Using the Modified Rayleigh Criteria, for an incidence angle of \(20^\circ\), select the Root Mean Square (RMS) height (in mm) that appears smooth to an L-band microwave sensor (23.5 cm wavelength) and rough to an X-band microwave sensor (3 cm wavelength).
Show Hint
Apply the Modified Rayleigh smooth limit \(h < \lambda/(25\cos\theta)\) for L-band and the rough limit \(h > \lambda/(4.4\cos\theta)\) for X-band, then find the RMS height that satisfies both.
Step 1: Set up the two threshold constants. With $\cos 20^\circ \approx 0.9397$, the smooth threshold for L-band is $\lambda_L/(25\cos\theta) = 235/23.49 \approx 10.0$ mm, and the rough threshold for X-band is $\lambda_X/(4.4\cos\theta) = 30/4.135 \approx 7.26$ mm. Step 2: Test option (A), $h = 8$ mm. Smooth check: is $8 < 10.0$? Yes, so it looks smooth to L-band. Rough check: is $8 > 7.26$? Yes, so it also looks rough to X-band. Both conditions hold. Step 3: Test option (B), $h = 11$ mm. Smooth check: is $11 < 10.0$? No. So 11 mm already fails to look smooth to the L-band sensor, and is rejected. Step 4: Test options (C) $h = 1$ mm and (D) $h = 3$ mm. Rough check for X-band: is $1 > 7.26$ or $3 > 7.26$? No for both, so both look smooth (not rough) to the X-band sensor as well, and both are rejected. Step 5: Conclude. Only option (A), $h = 8$ mm, satisfies both the smooth-to-L-band and rough-to-X-band conditions simultaneously. \[ \boxed{h = 8\text{ mm, option (A)}} \]
