Question

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).

• A. 8
• B. 11
• C. 1
• D. 3

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.

Answer 1

The Correct Option is A

Step 1: State the Modified Rayleigh (Peake and Oliver) Criteria.
A surface with RMS height \(h\) is classed as smooth for a sensor of wavelength \(\lambda\) and incidence angle \(\theta\) if \(h < \dfrac{\lambda}{25\cos\theta}\), and rough if \(h > \dfrac{\lambda}{4.4\cos\theta}\). A surface with height between these two limits is neither purely smooth nor purely rough.
Step 2: Compute the smooth-surface limit for the L-band sensor.
Here \(\lambda_L = 23.5\text{ cm} = 235\text{ mm}\) and \(\theta = 20^\circ\), so \(\cos 20^\circ \approx 0.9397\). \[ h_{smooth,\ max} = \frac{235}{25 \times 0.9397} = \frac{235}{23.49} \approx 10.0\text{ mm} \] So for the surface to appear smooth to the L-band sensor, \(h < 10.0\) mm.
Step 3: Compute the rough-surface limit for the X-band sensor.
Here \(\lambda_X = 3\text{ cm} = 30\text{ mm}\). \[ h_{rough,\ min} = \frac{30}{4.4 \times 0.9397} = \frac{30}{4.135} \approx 7.26\text{ mm} \] So for the surface to appear rough to the X-band sensor, \(h > 7.26\) mm.
Step 4: Combine both conditions and check the options.
Both conditions together require \(7.26\text{ mm} < h < 10.0\text{ mm}\). Checking the options: 11 mm fails the smooth condition (above 10.0 mm), and 1 mm and 3 mm both fail the rough condition (below 7.26 mm). Only 8 mm satisfies \(7.26 < 8 < 10.0\). \[ \boxed{h = 8\text{ mm, option (A)}} \]