Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 338 / 2 / b / ii / Solution

Count independent sinusoidal ripples by their spatial Fourier series indices. A mode with $n$ cycles across $D$ has $\Lambda =D/n$, so the given half-wave index is $j=2n$. The Nyquist spatial frequency permits $n\le N/2$. In two dimensions, a circular cutoff therefore contains approximately $\pi (N/2{)}^2$ integer wavevectors.

For a real wavefront error, the wavevectors $\mathbf{n}$ and $-\mathbf{n}$ describe the same ripple with conjugate coefficients, so count each pair once. The continuum mode-counting approximation givesThis counts one ripple with amplitude and phase per conjugate pair, not one real coefficient. Exact finite-grid counts are integers and have boundary corrections; the formula is the circular area estimate implicit in the question, with piston excluded.