One of the most basic characteristics of waves is the ability of two waves to combine into a single wave whose displacement is given by the principle of superposition. The pattern resulting from the superposition of two waves is often called interference. A standing wave is the interference pattern produced when two waves of equal frequency travel in opposite directions. In this section we will look at the interference of two waves traveling in the same direction.

FIGURE 17.16a shows two light waves impinging on a partially silvered mirror. Such a mirror partially transmits and partially reflects each wave, causing two overlapped light waves to travel along the x-axis to the right of the mirror. Or consider the two loudspeakers in FIGURE 17.16b. The sound wave from loudspeaker 2 passes just to the side of loudspeaker 1; hence two overlapped sound waves travel to the right along the x-axis. We want to find out what happens when two overlapped waves travel in the same direction along the same axis.

FIGURE 17.16 Two overlapped waves travel along the x-axis.

Full Alternative Text

Figure 17.16b shows a point on the x-axis where the overlapped waves are detected, either by your ear or by a microphone. This point is distance $x_1$ from loudspeaker 1 and distance $x_2$ from loudspeaker 2. (We will use loudspeakers and sound waves for most of our examples, but our analysis is valid for any wave.) What is the amplitude of the combined waves at this point?

Throughout this section, we will assume that the waves are sinusoidal, have the same frequency and amplitude, and travel to the right along the x-axis. Thus we can write the displacements of the two waves as

where ${\varphi }_1$ and ${\varphi }_2$ are the phases of the waves. Both waves have the same wave number $k=2\pi \text{/}\text{λ}$ and the same angular frequency $\omega =2\pi f$.

The phase constants ${\varphi }_{10}$ and ${\varphi }_{20}$ are characteristics of the sources, not the medium. FIGURE 17.17 shows snapshot graphs at $t=0$ of waves emitted by three sources with phase constants ${\varphi }_0=0\text{rad,}$ ${\varphi }_0=\pi \text{/}2\text{rad,}$ and ${\varphi }_0=\pi \text{rad}$. You can see that the phase constant tells us what the source is doing at $t=0$. For example, a loudspeaker at its center position and moving backward at $t=0$ has ${\varphi }_0=0\text{rad}$. Looking back at Figure 17.16b, you can see that loudspeaker 1 has phase constant ${\varphi }_{10}=0\text{rad}$ and loudspeaker 2 has ${\varphi }_{20}=\pi \text{rad}$.

FIGURE 17.17 Waves from three sources having phase constants ${\varphi }_0=0\text{rad,}$ ${\varphi }_0=\pi \text{/}2\text{rad,}$ and ${\varphi }_0=\pi \text{rad}$.

Full Alternative Text

Full Alternative Text

Let’s examine overlapped waves graphically before diving into the mathematics. FIGURE 17.18 shows two important situations. In part a, the crests of the two waves are aligned as they travel along the x-axis. In part b, the crests of one wave align with the troughs of the other wave. The graphs are slightly displaced from each other so that you can see what each wave is doing, but the physical situation is one in which the waves are traveling on top of each other.

FIGURE 17.18 Interference of two waves traveling along the x-axis.

Full Alternative Text

The two waves of Figure 17.18a have the same displacement at every point: $D_1\left(x\right)=D_2\left(x\right)$. Two waves that are aligned crest to crest and trough to trough are said to be in phase. Waves that are in phase march along “in step” with each other.

When we combine two in-phase waves, using the principle of superposition, the net displacement at each point is twice the displacement of each individual wave. The superposition of two waves to create a traveling wave with an amplitude larger than either individual wave is called constructive interference. When the waves are exactly in phase, giving $A=2a$, we have maximum constructive interference.

In Figure 17.18b, where the crests of one wave align with the troughs of the other, the waves march along “out of step” with $D_1\left(x\right)=-D_2\left(x\right)$ at every point. Two waves that are aligned crest to trough are said to be 180$\text{°}$ out of phase or, more generally, just out of phase. A superposition of two waves to create a wave with an amplitude smaller than either individual wave is called destructive interference. In this case, because $D_1=-D_2$, the net displacement is zero at every point along the axis. The combination of two waves that cancel each other to give no wave is called perfect destructive interference.