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Abstract
Rainbows and boat wakes may seem unrelated, but they share deep mathematical connections through ray folding, caustics, and Airy interference. This paper explores these principles, which are also relevant for explaining phenomena such as shimmering effects on the bottom of pools and twinkling stars. By revisiting Airy’s theories on wavefronts and caustics, we demonstrate their applications not only in optics and for water waves but also in quantum wave packets. Using concepts from undergraduate physics, we highlight the universal patterns that unify these diverse phenomena.
“Caminante no hay camino,
sino estelas en la mar.”
(Traveler, there is no road;
only a ship’s wake on the sea)
Antonio Machado[1]
“Why are there so many songs about rainbows
And what’s on the other side”
Paul Williams[2]
I Introduction
Rainbows and boat wakes are two examples of wave phenomena, the first related to electromagnetism
and the second involving gravity waves on a liquid surface.
Despite this shared property, the two phenomena may seem unrelated at first glance.
However, a deep mathematical and physical relation exists between them.
Although the physics of rainbows and boat wakes is well understood individually[3, 4], the pedagogical value of this work lies in emphasizing the unifying principles that connect them.
These principles include ray folding, caustics, and Airy interference, which
also play a central role in other everyday phenomena, such as
the shimmering patterns that can be observed on the bottom of a swimming pool, the twinkling of stars, and the reflections
of light inside a cup of tea.
The modern theory of the rainbow is due to George Biddle Airy[6], who improved upon the Cartesian-Newtonian ray optics theory. Specifically, ray optics failed to explain how the water droplet size influences the colors of the bow and could not account for the supernumerary arcs, the additional, fainter arcs separated by darker regions that appear inside the main rainbow (Fig. 1).
This phenomenon was documented as early as the 13th century [4].
Two important concepts were central to Airy’s treatment: the Huygens concept of wavefronts, and the mathematical concept of caustic surfaces—the envelope of a family of reflected or refracted light rays[7]. A common example of a caustic curve, a plane section of a caustic surface, are the bright arcs on the bottom of a swimming pool formed by light rays refracted at the surface.
Applying these concepts to rainbows, Airy derived an analytical expression for the light intensity at each point of the bow and showed that the brightest region lies at a radius smaller than what geometrical optics predicts.
In this paper, we will show using undergraduate-level concepts that Airy’s treatment of caustics can also successfully apply to boat wakes and quantum wave packets, revealing a universal principle that underpins both phenomena.
Figure 1: (Color online) Supernumerary arcs generated on a sunny day from drops produced by a flower mister. Photograph by one of the authors (A.G.R.)
II Caustics from cubic wavefronts.
In the context of geometric optics,
caustics are the envelopes of a family of rays along which light propagates.
The detailed analysis of caustics goes back to the studies of burning mirrors and lenses in the Middle Ages. Astronomer and musician Francesco Maurolico (1494–1575) noted that, due to spherical aberration, light rays hitting a spherical mirror do not converge at a single focal point, but rather on a surface[8]. The caustic
surface is the surface to which the light rays are tangent.
In this section we will discuss the main properties of caustics with a simple generic example using light rays which is applicable to the three physical realizations presented in the rest of the paper.
Let us consider a two-dimensional curve defining a starting wavefront.
Rays emerge perpendicularly at every point of the wavefront,
and they converge or diverge depending on the wavefront concavity.
Points of constant optic length along the rays define the time evolution of the wavefront.
We consider a situation in which the starting wavefront generates a combination of convergent and divergent rays. In other words the starting wavefront has an inflection point. We will now show how this wavefront gives rise to a caustic line.
The simplest continuous curve that exhibits this behavior is a cubic wavefront, which happens to arise naturally in the physical phenomena discussed in this paper. It is also the generic form of a smooth curve sufficiently close to an inflection point.
