Imagine a sinusoidal wave like that of Fig. 16-1b traveling in the positivedirection of an x axis. As the wave sweeps through succeeding elements (that is,very short sections) of the string, the elements oscillate parallel to the y axis. Attime t, the displacement y of the element located at position x is given byy(x, t) ym sin(kx vt). (16-2)Because this equation is written in terms of position x, it can be used to find thedisplacements of all the elements of the string as a function of time. Thus, it cantell us the shape of the wave at any given time and how that shape changes as thewave moves along the string.The names of the quantities in Eq. 16-2 are displayed in Fig. 16-3 and de-fined next. Before we discuss them, however, let us examine Fig. 16-4, whichshows five “snapshots” of a sinusoidal wave traveling in the positive directionof an x axis. The movement of the wave is indicated by the rightward progressof the short arrow pointing to a high point of the wave. From snapshot to snap-shot, the short arrow moves to the right with the wave shape, but the stringmoves only parallel to the y axis. To see that, let us follow the motion of the red-dyed string element at x 0. In the first snapshot (Fig. 16-4a), this element is atdisplacement y 0. In the next snapshot, it is at its extreme downward dis-placement because a valley (or extreme low point) of the wave is passingthrough it. It then moves back up through y 0. In the fourth snapshot, it is atits extreme upward displacement because a peak (or extreme high point) of thewave is passing through it. In the fifth snapshot, it is again at y 0, having com-pleted one full oscillation
Question
Imagine a sinusoidal wave like that of Fig. 16-1b traveling in the positivedirection of an x axis. As the wave sweeps through succeeding elements (that is,very short sections) of the string, the elements oscillate parallel to the y axis. Attime t, the displacement y of the element located at position x is given byy(x, t) ym sin(kx vt). (16-2)Because this equation is written in terms of position x, it can be used to find thedisplacements of all the elements of the string as a function of time. Thus, it cantell us the shape of the wave at any given time and how that shape changes as thewave moves along the string.The names of the quantities in Eq. 16-2 are displayed in Fig. 16-3 and de-fined next. Before we discuss them, however, let us examine Fig. 16-4, whichshows five “snapshots” of a sinusoidal wave traveling in the positive directionof an x axis. The movement of the wave is indicated by the rightward progressof the short arrow pointing to a high point of the wave. From snapshot to snap-shot, the short arrow moves to the right with the wave shape, but the stringmoves only parallel to the y axis. To see that, let us follow the motion of the red-dyed string element at x 0. In the first snapshot (Fig. 16-4a), this element is atdisplacement y 0. In the next snapshot, it is at its extreme downward dis-placement because a valley (or extreme low point) of the wave is passingthrough it. It then moves back up through y 0. In the fourth snapshot, it is atits extreme upward displacement because a peak (or extreme high point) of thewave is passing through it. In the fifth snapshot, it is again at y 0, having com-pleted one full oscillation
Solution
The given passage describes a sinusoidal wave traveling in the positive direction of an x-axis. The displacement of each element of the wave is parallel to the y-axis and can be represented by the equation y(x, t) = ym sin(kx - vt) (Eq. 16-2). This equation allows us to determine the shape of the wave at any given time and how it changes as it moves along the string.
In Fig. 16-4, five snapshots of the wave are shown, illustrating its rightward progression along the x-axis. The movement of the wave is indicated by a short arrow pointing to a high point of the wave. However, the string itself only moves parallel to the y-axis. For example, the red-dyed string element at x = 0 undergoes different displacements in each snapshot. In the first snapshot, it is at y = 0. In the next snapshot, it reaches its extreme downward displacement as a valley of the wave passes through it. It then moves back up through y = 0. In the fourth snapshot, it reaches its extreme upward displacement as a peak of the wave passes through it. Finally, in the fifth snapshot, it returns to y = 0, having completed one full oscillation.
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