To understand waves we must deal with functions of two variables. Until now, we have been concerned with quantities that depend only on time, such as $x\left(t\right)$ or $v\left(t\right)$. Functions of the one variable t are appropriate for a particle because a particle is only in one place at a time, but a wave is not localized. It is spread out through space at each instant of time. To describe a wave mathematically requires a function that specifies not only an instant of time (when) but also a point in space (where).

Rather than leaping into mathematics, we will start by thinking about waves graphically. Consider the wave pulse shown moving along a stretched string in FIGURE 16.3. (We will consider somewhat artificial triangular and square-shaped pulses in this section to make clear where the edges of the pulse are.) The graph shows the string’s displacement $\text{Δ}y$ at a particular instant of time $t_1$ as a function of position x along the string. This is a “snapshot” of the wave, much like what you might make with a camera whose shutter is opened briefly at $t_1$. A graph that shows the wave’s displacement as a function of position at a single instant of time is called a snapshot graph. For a wave on a string, a snapshot graph is literally a picture of the wave at this instant.

FIGURE 16.3 A snapshot graph of a wave pulse on a string.

FIGURE 16.4 shows a sequence of snapshot graphs as the wave of Figure 16.3 continues to move. These are like successive frames from a video. Notice that the wave pulse moves forward distance $\text{Δ}x=v\text{Δ}t$ during the time interval $\text{Δ}t$. That is, the wave moves with constant speed.

FIGURE 16.4 A sequence of snapshot graphs shows the wave in motion.

A snapshot graph tells only half the story. It tells us where the wave is and how it varies with position, but only at one instant of time. It gives us no information about how the wave changes with time. As a different way of portraying the wave, suppose we follow the dot marked on the string in Figure 16.4 and produce a graph showing how the displacement of this dot changes with time. The result, shown in FIGURE 16.5, is a displacement-versus-time graph at a single position in space. A graph that shows the wave’s displacement as a function of time at a single position in space is called a history graph. It tells the history of that particular point in the medium.

FIGURE 16.5 A history graph for the dot on the string in Figure 16.4.

You might think we have made a mistake; the graph of Figure 16.5 is reversed compared to Figure 16.4. It is not a mistake, but it requires careful thought to see why. As the wave moves toward the dot, the steep leading edge causes the dot to rise quickly. On the displacement-versus-time graph, earlier times (smaller values of t) are to the left and later times (larger t) to the right. Thus the leading edge of the wave is on the left side of the Figure 16.5 history graph. As you move to the right on Figure 16.5 you see the slowly falling trailing edge of the wave as it moves past the dot at later times.

The snapshot graph of Figure 16.3 and the history graph of Figure 16.5 portray complementary information. The snapshot graph tells us how things look throughout all of space, but at only one instant of time. The history graph tells us how things look at all times, but at only one position in space. We need them both to have the full story of the wave. An alternative representation of the wave is the series of graphs in FIGURE 16.6, where we can get a clearer sense of the wave moving forward. But graphs like these are essentially impossible to draw by hand, so it is necessary to move back and forth between snapshot graphs and history graphs.

FIGURE 16.6 An alternative look at a traveling wave.

EXAMPLE 16.2 Finding a history graph from a snapshot graph

FIGURE 16.7 is a snapshot graph at $t=0\text{s}$ of a wave moving to the right at a speed of $2.0\text{m}\text{/}\text{s}$. Draw a history graph for the position $x=8.0\text{m}$.

FIGURE 16.7 A snapshot graph at $t=0\text{s}$.

FIGURE 16.7 Full Alternative Text

MODEL This is a wave traveling at constant speed. The pulse moves 2.0 m to the right every second.

VISUALIZE The snapshot graph of Figure 16.7 shows the wave at all points on the x-axis at $t=0\text{s}$. You can see that nothing is happening at $x=8.0\text{m}$ at this instant of time because the wave has not yet reached $x=8.0\text{m}$. In fact, at $t=0\text{s}$ the leading edge of the wave is still 4.0 m away from $x=8.0\text{m}$. Because the wave is traveling at $2.0\text{m}\text{/}\text{s}$, it will take 2.0 s for the leading edge to reach $x=8.0\text{m}$. Thus the history graph for $x=8.0\text{m}$ will be zero until $t=2.0\text{s}$. The first part of the wave causes a downward displacement of the medium, so immediately after $t=2.0\text{s}$ the displacement at $x=8.0\text{m}$ will be negative. The negative portion of the wave pulse is 2.0 m wide and takes 1.0 s to pass $x=8.0\text{m,}$ so the midpoint of the pulse reaches $x=8.0\text{m}$ at $t=3.0\text{s}$. The positive portion takes another 1.0 s to go past, so the trailing edge of the pulse arrives at $t=4.0\text{s}$. You could also note that the trailing edge was initially 8.0 m away from $x=8.0\text{m}$ and needed 4.0 s to travel that distance at $2.0\text{m}\text{/}\text{s}$. The displacement at $x=8.0\text{m}$ returns to zero at $t=4.0\text{s}$ and remains zero for all later times. This information is all portrayed on the history graph of FIGURE 16.8.

FIGURE 16.8 The corresponding history graph at $x=8.0\text{m}$.

FIGURE 16.8 Full Alternative Text