We show an example of such a wavefront in
Fig. 2(a). Note that when the wave has propagated sufficiently far away
from the initial cubic line, the wavefronts “fold”, giving rise to two divergent wavefronts of equal optical paths (Fig. 2(b), dash-dotted lines), that meet at the caustic line (dashed line).
The folded wavefront also means that, for any direction emerging from the wavefront, two parallel rays exist,
as illustrated with the rays a and a^{\prime} in
Fig. 2(a).
Near the inflection point, the starting wavefront can be described with the equation y=-x^{3}/3d^{2}, with d a scale factor (see Figure 3(a)). Equivalently, in parametric form we have:
\mathbf{x}(x)=\left(x,-\frac{x^{3}}{3d^{2}}\right).The unit normals \hat{\mathbf{n}}\equiv(n_{x},n_{y})=(\sin\alpha,\cos\alpha) to the curve \mathbf{x}(x) correspond to the directions of the light rays. The angle α of the normal with the vertical is given by:
\alpha\simeq\tan\alpha=\frac{n_{x}}{n_{y}}=\left({x\over d}\right)^{2}.
Figure 2: (a): Rays normal to a cubic wavefront parametrized as \mathbf{x}=(x,-x^{3}/(3d^{2})). Rays a and a^{\prime}, originating at \mathbf{x} and -\mathbf{x}, are parallel and differ in path length by {}2\mathbf{x}\cdot\hat{\mathbf{n}}, where \hat{\mathbf{n}} is the normal to the wavefront at \mathbf{x}. Ray a belongs to a “fold”: it is part of a beam of rays
that are initially convergent and then become divergent upon propagation.(b): Wavefronts propagating from the initial wavefront shown in panel (a). The structure of these wavefronts, particularly the “fold” (giving rise to two superimposed divergent fronts (dotted)) and the caustic line (dashed), is similar to the one observed in the case of raindrops (Fig. 3b).
Considering the situation in which rays impact a very distant screen (or are observed by a distant eye), the coordinate on the screen is simply proportional to α.
Disregarding interference among rays,
the intensity observed for each direction α is given by the density I(\alpha) of rays as a function of α. The
quadratic dependence of α on x (for small x)
of Eq. 2 shows no solutions for \alpha<0; in other words, no rays will emerge with \alpha<0, meaning that the intensity is zero for negative angles. If we consider the number of rays per unit length of the wavefront I_{0} as uniform, the number of rays dN incident in an interval dx is I_{0}\sqrt{1+\alpha^{2}}dx. These rays emerge in the interval (\alpha,\alpha+d\alpha): dN=I(\alpha)d\alpha, with I(\alpha) the density of rays per unit angle. Using Equation
(2) we obtain, for small α
I(\alpha)=I_{0}\sqrt{1+\alpha\texttwosuperior}{dx\over d\alpha}\simeq\frac{I_{0}d}{2\alpha^{1/2}}.The above expression shows a divergent accumulation of rays
at the angular position of the caustic:\alpha\to 0^{+}.
In the next section we will see how the previous generic description applies to the case of the rainbow.
We will then detail the additional improvements that led Airy to quantitatively describe the light intensity in a rainbow.
III Airy’s (and Young’s) theory of the rainbow.
In the geometric optics description, the primary rainbow forms when sunlight undergoes two refractions and one reflection inside a spherical water droplet (Fig. 3).
The key point in the formation of the rainbow is that the angle \theta between the incident ray and the ray emerging from the droplet varies non-monotonically with the distance b of the incident ray to the droplet’s center. This non-monotonic dependence gives rise to a caustic.
Figure 3: (a) Rays travelling through a drop, to form a rainbow. The caustic ray of minimum deviation is indicated by the thicker continuous line.
