# Amplitude/Magnitude of waves in a beam

My textbook, Optics, Fifth Edition, by Hecht, says the following in a section on plane waves:

Another approach to visualizing the harmonic plane wave is shown in Fig. 2.23, which depicts two slices across an ideal cylindrical beam. The light is imagined to be composed of an infinitude of sinusoidal wavelets all of the same frequency moving forward in lockstep along parallel paths. The two slices are separated by exactly one wavelength and catch the sinusoids at the place in their cycles where they are all at a crest. The two surfaces of constant phase are flat discs and the beam is said to consist of “plane waves.” Had either slice been shifted a bit along the length of beam the magnitude of the wave on that new front would be different, but it still would be planar. In fact, if the location of the slice is held at rest as the beam progresses through it, the magnitude of the wave there would rise and fall sinusoidally. Notice that each wavelet in the diagram has the same amplitude (i.e., maximum magnitude). In other words, the composite plane wave has the same “strength” everywhere over its face. We say that it is therefore a homogeneous wave. I found the following excerpt confusing:

Had either slice been shifted a bit along the length of beam the magnitude of the wave on that new front would be different, but it still would be planar. In fact, if the location of the slice is held at rest as the beam progresses through it, the magnitude of the wave there would rise and fall sinusoidally.

If my understanding is correct, the magnitude of a wave is the amplitude of the harmonic wavefunction $$\psi(x, t) = A \sin(kx - \omega t) = A \sin(\varphi)$$, which is $$A$$. And, again, if my understanding is correct, this is a constant for any individual wave $$\psi$$. So, if this is all true, then how does it make sense to say, as it seemingly does in the aforementioned excerpt, that the magnitude/amplitude changes as either slice is shifted along the length of the beam, or that, if the location of the slice is held at rest as the beam progresses through it, then the magnitude/amplitude of the wave at the slices would vary? It seems to me that, since the magnitude/amplitude is a constant, inherent property of the wave itself, the magnitude/amplitude should be constant, regardless of which arbitrary part of the wave is being examined?

This is especially confusing given the explanation that immediately follows this part, since it seems to agree with what I just explained:

Notice that each wavelet in the diagram has the same amplitude (i.e., maximum magnitude). In other words, the composite plane wave has the same “strength” everywhere over its face. We say that it is therefore a homogeneous wave.

Am I misunderstanding something? I would greatly appreciate it if people could please take the time to clarify this.

• Thanks for the answer. So you're saying that the author is using "magnitude" to refer to the value of $\psi(x, t)$ for various $x, t$? For instance, if we had "maximum magnitude", this would be where the sine term equals 1, and so we have that $\psi = A$ (the wave is equal to the amplitude $A$)? Am I interpreting you correctly? Nov 27 '19 at 3:16