Why is the half-wave dipole the most used antenna design? When producing em waves using a dipole antenna (of length L), you could theoretically use any L and adjust the frequency of the oscillating voltage to get the desired wavelength. Then why are most antennas half a wavelength long? I'd also like to know why it's useful in the context of receiving em waves. Thanks.
 A: An antenna, in general, is NOT a resonator (contra @oliver). The purpose of the antenna is to be a direction dependent impedance transformer that matches the wave impedance of the transmission line connecting the transmitter/receiver to space (air) having impedance $\sqrt{\frac{\mu_0}{\epsilon_0}}=377\Omega$ in all direction. In other words, the apparent impedance driving the antenna in the desired direction is matched to the line; waves that are to be in or from undesired direction be completely mismatched (reflected). Being resonant can help with impedance matching be achieved at a given frequency/direction. If wider bandwidth than that of a simple resonant curve is required then different radiators need be employed, for example, a horn that is being an aperture radiator can easier be made wider bandwidth.
It happens that a "half-wave dipole* has about $70\Omega$ input impedance for radiation in its "mid-plane" in which it radiates uniformly, and $70\Omega$ is a convenient number to match to $50\Omega - 75\Omega$ the impedance level of most RF equipment. If the length of the antenna is longer relative to its wavelength then the radiation breaks up into several narrower lobes with two practical consequences, (1) matching gets more difficult (2) a complicated antenna shape is difficult to maintain because it becomes more sensitive to environmental effects.
A: An antenna is a resonator. If you are not feeding it with its natural frequency, it is not going to oscillate with sufficient amplitude. Of course you can always think about increasing voltage, but usually everything in technology is about efficiency. Imagine you would have to carry a heavy car battery and a high voltage generator with your smartphone in order to get enough sending power.
Remember your childhood days, when you had to learn how to ramp up a swing? If you did not move your feet and trunk in the right "rythm", the oscillation would just starve. That's exactly what would happen if you feed the antenna with a frequency far off its resonance.
With receiving it's the same as with sending. If the frequency of the incoming wave does not match the antenna, the signal will be very low. It's actually much worse for reception because you cannot just crank up the voltage. You can increase amplification, but that will usually result in a worse signal-to-noise ratio.
A: A half-wave dipole is fed with the signal at its mid-point.
Consider the current distribution along the antenna. Some current has to flow in and out at the middle, or power cannot be transferred. But at the ends the current has nowhere to flow. So the current distribution along the dipole takes on a distinct "hump" shape, basically half a sine wave. And that is the clue - half a wave.
Doubling the length of the antenna to a whole wave would de-tune it so that, at that frequency, the current in the middle would be as zero as the current at the ends and no power would be transferred.
In practice, an antenna standing on the ground will be only a quarter-wave long. The ground acts like a mirror, creating a virtual reflection of the antenna so it looks a half-wave long. You can see the effect optically if you stand on the shore of a lake on a still, calm day and the opposite shore is reflected upside down in the water.
