Wavelength and Frequency

To completely describe a wave on a string (and the motion of any element along its length), we need a function that gives the shape of the wave. This means that we need a relation in the form y = h(x, t), in which y is the transverse displacement of any string element as a function h of the time t and the position x of the element along the string. In general, a sinusoidal shape like the wave in Fig. 16-1b can be described with h being either a sine or cosine function; both give the same general shape for the wave. In this lesson we use the sine function.

Imagine a sinusoidal wave like that of Fig. 16-1b traveling in the positive direction 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. At time t, the displacement y of the element located at position x is given by

Because this equation is written in terms of position x, it can be used to find the displacements of all the elements of the string as a function of time. Thus, it can tell us the shape of the wave at any given time and how that shape changes as the wave moves along the string.

The names of the quantities in Eq. 16-2 are displayed in Fig. 16-4 and defined next. Before we discuss them, however, let us examine Fig. 16-5, which shows five “snapshots” of a sinusoidal wave traveling in the positive direction of an x axis. The movement of the wave is indicated by the rightward progress of the short arrow pointing to a high point of the wave. From snapshot to snapshot, the short arrow moves to the right with the wave shape, but the string moves 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. 16-5a), this element is at displacement y = 0. In the next snapshot, it is at its extreme downward displacement because a valley (or extreme low point) of the wave is passing through it. It then moves back up through y = 0. In the fourth snapshot, it is at its extreme upward displacement because a peak (or extreme high point) of the wave is passing through it. In the fifth snapshot, it is again at y = 0, having completed one full oscillation.

Fig. 16-4   The names of the quantities in Eq. 16-2, for a transverse sinusoidal wave.

Amplitude and Phase

The amplitude y_m of a wave, such as that in Fig. 16-5, is the magnitude of the maximum displacement of the elements from their equilibrium positions as the wave passes through them. (The subscript m stands for maximum.) Because y_m is a magnitude, it is always a positive quantity, even if it is measured downward instead of upward as drawn in Fig. 16-5a.

The phase of the wave is the argument kx − ωt of the sine in Eq. 16-2. As the wave sweeps through a string element at a particular position x, the phase changes linearly with time t. This means that the sine also changes, oscillating between +1 and −1. Its extreme positive value (+1) corresponds to a peak of the wave moving through the element; at that instant the value of y at position x is y_m. Its extreme negative value (−1) corresponds to a valley of the wave moving through the element; at that instant the value of y at position x is −y_m. Thus, the sine function and the time-dependent phase of a wave correspond to the oscillation of a string element, and the amplitude of the wave determines the extremes of the element’s displacement.

Wavelength and Angular Wave Number

The wavelength λ of a wave is the distance (parallel to the direction of the wave’s travel) between repetitions of the shape of the wave (or wave shape). A typical wavelength is marked in Fig. 16-5a, which is a snapshot of the wave at time t = 0. At that time, Eq. 16-2 gives, for the description of the wave shape,

By definition, the displacement y is the same at both ends of this wavelength—that is, at x = x₁ and x = x₁ + λ. Thus, by Eq. 16-3,

A sine function begins to repeat itself when its angle (or argument) is increased by 2π rad, so in Eq. 16-4 we must have kλ = 2π, or

We call k the angular wave number of the wave; its SI unit is the radian per meter, or the inverse meter. (Note that the symbol k here does not represent a spring constant as previously.)

Notice that the wave in Fig. 16-5 moves to the right by λ from one snapshot to the next. Thus, by the fifth snapshot, it has moved to the right by 1λ.

Fig. 16-5   Five “snapshots” of a string wave traveling in the positive direction of an x axis. The amplitude y_m is indicated. A typical wavelength λ, measured from an arbitrary position x₁, is also indicated.

Period, Angular Frequency, and Frequency

Figure 16-6 shows a graph of the displacement y of Eq. 16-2 versus time t at a certain position along the string, taken to be x = 0. If you were to monitor the string, you would see that the single element of the string at that position moves up and down in simple harmonic motion given by Eq. 16-2 with x = 0:

Here we have made use of the fact that sin(−α) = −sin α, where α is any angle. Figure 16-6 is a graph of this equation; it does not show the shape of the wave.

We define the period of oscillation T of a wave to be the time any string element takes to move through one full oscillation. A typical period is marked on the graph of Fig. 16-6. Applying Eq. 16-6 to both ends of this time interval and equating the results yield

Fig. 16-6   A graph of the displacement of the string element at x = 0 as a function of time, as the sinusoidal wave of Fig. 16-5 passes through the element. The amplitude y_m is indicated. A typical period T, measured from an arbitrary time t₁, is also indicated.

This can be true only if ωT = 2π, or if

We call ω the angular frequency of the wave; its SI unit is the radian per second.

Look back at the five snapshots of a traveling wave in Fig. 16-5. The time between snapshots is T. Thus, by the fifth snapshot, every string element has made one full oscillation.

Fig. 16-7   A sinusoidal traveling wave at t = 0 with a phase constant  of (a) 0 and (b) π/5 rad.

The frequency f of a wave is defined as 1/T and is related to the angular frequency ω by

Like the frequency of simple harmonic motion in Lesson 15, this frequency f is a number of oscillations per unit time—here, the number made by a string element as the wave moves through it. As in Lesson 15, f is usually measured in hertz or its multiples, such as kilohertz.

 CHECK POINT 1 The figure is a composite of three snapshots, each of a wave traveling along a particular string. The phases for the waves are given by (a) 2x − 4t, (b) 4x − 8t, and (c) 8x − 16t. Which phase corresponds to which wave in the figure?

Phase Constant

When a sinusoidal traveling wave is given by the wave function of Eq. 16-2, the wave near x = 0 looks like Fig. 16-7a when t = 0. Note that at x = 0, the displacement is y = 0 and the slope is at its maximum positive value. We can generalize Eq. 16-2 by inserting a phase constant  in the wave function:

The value of  can be chosen so that the function gives some other displacement and slope at x = 0 when t = 0. For example, a choice of  = +π/5 rad gives the displacement and slope shown in Fig. 16-7b when t = 0. The wave is still sinusoidal with the same values of y_m, k, and ω, but it is now shifted from what you see in Fig. 16-7a (where  = 0).