17.7.1 Identical Sources

For two identical sources (i.e., sources that oscillate in phase with $Δ ϕ_{0} = 0$ ), the conditions for constructive and destructive interference are simple:

\begin{array}{l} Constructive: Δ r = m λ \\ (identical sources) \\ Destructive: Δ r = (m + \frac{1}{2}) λ \end{array}

(17.38)

The waves from two identical sources interfere constructively at points where the path-length difference is an integer number of wavelengths because, for these values of $Δ r$ , crests are aligned with crests and troughs with troughs. The waves interfere destructively at points where the path-length difference is a half-integer number of wavelengths because, for these values of $Δ r$ , crests are aligned with troughs. These two statements are the essence of interference.

NOTE Equations 17.38 apply only if the sources are in phase. If the sources are not in phase, you must use the more general Equations 17.37 to locate the points of constructive and destructive interference.

Wave fronts are spaced exactly one wavelength apart; hence we can measure the distances $r_{1}$ and $r_{2}$ simply by counting the rings in the wave-front pattern. In FIGURE 17.26, which is based on Figure 17.25, point A is distance $r_{1} = 3 λ$ from the first source and $r_{2} = 2 λ$ from the second. The path-length difference is $Δ r_{A} = 1 λ$ , the condition for the maximum constructive interference of identical sources. Point B has $Δ r_{B} = \frac{1}{2} λ$ , so it is a point of maximum destructive interference.

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

FIGURE 17.26 Full Alternative Text

NOTE Interference is determined by $Δ r$ , the path-length difference, rather than by $r_{1}$ or $r_{2}$ .

We can now locate the points of maximum constructive interference by drawing a line through all the points at which $Δ r = 0$ , another line through all the points at which $Δ r = λ$ , and so on. These lines, shown in red in FIGURE 17.27, are called antinodal lines. They are analogous to the antinodes of a standing wave, hence the name. An antinode is a point of maximum constructive interference; for circular waves, oscillation at maximum amplitude occurs along a continuous line. Similarly, destructive interference occurs along lines called nodal lines. The amplitude is a minimum along a nodal line, usually close to zero, just as it is at a node in a standing-wave pattern.

The figure illustrates nodal and antinode points on overlapping circular fronts.

FIGURE 17.27 Full Alternative Text

17.7.1 Identical Sources

FIGURE 17.26 For identical sources, the path-length difference $Δ r$ determines whether the interference at a particular point is constructive or destructive.

FIGURE 17.27 The points of constructive and destructive interference fall along antinodal and nodal lines.