Reflections from rough surfaces

Reflections from rougher surfaces cause different types of interference pattern. We shall use the Cornu spiral to help us once more. The Cornu

Distance (wavelengths)

Relative signal strength (dB)

0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40

Figure 4.12 Multipath fading produced by six signals with one signal approximately 20 dB stronger than the other five.

–1.2 –0.8 –0.4 0 0.4 0.8 1.2 –1.2 –0.8 –0.4 0 0.4 0.8 1.2 (a) –1.2 –0.8 –0.4 0 0.4 0.8 1.2 –1.2 –0.8 –0.4 0 0.4 0.8 1.2 (b) –1.2 –0.8 –0.4 0 0.4 0.8 1.2 –1.2 –0.8 –0.4 0 0.4 0.8 1.2 (c)

Figure 4.13 Distortion of the Cornu spiral for random phase variations of up to 0(a), 0.25(b), 0.5(c), 1.0(d), 3.0(e) and 10.0(f) radians between successive wavelets.

–1.2 –0.8 –0.4 0 0.4 0.8 1.2 –1.2 –0.8 –0.4 0 0.4 0.8 1.2 (d) –1.2 –0.8 –0.4 0 0.4 0.8 1.2 (e) –1.2 –0.8 –0.4 0 0.4 0.8 1.2 –1.2 –0.8 –0.4 0 0.4 0.8 1.2 (f) –1.2 –0.8 –0.4 0 0.4 0.8 1.2 Figure 4.13 (cont.) r e f l e c t i o n s f r o m r o u g h s u r f a c e s 103

spiral provides us with a picture of how the received signal is produced when phase shifts between the contributions from the individual wave- lets are due to path-length differences alone. If the surface is rough, the path-length difference (and hence the phase shift) will be randomised to a certain extent by the roughness of the surface. Figure4.13shows the effect of introducing an increasing random phase shift between one wavelet and the next. It is seen that the spiral distorts.

Where the random phase shift is kept to less than half of one radian (corresponding to a random path-length shift of up to about one twelfth of a wavelength) the overall resultant does not reduce much (the mag- nitude of the resultant reflected signal is indicated by the distance from one end of the ‘spiral’ to the other). However, if the variation is a maximum of 3 radians, the resultant is considerably reduced, and, where the phase shift is as much as 10 radians (corresponding to a path-length variation of about 1.5 wavelengths), the spiral has collapsed in on itself, giving a very small resultant reflected field strength. Further increases in phase shift have little effect. Even though the material itself may be perfectly reflective, the resultant reflected field will typically be at least 10 dB below that which would have been obtained if the surface had been perfectly smooth. When thinking of this, it is useful to consider the intensity of reflections from a polished metal and compare it with that from a roughened or ‘brushed’ surface such as ‘brushed aluminium’. The situation with a rough surface is further complicated by the fact that, if the receiver moves, the reflection point changes and the nature of the reflecting surface changes (unlike with a perfectly smooth reflecting surface, for which all points of the surface lie in the same plane). This means that the nature of any interference pattern that the receiver moves through is very difficult to predict, except in general terms. It will gen- erally be more noise-like and less regular, with a lower maximum depth of fade than occurs with a single, coherent reflected wave. The severity of the multipath environment can be said to be lower when the reflecting surface is rough. In reality, no surface is absolutely smooth or absolutely rough. In deciding whether the surface is rough or smooth for our pur- poses, we use what is known as the ‘Rayleigh criterion’. This considers the geometry of the incident wave and the reflecting surface. The

roughness of the surface is defined by its standard deviation,σ, relative to a perfect plane. A constant, C, is then derived:

C ¼4pr cos_k , ð4:13Þ

where is the angle of incidence.

Notice that the angle of incidence is a parameter. Even rough surfaces at near ‘grazing’ incidence appear to be quite smooth. If the surface is smooth you can expect peaks and troughs to form. If the surface is rough you do not get regular peaks and troughs but, rather, the interference pattern becomes more noise-like. There is, however, a transition region between the two, for which you will observe ever-weakening peaks and troughs as the roughness increases.

Generally, if C < 0.1 then the surface can be regarded as smooth, but if C > 10 then the surface is regarded as rough and the reflections will be diffuse. Between the values of 0.1 and 10 lies a transition region. In our experiments with the Cornu spiral a transition was certainly observed, but it is of interest to see how the value of C relates to values of variation of phase. If an ‘idealised’ rough surface is considered to have two different levels, separated by a distance, d, as shown in figure4.14, then one part of the wave will travel an extra distance. This extra distance equals 2d cos and the phase difference,
, between these two parts is given by


¼4pd cos _k : ð4:14Þ

d

C

Figure 4.14 The path geometry for reflection from an idealised rough surface.

The similarity between this and the constant used for the Rayleigh criterion is clear. In the experiments with the Cornu spiral we assumed a random variation in phase between certain limits. The Rayleigh criterion assumes a normal distribution of surface heights with a standard devi- ation defined. However, our experiments with the Cornu spiral showed that if a random variation of 10 radians in phase existed then the resultant of the reflected signal collapsed to a negligible level compared with that of a reflection from a smooth surface. This tallies well with the standard deviation of 10 radians that is set as the threshold for regarding the surface as rough. Further, the Cornu spiral for a variation in phase limited to 0.1 radians still had the appearance of a good spiral and justifies the statement that the surface could be regarded as smooth. Whether it is better to consider random variations within a certain limit or a standard deviation of surface height is a matter of opinion. It should be borne in mind, however, that a normal distribution that is character- ised by its standard distribution does stretch between ∞.

The Rayleigh criterion necessarily involves a conversion from the roughness of a surface to the phase variation caused. Diagrams such as figure4.14that are used to support this argument often show the surface not to be rough as such but rather stepping between two levels and composed of smooth panels. For a surface to be rough, the size of each of the panels must be small, otherwise the signal at the receiving point may be dominated by a single (smooth) reflecting panel.