= 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 $n\leq N/2$. 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 gives
$$
\boxed{M\simeq\frac12\pi\left(\frac N2\right)^2=\frac{\pi N^2}8=\frac\pi4\left(\frac{N^2}2\right).}
$$
This 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.
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