We also show two additional rays emerging along the same direction, that entered the drop on both sides of the caustic ray (dashed and dotted lines). Since the optical lengths P_{1}Q_{1} and P_{2}Q_{2} are different these two rays interfere. We show in red the virtual cubic wavefront from which two –in phase– rays (dash-dotted) originate and give the same path difference. (b) Wavefront description. In rainbows, the sunlight arrives on a droplet with planar wavefronts which are distorted after being submitted to one or several reflections within the droplet. Note the “folded” nature of the wavefront that comes out of the droplet.
The observation directions of the main bow and of the first two supernumeraries (according to Young, see the discussion in text) are indicated.
Specifically, if we define the dimensionless length z\equiv\frac{b}{R} where R is the radius of the droplet, a straightforward geometric calculation using Snell’s law yields
\theta(z)=4\sin^{-1}(z/n)-2\sin^{-1}z,where n is the refractive index of the droplet. This function has a minimum for
z_{0}=\sqrt{4-n^{2}\over 3},which, for n\simeq 1.33, gives the well-known angle of the principal rainbow \theta_{0}\simeq 42^{\circ}.
In other words, the rainbow appears at an angle corresponding to a sharp boundary between a dark and a bright region where light rays concentrate. This angle therefore corresponds to a caustic, as described in the previous section.
Differences in the value of n for the different wavelengths are responsible for the spread of colors in the rainbow.
If we stay within the geometric optics description, the intensity decreases monotonically as the viewing angle deviates from \theta_{0}.
To compute the ray intensity as in Equation (3), where we now have \alpha=\theta-\theta_{0},
we expand the angle between rays around the extremum
\alpha\simeq\frac{1}{2}\theta^{\prime\prime}(z_{0})\delta^{2},with \delta\equiv(z-z_{0}) and
\theta^{\prime\prime}(z_{0})=-{9\over 2}{\sqrt{4-n^{2}}\over({n^{2}-1})^{3/2}},and obtain for the ray intensity I close to the caustic:
I(\alpha)=\frac{I_{0}}{\left|d\alpha/dz\right|}=\frac{I_{0}}{|\theta^{\prime\prime}(z_{0})|\delta}=\frac{I_{0}}{\sqrt{2\theta^{\prime\prime}(z_{0})\alpha}}.In the present case we have a family of initially parallel rays (perpendicular to the vertical line P_{1}P_{2} of Figure 3) that emerges from the droplet forming a small angle \alpha\sim{1\over 2}\theta^{\prime\prime}(z_{0})\delta^{2} with the direction of the caustic ray.
For each value of the exit angle α, there are two light rays contributing, as discussed in the previous section.
In the latter case, the dependence of α on x (Eq. (2)) originated in the form of the cubic wavefront:y=-x^{3}/(3d^{2}).
Note that, since Eqs. (1) and (2) can be written in
terms of a dimensionless variable \tilde{\delta}\equiv x/d (as y=-d\tilde{\delta}^{3}/3 and \alpha=\tilde{\delta}^{2}), the dependence of α on δ for the rainbow (Eq. (6)) can be thought as originating from a virtual cubic wavefront (Fig. 3a)
given by
y=-{R\over 6}\theta^{\prime\prime}(z_{0})\delta^{3},where y is measured along the direction of the rainbow´s main bow (See Figure 3a).
The existence of two rays for each observation direction, combined with the assumption of the wave-like nature of light, provided the key ingredients to address the phenomenon of “supernumerary arcs.” This problem was first tackled in the early 19th century by Thomas Young and later refined by George Airy, as we will discuss in the following sections.
III.1 Young’s theory
Thomas Young provided the first
explanation linking the supernumerary arcs to interference effects[10].
Young attributed the dark spaces between supernumerary arcs to destructive interference between the two rays emerging from each water droplet in the same direction.
