Let's assume we have a transmitter, a feed line, and a standing wave antenna (dipole, etc.). The transmitter is matched with the feed line, and the feed line is matched with the antenna, which is why the traveling wave in the feed line does not reflect from the antenna's feed point and passes without any obstacle. That is, there is no standing wave in the feed line, and this is good (radio amateurs appreciate this).
Now our antenna creates a standing wave. But if the wave doesn't see an obstacle when it moves towards the antenna, why doesn't the same happen in the reverse direction, when the wave goes from the tip of the antenna and reaches its base? Why does the wave bounce multiple times inside the antenna and why it doesn't move back into the feed line if there is a perfect impedance match at the base? Yet a standing wave is still created in the antenna. The question is, how?