Waves/Plane Superposition

Waves : 2 and 3 Dimension Waves
1 - 2 - 3 - 4 - 5 - 6 - 7
Problems

Superposition of Plane Waves

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We now study of wave packets in two dimensions by asking what the superposition of two plane sine waves looks like. If the two waves have different wavenumbers, but their wave vectors point in the same direction, the results are identical to those presented in the previous chapter, except that the wave packets are indefinitely elongated without change in form in the direction perpendicular to the wave vector. The wave packets produced in this case march along in the direction of the wave vectors and thus appear to a stationary observer like a series of passing pulses with broad lateral extent.

Superimposing two plane waves which have the same frequency results in a stationary wave packet through which the individual wave fronts pass. This wave packet is also elongated indefinitely in some direction, but the direction of elongation depends on the dispersion relation for the waves being considered. One can think of such wave packets as steady beams, which guide the individual phase waves in some direction, but don't themselves change with time. By superimposing multiple plane waves, all with the same frequency, one can actually produce a single stationary beam, just as one can produce an isolated pulse by superimposing multiple waves with wave vectors pointing in the same direction.

Two Waves of Identical Wavelength

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If the frequency of a wave depends on the magnitude of the wave vector, but not on its direction, the wave's dispersion relation is called isotropic. In the isotropic case two waves have the same frequency only if the lengths of their wave vectors, and hence their wavelengths, are the same. The first two examples in figure 2.6 satisfy this condition. In this section we investigate the beams produced by superimposed isotropic waves.

We superimpose two plane waves with wave vectors   and  . The lengths of the wave vectors in both cases are  :

  (3.15)

If  , then both waves are moving approximately in the   direction. An example of such waves would be two light waves with the same frequencies moving in slightly different directions.

 
Figure 2.7: Wave fronts and wave vectors of two plane waves with the same wavelength but oriented in different directions. The vertical bands show regions of constructive interference where wave fronts coincide. The vertical regions in between the bars have destructive interference, and hence define the lateral boundaries of the beams produced by the superposition. The components   and   of one of the wave vectors are shown.

Applying the trigonometric identity for the sine of the sum of two angles (as we have done previously), equation (3.15) can be reduced to

  (3.16)

This is in the form of a sine wave moving in the   direction with phase speed   and wavenumber  , modulated in the   direction by a cosine function. The distance   between regions of destructive interference in the   direction tells us the width of the resulting beams, and is given by  , so that

  (3.17)

Thus, the smaller  , the greater is the beam diameter. This behavior is illustrated in figure 2.7.

 
Figure 2.8: Example of beams produced by two plane waves with the same wavelength moving in different directions. The wave vectors of the two waves are  . Regions of positive displacement are illustrated by vertical hatching, while negative displacement has horizontal hatching.

Figure 2.8 shows an example of the beams produced by superposition of two plane waves of equal wavelength oriented as in figure 2.7. It is easy to show that the transverse width of the resulting wave packet satisfies equation (2.17).

Two Waves of Differing Wavelength

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In the third example of figure 2.6, the frequency of the wave depends only on the direction of the wave vector, independent of its magnitude, which is just the reverse of the case for an isotropic dispersion relation. In this case different plane waves with the same frequency have wave vectors which point in the same direction, but have different lengths.

More generally, one might have waves for which the frequency depends on both the direction and magnitude of the wave vector. In this case, two different plane waves with the same frequency would typically have wave vectors which differed both in direction and magnitude.

 
Figure 2.9: Wave fronts and wave vectors of two plane waves with different wavelengths oriented in different directions. The slanted bands show regions of constructive interference where wave fronts coincide. The slanted regions in between the bars have destructive interference, and as previously, define the lateral limits of the beams produced by the superposition.

