How does a buck converter's output ripple waveform become sinusoidal

[newbie_hs]

Below image is from Fundamentals of Power Electronics.

The ripple component of the inductor current will be flowing through the capacitor as shown in the ic(t) graph. This one I understood.

May I know how the ripple waveform becomes sinusoidal? What is the mathematical equation behind it?

 

[Easy peasy]

it's not completely sinusiodal - you are effectively filtering a square wave of voltage with an LC filter ( damped by load ).

 

[FvM]

Integral of sawtooth is parabolic arc curve, quite similar to sine.