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$$\sqrt{1 - (-1 + x\bmod 2)^2}\cdot\operatorname{sign}(-2 + x\bmod 4)$$

Like half-circles connected to each other to look like waves:

plot http://vi-server.org/vi/_/func.png

Its plot looks smooth, but the function is actually not smooth.

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    This is like proving that pink unicorns do not exist, of course... but my guess is that your function does not have a standard name.2011-04-28
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    Henceforth it shall be called the Vio periodic function!2011-04-28
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    I've seen the version where all the semicircles are above the horizontal axis... this one though, maybe you get dibs on naming it. :)2011-04-28
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    Looks like "poor man's sine that can be drawn with a compass".2011-04-28
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    @Raskolnikov, (actual nick is "_Vi", but is's not StackExchange-compliant)2011-04-28
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    @Mariano Suárez-Alvarez, But there can related named function (e.g. negated, with other period).2011-04-28
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    I don't know if it has a name, but it is a very ugly thing.2011-04-28
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    @Jonas T, Why ugly? It's even continuous. And circle is not ugly, so that semi-circle is unlikely to be ugly, and this is the most logical way to make a "semicircle function".2011-04-28

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