Mathematically, we can represent the superposition of these two waves as a generalization of equation (2.15):

  (3.18)

In this equation we have given the first wave vector a   component   while the second wave vector has  . As a result, the first wave has overall wavenumber   while the second has  , so that  . Using the usual trigonometric identity, we write equation (2.18) as

  (3.19)

To see what this equation implies, notice that constructive interference between the two waves occurs when  , where   is an integer. Solving this equation for   yields  , which corresponds to lines with slope  . These lines turn out to be perpendicular to the vector difference between the two wave vectors,  . The easiest way to show this is to note that this difference vector is oriented so that it has a slope  . Comparison with the   slope of the lines of constructive interference indicates that this is so.

 
Figure 2.10: Example of beams produced by two plane waves with wave vectors differing in both direction and magnitude. The wave vectors of the two waves are   and  . Regions of positive displacement are illustrated by vertical hatching, while negative displacement has horizontal hatching.

An example of the production of beams by the superposition of two waves with different directions and wavelengths is shown in figure 2.10. Notice that the wavefronts are still horizontal, as in figure 2.8, but that the beams are not vertical, but slant to the right.

 
Figure 2.11: Illustration of factors entering the addition of two plane waves with the same frequency. The wave fronts are perpendicular to the vector average of the two wave vectors,  , while the lines of constructive interference, which define the beam orientation, are oriented perpendicular to the difference between these two vectors,  .

Figure 2.11 summarizes what we have learned about adding plane waves with the same frequency. In general, the beam orientation and the lines of constructive interference are not perpendicular to the wave fronts. This only occurs when the wave frequency is independent of wave vector direction.

Many Waves with the Same Wavelength

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As with wave packets in one dimension, we can add together more than two waves to produce an isolated wave packet. We will confine our attention here to the case of an isotropic dispersion relation in which all the wave vectors for a given frequency are of the same length.

 
Figure 2.12: Illustration of wave vectors of plane waves which might be added together.

Figure 2.12 shows an example of this in which wave vectors of the same wavelength but different directions are added together. Defining   as the angle of the  th wave vector clockwise from the vertical, as illustrated in figure 2.12, we could write the superposition of these waves at time   as  

  (3.20)

where we have assumed that   and  . The parameter   is the magnitude of the wave vector and is the same for all the waves. Let us also assume in this example that the amplitude of each wave component decreases with increasing  :

  (3.21)

The exponential function decreases rapidly as its argument becomes more negative, and for practical purposes, only wave vectors with   contribute significantly to the sum. We call   the spreading angle.

 
Figure 2.13: Plot of the displacement field   from equation (2.20) for   and  .

Figure 2.13 shows what   looks like when   and  . Notice that for   the wave amplitude is only large for a small region in the range  . However, for   the wave spreads into a broad semicircular pattern.

 
Figure 2.14: Plot of the displacement field   from equation (2.20) for   and  .

Figure 2.14 shows the computed pattern of   when the spreading angle  . The wave amplitude is large for a much broader range of   at   in this case, roughly  . On the other hand, the subsequent spread of the wave is much smaller than in the case of figure 2.13.

We conclude that a superposition of plane waves with wave vectors spread narrowly about a central wave vector which points in the   direction (as in figure 2.14) produces a beam which is initially broad in   but for which the breadth increases only slightly with increasing  . However, a superposition of plane waves with wave vectors spread more broadly (as in figure 2.13) produces a beam which is initially narrow in   but which rapidly increases in width as   increases.

The relationship between the spreading angle   and the initial breadth of the beam is made more understandable by comparison with the results for the two-wave superposition discussed at the beginning of this section. As indicated by equation (2.17), large values of  , and hence  , are associated with small wave packet dimensions in the   direction and vice versa. The superposition of two waves doesn't capture the subsequent spread of the beam which occurs when many waves are superimposed, but it does lead to a rough quantitative relationship between   (which is just   in the two wave case) and the initial breadth of the beam. If we invoke the small angle approximation for   so that  , then   and equation (2.17) can be written  . Thus, we can find the approximate spreading angle from the wavelength of the wave   and the initial breadth of the beam  :

  (3.22)