17.7 Interference in Two and Three Dimensions

Ripples on a lake move in two dimensions. The glow from a lightbulb spreads outward as a spherical wave. A circular or spherical wave, illustrated in FIGURE 17.24, can be written

D (r, t) = a sin (k r - ω t + ϕ_{0})

(17.34)

where r is the distance measured outward from the source. Equation 17.34 is our familiar wave equation with the one-dimensional coordinate x replaced by a more general radial coordinate r. Recall that the wave fronts represent the crests of the wave and are spaced by the wavelength $λ$ .

The figure illustrates a circular wave front for a sound source.

FIGURE 17.24 Full Alternative Text

What happens when two circular or spherical waves overlap? For example, imagine two paddles oscillating up and down on the surface of a pond. We will assume that the two paddles oscillate with the same frequency and amplitude and that they are in phase. FIGURE 17.25 shows the wave fronts of the two waves. The ripples overlap as they travel, and, as was the case in one dimension, this causes interference. An important difference, though, is that amplitude decreases with distance as waves spread out in two or three dimensions—a consequence of energy conservation—so the two overlapped waves generally do not have equal amplitudes. Consequently, destructive interference rarely produces perfect cancellation.

The figure illustrates the overlapping ripple patterns of two identical sources.

FIGURE 17.25 Full Alternative Text

Maximum constructive interference occurs where two crests align or two troughs align. Several locations of constructive interference are marked in Figure 17.25. Intersecting wave fronts are points where two crests are aligned. It’s a bit harder to visualize, but two troughs are aligned when a midpoint between two wave fronts is overlapped with another midpoint between two wave fronts. Maximum, but usually not perfect, destructive interference occurs where the crest of one wave aligns with a trough of the other wave. Several points of destructive interference are also indicated in Figure 17.25.

A picture on a page is static, but the wave fronts are in motion. Try to imagine the wave fronts of Figure 17.25 expanding outward as new circular rings are born at the sources. The waves will move forward half a wavelength during half a period, causing the crests in Figure 17.25 to be replaced by troughs while the troughs become crests.

The important point to recognize is that the motion of the waves does not affect the points of constructive and destructive interference. Points in the figure where two crests overlap will become points where two troughs overlap, but this overlap is still constructive interference. Similarly, points in the figure where a crest and a trough overlap will become a point where a trough and a crest overlap—still destructive interference.

The mathematical description of interference in two or three dimensions is very similar to that of one-dimensional interference. The net displacement of a particle in the medium is

D = D_{1} + D_{2} = a_{1} sin (k r_{1} - ω t + ϕ_{10}) + a_{2} sin (k r_{2} - ω t + ϕ_{20})

(17.35)

The only differences between Equation 17.35 and the earlier one-dimensional Equation 17.24 are that the linear coordinates have been changed to radial coordinates and we’ve allowed the amplitudes to differ. These changes do not affect the phase difference, which, with x replaced by r, is now

Δ ϕ = 2 π \frac{Δ r}{λ} + Δ ϕ_{0}

(17.36)

The term $2 π (Δ r / λ)$ is the phase difference that arises when the waves travel different distances from the sources to the point at which they combine. $Δ r$ itself is the path-length difference. As before, $Δ ϕ_{0}$ is any inherent phase difference of the sources themselves.

A photograph of two overlapping water waves that create an interference pattern. — Two overlapping water waves create an interference pattern.

Maximum constructive interference occurs, just as in one dimension, at those points where $cos (Δ ϕ / 2) = \pm 1$ . Similarly, maximum destructive interference occurs at points where $cos (Δ ϕ / 2) = 0$ . The conditions for constructive and destructive interference are

\begin{array}{l} Maximum constructive interference: \\ Δ ϕ = 2 π \frac{Δ r}{λ} + Δ ϕ_{0} = m \cdot 2 π \\ m = 0, 1, 2, . . \\ Maximum destructive interference: \\ Δ ϕ = 2 π \frac{Δ r}{λ} + Δ ϕ_{0} = (m + \frac{1}{2}) \cdot 2 π \end{array}

(17.37)

17.7 Interference in Two and Three Dimensions

FIGURE 17.24 A circular or spherical wave.

FIGURE 17.25 The overlapping ripple patterns of two identical sources. Several points of constructive and destructive interference are noted.