Waves obey the principle of superposition: If two waves overlap, their displacements add—whether it’s the displacement of air molecules in a sound wave or the electric field in a light wave. You learned in 《 SECTIONS 17.5–17.7 that constructive interference occurs when the crests of two waves overlap, adding to produce a wave with a larger amplitude and thus a greater intensity. Conversely, the overlap of the crest of one wave with the trough of another wave produces a wave with reduced amplitude (perhaps even zero) and thus a lesser intensity. This is destructive interference.

FIGURE 33.3 shows two in-phase sources of sinusoidal waves. The circular rings, you will recall, are the wave crests, and they are spaced one wavelength $\text{λ}$ apart. Although Figure 33.3 is a snapshot, frozen in time, you should envision both sets of rings propagating outward at the wave speed as the sources oscillate.

FIGURE 33.3 Constructive and destructive interference occur along antinodal and nodal lines.

We can measure distances simply by counting rings. For example, you can see that the dot on the right side is exactly three wavelengths from the top source and exactly four wavelengths from the bottom source, making $r_1=3\text{λ}$ and $r_2=4\text{λ}$. At this point, the path-length difference is $\text{Δ}r=r_2-r_1=\text{λ}$. Points where $\text{Δ}\mathit{r}$ is an integer number of wavelengths are points of constructive interference. The waves may have traveled difference distances, but crests align with crests and troughs align with troughs to produce a wave with a larger amplitude.

If the path-length difference is a half-integer number of wavelengths, then the crest of one wave will arrive with the trough of the other. Thus points where $\text{Δ}\mathit{r}$ is a half-integer number of wavelengths are points of destructive interference. Mathematically, the conditions for interference are:

Destructive interference is perfect destructive interference, with zero intensity, only if both waves have exactly the same amplitude.

The set of all points for which $\text{Δ}r=m\text{λ}$ is called an antinodal line. Maximum constructive interference is happening at every point along this line. If these are sound waves, you will hear maximum loudness by standing on an antinodal line. If these are light waves, you will see maximum brightness everywhere an antinodal line touches a viewing screen.

Similarly, maximum destructive interference occurs along nodal lines—analogous to the nodes of a one-dimensional standing wave. If these are light waves, a viewing screen will be dark everywhere it intersects a nodal line.