Section 29–3 Sinusoidal waves

(Angular frequency w / Wave number k / w-k relationship)

 

The three interesting points discussed in this section are angular frequency (angular temporal frequency), wave number (angular spatial frequency), and the relationship between angular frequency and wave number.

 

1. Angular frequency:

“The angular frequency ω can be defined as the rate of change of phase with time (radians per second). (Feynman et al., 1963, section 29–3 Sinusoidal waves).”

 

Feynman defines angular frequency (ω) as the rate of change of phase with time (radians per second). For a sinusoidal wave, angular frequency can be simply described as the ratio of an angular displacement to the time taken (e.g., ω = 2p/T in which T is the period of oscillation). Alternatively, we may use the term angular temporal frequency because ω is related to time and it helps to distinguish the concept of wave number that is the rate of change of phase with distance. Similarly, the term acceleration could be defined either as the rate of change of velocity “with time” or “with distance.” To be more specific, we can define angular temporal frequency as the rate of change of an oscillator’s phase (or angular displacement) with respect to time that is applicable to a sinusoidal wave.

 

To recall the concept of phase, we can refer to Chapter 21 in which Feynman suggests: “The constant Δ is sometimes called the phase of the oscillation, but that is a confusion, because other people call ω₀t+Δ the phase, and say the phase changes with time. We might say that Δ is a phase shift from some defined zero. Let us put it differently. Different Δ’s correspond to motions in different phases. That is true, but whether we want to call Δ the phase, or not, is another question...” In short, a phase is a state of an oscillator pertaining to its position and direction of motion. The term phase has been used traditionally to describe the successive stages of various cycles, such as the periodical appearance of the Moon as it orbits the Earth.

 

2. Wave number:

“… we can define a quantity called the wave number, symbolized as k. This is defined as the rate of change of phase with distance (radians per meter). (Feynman et al., 1963, section 29–3 Sinusoidal waves).”

 

Feynman defines wave number (k) as the rate of change of phase with distance (radians per meter). However, this term is a misnomer because it is not strictly a number and k is sometimes used as a complex function. In dispersive media (e.g., water), k is a complex function of frequency that has both real and imaginary parts. Thus, a better term could be angular spatial frequency instead of angular wavenumber or wave number. Specifically, we can define angular spatial frequency as the rate of change of an oscillator’s phase with respect to distance that is applicable to a sinusoidal wave (instead of a wave packet). From an operational perspective, k can be measured as the number of the oscillator’s cycles per unit distance (e.g., k = 2p/l).

 

According to Feynman, the formula for a cosine wave moving in a direction x with a wave number k and an angular frequency ω will be written in general as cos (ωt−kx). Perhaps he could have elaborated that it can be written as a sine function or cosine function, and it can be written as either cos (ωt−kx) or cos (kx−ωt) because cos q = cos (−q). In general, cos (kx−ωt) may refer to a wave moving to the right, whereas cos (kx+ωt) may refer to another wave moving to the left. If we fix t as a specific instant in time (e.g., t = 0), we can have a snapshot graph that shows how a wave’s vertical displacement changes as a function of x. If we fix x as a specific point in space (e.g., x = 0), we can have a history graph that shows how a wave’s vertical displacement changes with time.

 

3. w-k relationship:

“Now in our particular wave there is a definite relationship between the frequency and the wavelength, but the above definitions of k and ω are actually quite general (Feynman et al., 1963, section 29–3 Sinusoidal waves).”

 

Feynman clarifies that there is a definite relationship between the frequency and the wavelength in our particular wave, but the definitions of ω and k provided in this section are quite general. That is, ω and k are not related in the same way in other physical circumstances. Perhaps Feynman could simply explain that the definite relationship shown is applicable only to sinusoidal waves that have only one frequency and one wavelength. In other words, ω and k are related in a complicated manner for a wave packet. This leads to the concept of group velocity and phase velocity that will be covered in chapter 48 when Feynman emphasizes that “[t]he group velocity is the derivative of ω with respect to k, and the phase velocity is ω/k.”

 

In the end of the section, Feynman mentions that equation (29.1) is a legitimate formula because it is applicable to the “wave zone” (the region that is beyond a few wavelengths). However, the term “wave zone” is not commonly used in this context. More interestingly, in section 34-7 The ω, k four-vector, Feynman says, “if ω is thought of as being like t, and k is thought of as being like x divided by c², then the new ω′ will be like t′, and the new k′ will be like x′/c².” That is, the angular frequency ω of a sinusoidal wave and its wave number k transform in the same way as space and time under the Lorentz transformation. Thus, it is worthwhile mentioning that the wave number k and the angular frequency ω are interrelated to the extent they are analogous to the space and time in special relativity.

 

Review Questions:

1. Would you use the term angular frequency and define it as the rate of change of phase with time?

2. Would you use the term wave number and define it as the rate of change of phase with distance?

3. How would you describe the relationship between the angular frequency and wave number of an oscillator?

 

The moral of the lesson: the angular frequency (ω) is the rate of change of phase with time, whereas the wave number (k) is the rate of change of phase with distance; ω = 2p/T = 2pf and k = 2p/l implies ω/k = fl = c (for a sinusoidal wave).

 

Reference:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.