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. Such an example is illustrated in figures 2.9 and 2.10. We now investigate the superposition of non-isotropic waves with the same frequency.

Figure: Wave fronts and wave vectors ( $\mbox{\bf k}_1$ and $\mbox{\bf k}_2$) 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. The quantities $\mbox{\bf k}_0$ and $\Delta \mbox{\bf k}$ are also shown.

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, $\mbox{\bf k}_2 - \mbox{\bf k}_1 = ( 2 \Delta k_x , -2 \Delta k_y )$. 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: 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 $\mbox{\bf k}_1 = ( -0.1 , 1.0)$ and $\mbox{\bf k}_2 = ( 0.1, 0.9)$. 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: 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, $\mbox{\bf k}_0 = ( \mbox{\bf k}_1 + \mbox{\bf k}_2 )/2$, while the lines of constructive interference, which define the beam orientation, are oriented perpendicular to the difference between these two vectors, $\mbox{\bf k}_2 - \mbox{\bf k}_1$.

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.