I understand they managed to emit a sound wave that came back, but how in the world did they digitize that?
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The first operational radar screens didn’t produce a dot, but instead a graph showing the strength of return signals against distance. [Here’s an example display image](https://en.wikipedia.org/wiki/File:Chain_Home_screen_shot_-NEDAD.2013.047.058A.jpg) where strong returns cause downward spikes. The operator had to manually vary the radar antenna configuration (there were multiple huge, fixed antennas) to make the spike of interest as large as possible and that would give the direction.
The display was essentially an oscilloscope where an electron beam could be steered across the phosphor coated vacuum tube, exactly like an old TV but much simpler. In this case the beam was repeatedly swept left to right at a fixed rate, synchronised with the outgoing radar pulses, so that the horizontal line matched the scale of miles needed. The strength of return radar signals was used to drive the beam downwards to form the spikes you can see. This was all analogue, vacuum-tube electronics developed in the 1930s, long before the invention of the transistor. If you wanted a record, you photographed the screen.
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