Two such rays are indicated in Fig. 2 as a and a^{\prime}. Their difference in optical path \ell is given as
\displaystyle\delta\ell=|2\mathbf{x}(x)\cdot\hat{\mathbf{n}}| \\ \displaystyle= \\ \displaystyle\left|2\left(x\sin\alpha-{x^{3}\over 3d^{2}}\cos\alpha\right)\right| \\ \displaystyle\simeq \\ \displaystyle{4\over 3}d\alpha^{3/2},where where \hat{\mathbf{n}} is the normal to the wavefront at \mathbf{x}, and where we used \sin\alpha\sim\alpha and \cos\alpha\sim 1.
In the case of the virtual cubic front emerging from the droplet:
\delta\ell={4\over 3}R\sqrt{2\over\theta^{\prime\prime}(z_{0})}\alpha^{3/2}.Constructive interference arises when the path difference satisfies\delta\ell=m\lambda,
with λ denoting the wavelength of light and m an integer.
The intensity is then modulated by a factor of\cos^{2}\left({\pi\delta\ell}/{\lambda}\right).
The
Young’s result for the intensity is obtained as (using Eqs. 12 and 8)
I(\alpha)=\frac{I_{0}}{\sqrt{2\theta^{\prime\prime}(z_{0})\alpha}}\cos^{2}\left(\frac{4\pi R}{3\lambda}\sqrt{2\over\theta^{\prime\prime}(z_{0})}\alpha^{3/2}\right).We plot this intensity in Figure 4.
Notice the existence of supernumerary arcs, separated by points of zero intensity, corresponding to the destructive
interference condition. The main arc appears as \alpha\to 0^{+}, where the intensity diverges. This unrealistic
behavior is superseded by Airy’s theory that we discuss in the next section.
III.2 Airy’s theory of supernumerary arcs
Whereas Young treated the problem of the supernumerary arcs as an interference effect between two parallel rays that propagate perpendicular to the source wave front, Airy considered that every point on the wavefront generates rays in all possible directions (i.e., for all values of α) and computed the amplitude U as the sum of the contributions from all source points.
Following Airy, we consider rays with a given α that emerge from every point of the cubic front (Fig. 2). All of them are focused at the same point when observed from a very large distance \boldsymbol{\rho}=\rho\hat{\boldsymbol{n}}, and we should account for the interference among all of them.
The optical path of these rays is given by
\ell=|\boldsymbol{\rho}-\mathbf{x}|\simeq\rho-\hat{\boldsymbol{n}}\cdot\mathbf{x}\simeq\rho-\alpha x+{1\over 3}{x^{3}\over d^{2}},where \rho is the reference optical length of the ray corresponding to x=0 once it has propagated to the observation point.
The amplitude U(\alpha) is therefore given by the superposition of plane waves that originate from different points on the wavefront. This is essentially the content of Huygens’s principle.
We obtain
\displaystyle U(\alpha) \\ \displaystyle= \\ \displaystyle e^{ik\rho}\int_{-\infty}^{\infty}dxe^{-ik\left(\alpha x-{1\over 3}{x^{3}\over d^{2}}\right)} \\ \displaystyle= \\ \displaystyle 2\pi\left(d^{2}\over k\right)^{1/3}e^{ik\rho}{\rm{Ai}}\left(-(kd)^{2/3}\alpha\right)where k\equiv 2\pi/\lambda, d=R\sqrt{2/|\theta^{\prime\prime}(z_{0})|} and {\rm{Ai}}(x)\equiv\frac{1}{\pi}\int_{0}^{\infty}du\cos(ux+u^{3}/3)
the celebrated Airy function.
Figure 4: Rainbow intensities as a function of the angle \alpha\equiv\theta-\theta_{0} from the “classical” angle \theta_{0}\simeq 42^{\circ} , for the different approaches presented in the text. The dashed line is the result of Eq. (8), which completely disregards the wave nature of light. Young’s curve corresponds to Eq. (13), and Airy’s result is the one of Eq. (16). Notice that away from \alpha=0, Young’s and Airy’s approaches give a remarkably similar form, although the maxima are shifted. This shift is related to the so-called Gouy’s phase[16], which is a phase shift that occurs when rays go through a caustic or a focus. See the Supplementary Material for a brief account of this effect.
