The classical physics ideas of particles and waves are mutually exclusive. An object can be one or the other, but not both. These classical models fail to describe the wave–particle duality seen at the atomic level. An alternative model with both particle and wave characteristics is a wave packet.

Consider the wave shown in FIGURE 39.11. Unlike the sinusoidal waves we have considered previously, which stretch through time and space, this wave is bunched up, or localized. The localization is a particle-like characteristic. The oscillations are wave-like. Such a localized wave is called a wave packet.

FIGURE 39.11 History graph of a wave packet with duration $\text{Δ}t$.

A wave packet travels through space with constant speed $v$, just like a photon in a light wave or an electron in a force-free region. A wave packet has a wavelength, hence it will undergo interference and diffraction. But because it is also localized, a wave packet has the possibility of making a “dot” when it strikes a detector. We can visualize a light wave as consisting of a very large number of these wave packets moving along together. Similarly, we can think of a beam of electrons as a series of wave packets spread out along a line.

Wave packets are not a perfect model of photons or electrons (we need the full treatment of quantum physics to get a more accurate description), but they do provide a useful way of thinking about photons and electrons in many circumstances.

You might have noticed that the wave packet in Figure 39.11 looks very much like one cycle of a beat pattern. You will recall that beats occur if we superimpose two waves of frequencies $f_1$ and $f_2$ where the two frequencies are very similar: $f_1\approx f_2$. FIGURE 39.12, which is copied from Chapter 17 where we studied beats, shows that the loud, soft, loud, soft,. . . pattern of beats corresponds to a series of wave packets.

FIGURE 39.12 Beats are a series of wave packets.

In Chapter 17, the beat frequency (number of pulses per second) was found to be

where $\text{Δ}f$ is the range of frequencies that are superimposed to form the wave packet. Figure 39.12 defines $\text{Δ}t$ as the duration of each beat or each wave packet. This interval of time is equivalent to the period $T_{\text{beat}}$ of the beat. Because period and frequency are inverses of each other, the duration $\text{Δ}t$ is

We can rewrite this as

Equation 39.20 is nothing new; we are simply writing what we already knew in a different form. Equation 39.20 is a combination of three things: the relationship $f=1\text{/}T$ between period and frequency, writing $T_{\text{beat}}$ as $\text{Δ}t$, and the specific knowledge that the beat frequency $f_{\text{beat}}$ is the difference $\text{Δ}f$ of the two frequencies contributing to the wave packet. As the frequency separation gets smaller, the duration of each beat gets longer.

When we superimpose two frequencies to create beats, the wave packet repeats over and over. A more advanced treatment of waves, called Fourier analysis, reveals that a single, nonrepeating wave packet can be created through the superposition of many waves of very similar frequency. FIGURE 39.13 illustrates this idea. At one instant of time, all the waves interfere constructively to produce the maximum amplitude of the wave packet. At other times, the individual waves get out of phase and their superposition tends toward zero.

FIGURE 39.13 A single wave packet is the superposition of many component waves of similar wavelength and frequency.

Suppose a single nonrepeating wave packet of duration $\text{Δ}t$ is created by the superposition of many waves that span a range of frequencies $\text{Δ}f$. We’ll not prove it, but Fourier analysis shows that for any wave packet

The relationship between $\text{Δ}f$ and $\text{Δ}t$ for a general wave packet is not as precise as Equation 39.20 for beats. There are two reasons for this:

1. Wave packets come in a variety of shapes. The exact relationship between $\text{Δ}f$ and $\text{Δ}t$ depends somewhat on the shape of the wave packet.

2. We have not given a precise definition of $\text{Δ}t$ and $\text{Δ}f$ for a general wave packet. The quantity $\text{Δ}t$ is “about how long the wave packet lasts,” while $\text{Δ}f$ is “about the range of frequencies needing to be superimposed to produce this wave packet.” For our purposes, we will not need to be any more precise than this.

Equation 39.21 is a purely classical result that applies to waves of any kind. It tells you the range of frequencies you need to superimpose to construct a wave packet of duration $\text{Δ}t$. Alternatively, Equation 39.21 tells you that a wave packet created as a superposition of various frequencies cannot have an arbitrarily short duration but must last for a time interval $\text{Δ}t\approx 1\text{/}\text{Δ}f$.

EXAMPLE 39.3 Creating radio-frequency pulses

A short-wave radio station broadcasts at a frequency of 10.000 MHz. What is the range of frequencies of the waves that must be superimposed to broadcast a radio-wave pulse lasting $0.800\mu \text{s}?$

MODEL A pulse of radio waves is an electromagnetic wave packet, hence it must satisfy the relationship $\text{Δ}f\text{Δ}t\approx 1$.

VISUALIZE FIGURE 39.14 shows the pulse.

FIGURE 39.14 A pulse of radio waves.

FIGURE 39.14 Full Alternative Text

SOLVE The period of a 10.000 MHz oscillation is $0.100\mu \text{s}$. A pulse $0.800\mu \text{s}$ in duration is 8 oscillations of the wave. Although the station broadcasts at a nominal frequency of 10.000 MHz, this pulse is not a pure 10.000 MHz oscillation. Instead, the pulse has been created by the superposition of many waves whose frequencies span

$$\text{Δ}f\approx {\displaystyle \frac{1}{\text{Δ}t}}={\displaystyle \frac{1}{0.800\times {10}^{-6}\text{s}}}=1.250\times {10}^6\text{Hz}=1.250\text{MHz}$$

This range of frequencies will be centered at the 10.000 MHz broadcast frequency, so the waves that must be superimposed to create this pulse span the frequency range

$$9.375\text{MHz}\le f\le 10.625\text{MHz}$$