're talking about elements, in and of themselves, then we have to allow that we're talking about voltages and currents in different locations, like how the impedance of a dipole's elements might correspond to the voltage at the tips divided by the current at the feedpoint. We probably lose phase information in the process, or at least we must be very careful to add the phase shift back in (from distance between points / speed of light), and hope to come up with a still-reasonable number!
And even if the discontinuity is small, if it's low loss -- and you can afford to use more material -- you can still do all of this. Fiber optics are just dielectric waveguides. You can make practical microwave antennas with plastic shapes. Mirrors can be made by stacking alternating dielectrics of particular thicknesses: a lowpass or bandpass filter, which, like an electronic filter, reflects incident (out-of-band) energy.
Tim