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Sharp corners

The corner is not PERFECTLY sharp, of course, on a real string, but it really is a long way from being a sine wave. As you say, bending stiffness of the string stops it having an ideally sharp corner. But there are plenty of measurements of real bowed strings vibrating, and also high-speed videos showing the Helmholtz motion directly.
The motion of a bowed string brings in different physics from the vibration of a plucked string such as a guitar. When you pluck a string you certainly excite many overtones of the string, and these are roughly harmonically spaced, but they each vibrate independently of each other: a characteristic of linear systems. But when you bow a string carefully and steadily the nonlinear interaction with friction at the bow produces a periodic motion consisting of exact harmonics, even though these will not precisely match the natural overtone frequencies of the string. The motion has been analysed quite extensively, and it tells you things about how a player might be able to vary the tone of a note, and how a string-maker might design strings to produce different balances of sound.

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