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: 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 $\alpha_i$ 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 $k = \vert \mbox{\bf k} \vert$ 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 $w$:

$$\alpha_{max} \approx \lambda / (2w) ~~~ \mbox{(single slit spreading angle)} .$$ (3.22)