Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{\sqrt{2}}{2}$ and major axis length 4.\\
(1) Find the equation of $C$;\\
(2) A line $l$ passing through the point $(0, -2)$ intersects $C$ at points $A, B$. Let $O$ be the origin. If the area of $\triangle OAB$ is $\sqrt{2}$, find $|AB|$.