Waves/Plane Superposition

Superposition of Plane Waves
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
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 $$\mbox{k}_1 = (- \Delta k_x, k_y )$$ and $$\mbox{k}_2 = ( \Delta k_x , k_y )$$. The lengths of the wave vectors in both cases are $$k_1 = k_2 = ( \Delta k_x^2 + k_y^2 )^{1/2}$$:
 * $$ A = \sin ( -\Delta k_x x + k_y y - \omega t ) + \sin (\Delta k_x x + k_y y - \omega t ) . $$ (3.15)

If $$\Delta k_x << k_y$$, then both waves are moving approximately in the $$y$$ 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 $$\Delta k_x$$ and $$k_y$$ 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
 * $$ A = 2 \sin (k_y y - \omega t) \cos (\Delta k_x x) . $$ (3.16)

This is in the form of a sine wave moving in the $$y$$ direction with phase speed $$c_{phase} = \omega / k_y$$ and wavenumber $$k_y$$, modulated in the $$x$$ direction by a cosine function. The distance $$w$$ between regions of destructive interference in the $$x$$ direction tells us the width of the resulting beams, and is given by $$\Delta k_x w = \pi$$, so that
 * $$ w = \pi / \Delta k_x . $$ (3.17)

Thus, the smaller $$\Delta k_x$$, 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 $$\mbox{k} = (\pm 0.1, 1.0)$$. 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
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.



Mathematically, we can represent the superposition of these two waves as a generalization of equation (2.15):
 * $$ A = \sin [ - \Delta k_x x + (k_y + \Delta k_y ) y - \omega t ] + \sin [ \Delta k_x x + (k_y - \Delta k_y ) y - \omega t ] . $$ (3.18)

In this equation we have given the first wave vector a $$y$$ component $$k_y + \Delta k_y$$ while the second wave vector has $$k_y - \Delta k_y$$. As a result, the first wave has overall wavenumber $$k_1 = [ \Delta k_x^2 + (k_y + \Delta k_y )^2 ]^{1/2}$$ while the second has $$k_2 = [ \Delta k_x^2 + (k_y - \Delta k_y )^2 ]^{1/2}$$, so that $$k_1 \ne k_2$$. Using the usual trigonometric identity, we write equation (2.18) as
 * $$ A = 2 \sin (k_y y - \omega t ) \cos ( - \Delta k_x x + \Delta k_y y ) . $$ (3.19)

To see what this equation implies, notice that constructive interference between the two waves occurs when $$- \Delta k_x x + \Delta k_y y = m \pi$$, where $$m$$ is an integer. Solving this equation for $$y$$ yields $$y = ( \Delta k_x / \Delta k_y )x + m \pi / \Delta k_y$$, which corresponds to lines with slope $$\Delta k_x / \Delta k_y$$. These lines turn out to be perpendicular to the vector difference between the two wave vectors, $$\mbox{k}_2 - \mbox{k}_1 = ( 2 \Delta k_x, -2 \Delta k_y )$$. The easiest way to show this is to note that this difference vector is oriented so that it has a slope $$- \Delta k_y / \Delta k_x$$. Comparison with the $$\Delta k_x / \Delta k_y$$ 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 $$\mbox{k}_1 = ( -0.1, 1.0)$$ and $$\mbox{k}_2 = ( 0.1, 0.9)$$. 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, $$\mbox{k}_0 = ( \mbox{k}_1 + \mbox{k}_2 )/2$$, while the lines of constructive interference, which define the beam orientation, are oriented perpendicular to the difference between these two vectors, $$\mbox{k}_2 - \mbox{k}_1$$.

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
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 shows an example of this in which wave vectors of the same wavelength but different directions are added together. Defining $$\alpha_i$$ as the angle of the $$i$$th wave vector clockwise from the vertical, as illustrated in figure 2.12, we could write the superposition of these waves at time $$t = 0$$ as $$A = \sum_i A_i \sin ( k_{xi} x + k_{yi} y )$$
 * $$= \sum_i A_i \sin [kx \sin ( \alpha_i ) + ky \cos ( \alpha_i ) ]$$ (3.20)

where we have assumed that $$k_{xi} = k \sin ( \alpha_i )$$ and $$k_{yi} = k \cos ( \alpha_i )$$. The parameter $$k = \vert \mbox{k} \vert$$ 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 $$\vert \alpha_i \vert$$:
 * $$ A_i = \exp [-( \alpha_i / \alpha_{max} )^2 ] . $$ (3.21)

The exponential function decreases rapidly as its argument becomes more negative, and for practical purposes, only wave vectors with $$\vert \alpha_i \vert \le \alpha_{max}$$ contribute significantly to the sum. We call $$\alpha_{max}$$ the spreading angle.

Figure 2.13: Plot of the displacement field $$A(x,y)$$ from equation (2.20) for $$\alpha _{max} = 0.8$$ and $$k = 1$$.

Figure 2.13 shows what $$A(x,y)$$ looks like when $$\alpha_{max} = 0.8 \mbox{ radians}$$ and $$k = 1$$. Notice that for $$y = 0$$ the wave amplitude is only large for a small region in the range $$-4 < x < 4$$. However, for $$y > 0$$ the wave spreads into a broad semicircular pattern.

Figure 2.14: Plot of the displacement field $$A(x,y)$$ from equation (2.20) for $$\alpha _{max} = 0.2$$ and $$k = 1$$.

Figure 2.14 shows the computed pattern of $$A(x,y)$$ when the spreading angle $$\alpha_{max} = 0.2 \mbox{ radians}$$. The wave amplitude is large for a much broader range of $$x$$ at $$y = 0$$ in this case, roughly $$-12 < x < 12$$. 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 $$y$$ direction (as in figure 2.14) produces a beam which is initially broad in $$x$$ but for which the breadth increases only slightly with increasing $$y$$. 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 $$x$$ but which rapidly increases in width as $$y$$ increases.

The relationship between the spreading angle $$\alpha_{max}$$ 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 $$k_x$$, and hence $$\alpha $$, are associated with small wave packet dimensions in the $$x$$ 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 $$\alpha_{max}$$ (which is just $$\tan^{-1} (k_x / k_y )$$ in the two wave case) and the initial breadth of the beam. If we invoke the small angle approximation for $$\alpha = \alpha_{max}$$ so that $$\alpha_{max} = \tan^{-1} (k_x / k_y ) \approx k_x / k_y \approx k_x / k$$, then $$k_x \approx k \alpha_{max}$$ and equation (2.17) can be written $$w = \pi / k_x \approx \pi / (k \alpha_{max} ) = \lambda / (2 \alpha_{max} )$$. Thus, we can find the approximate spreading angle from the wavelength of the wave $$\lambda $$ and the initial breadth of the beam $$w$$:
 * $$ \alpha_{max} \approx \lambda / (2d) \quad \mbox{(single slit spreading angle)} . $$ (3.22)