I am currently studying Optics, fifth edition, by Hecht. In chapter 2.9 Spherical Waves, the author says the following:
$$\dfrac{\partial^2}{\partial{r}^2}(r \psi) = \dfrac{1}{v^2} \dfrac{\partial^2}{\partial{t}^2} (r \psi) \tag{2.71}$$ Notice that this expression is now just the one-dimensional differential wave equation, Eq. (2.11), where the space variable is $r$ and the wavefunction is the product $(r \psi)$. The solution of Eq. (2.71) is then simply $$r \psi(r, t) = f(r - vt)$$ or $$\psi(r, t) = \dfrac{f(r - vt)}{r} \tag{2.72}$$ This represents a spherical wave progressing radially outward from the origin, at a constant speed $v$, and having an arbitrary functional form $f$. Another solution is given by $$\psi(r, t) = \dfrac{g(r + vt)}{r}$$ and in this case the wave is converging toward the origin. The fact that this expression blows up at $r = 0$ is of little practical concern. A special case of the general solution $$\psi(r, t) = C_1\dfrac{f(r - vt)}{r} + C_2 \dfrac{g(r + vt)}{r} \tag{2.73}$$ is the harmonic spherical wave $$\psi(r, t) = \left( \dfrac{\mathcal{A}}{r} \right) \cos k(r \mp vt) \tag{2.74}$$ or $$\psi(r, t) = \left( \dfrac{\mathcal{A}}{r} \right) e^{ik(r \mp vt)} \tag{2.75}$$ wherein the constant $\mathcal{A}$ is called the source strength. At any fixed value of time, this represents a cluster of concentric spheres filling all space. Each wavefront, or surface of constant phase, is given by $$kr = \text{constant}$$ Notice that the amplitude of any spherical wave is a function of $r$, ware the term $r^{-1}$ serves as an attenuation factor. Unlike the plane wave, a spherical wave decreases in amplitude, thereby changing its profile, as it expands and moves out from the origin. Figure 2.27 illustrates this graphically by showing a "multiple exposure" of a spherical pulse at four different times. The pulse has the same extent in space at any point along any radius $r$; that is, the width of the pulse along the $r$-axis is a constant.
I don't understand this part:
The pulse has the same extent in space at any point along any radius $r$; that is, the width of the pulse along the $r$-axis is a constant.
I don't understand what is meant by "the pulse has the same extent in space at any point along any radius $r$". Is the author claiming that the pulse at $r = t_1$ has the same width as the pulse at $r = t_4$? That doesn't look to be true to me.
I would greatly appreciate it if people would please take the time to explain this.
