This question is inspired by this physics question, but it is purely mathematical.
Let $\{f_i\}$ be a set of sine waves $f_i=a\sin(kx+\phi_i)$ where $i\in\{1,2,\dots,N\}$ and the $\phi_i\in[0,2\pi/k]$ are chosen randomly with a uniform probability distribution on $[0,2\pi/k].$ Let $F=\sum_{i=1}^Nf_i=A\sin(kx+\Phi).$
What is the expected amplitude of F as a function of a, k, and N?
Put differently, $\langle A\rangle(a,k,N)=?$
It is clear that $\langle A\rangle(a,k,N) = a\langle A\rangle(1,1,N)$, so we restrict to $a = 1$ and $k = 1$.
Note that $\sin(x + \phi) = a\sin(x) + b\cos(x)$ for $a^2 + b^2 = 1$ and $(\sin2\phi,\cos2\phi) = (2ab, a^2 - b^2)$, so that instead of uniformly drawing $\phi$, we can uniformly draw a point on the unit circle.
Drawing $N$ such points and summing them as elements of $\mathbb R^2$, gives us an expression of the form $A\sin(x) + B\cos(x)$, whose amplitude is $\sqrt{A^2 + B^2}$.
This shows that the expected amplitude you're looking for is the same as the expected distance of a random walk in the plane with unit step size and arbitrary direction.
Asymptotically for $N\to\infty$, this expectation is $\sqrt N$, see e.g. mathworld. For low $N$ this is actually pretty hard to compute, see this article by Borwein, Nuyens and Straub for some results and conjectures.