To better understand the form of the Airy function [11] (see also Appendix I), consider the case where \alpha>0. In this scenario, the argument of the exponential (denoted as \Theta) in Eq. (15) has two critical points, u_{\pm}=\pm d\sqrt{\alpha}, where d\Theta/du=0. Since \Theta is stationary at u_{\pm}, the regions around these points provide the dominant contributions to the Airy function.
Approximating the integral by focusing on these contributions is known as the method of stationary phase, or saddle point approximation (see Appendix I and Ref.[11]).
The contributions from the regions around u_{+} and u_{-} differ in phase, with a phase difference depending on the value of α. As a result, the Airy function oscillates when \alpha>0.
On the other hand, for \alpha<0, there are no points of stationary phase, and \Theta varies significantly with u throughout, leading the Airy function to be exponentially attenuated. The Airy integral also provides a smooth transition between the oscillatory regime for \alpha>0 and the exponentially decaying regime for \alpha<0.
In Figure 4 we show the intensity according to Airy’s (Eq. (16)) (the intensity is obtained as I=|U|^{2}) and Young’s (Eq. (13)) results. One important success of Airy’s treatment is to provide a finite intensity at the caustic edge.
In addition, it shows that the maximum intensity does not coincide with the predictions of geometric optics. It also accurately
accounts for the angular position of the supernumerary arcs that depends on the radius of the droplet, and are incorrectly predicted in Young’s treatment (see the Supplementary Material for a discussion of this difference).
The droplet radius R affects directly some observable characteristics of the rainbow.
Airy’s result shows that each wavelength produces a disk of light that is brightest at its caustic edge and fades inward.
The angular width \Delta\alpha of the disk is approximately (Eq. (13) and Fig. 4) \Delta\alpha\simeq(\lambda/R)^{2/3}.
These disks overlap substantially for different wavelengths, all the more as R is small, mixing colors and reducing spectral purity. As R increases, the arcs of different colors become more sharply defined and less overlapping, resulting in a brighter and more vividly colored rainbow.
This is why the most vivid rainbows typically appear during the warmer months, especially after an afternoon thunderstorm,
as such conditions favor the formation of large raindrops.
In fact, thunderstorms involve intense upward air currents (convection), which suspend droplets in the atmosphere longer, allowing them to grow larger by merging with smaller droplets.
The dependence on the raindrop radius also explains the fact that
supernumerary arcs are more prominent at the bow’s apex[5]. In fact, since raindrops increase in size while falling, they are the smallest at the central point of the rainbow, and this produces the largest angular separation between supernumerary arcs, according to Airy’s formula.
Airy’s treatment is widely regarded as the definitive theory of the rainbow, primarily because it explains most of observed properties under realistic conditions. However, it is important to note that Airy’s theory still describes light propagation in terms of rays. This approach has its limitations, especially when the size of raindrops becomes comparable to the wavelength of the incident light. In such cases, a more complete wave theory—specifically, one based on the Maxwell equations of the electromagnetic field—is needed. For a spherical drop, this is captured by Mie’s scattering theory[19], which, among other things, predicts that rainbows tend to appear fuzzier and more whitish when the raindrops are very small.
IV Airy oscillations in boat wakes
The treatment of the rainbow presented in the previous section is found in a variety of texts.
In contrast, the analysis of boat wakes (originally considered by Kelvin[13])
is less common and has even recently sparked controversies[14].
The application of Airy’s theory to the Kelvin wake problem is also rarely discussed, although it can be found in Ref. [15]. We present the Kelvin wake problem
in a way that highlights its analogy with the formation of rainbows.
Consider a nearly point-like source that creates a disturbance on the surface (Oxy) of a deep liquid. The source moves with uniform velocity v along a direction that we define to be the x axis. We want to determine the most general expression for the perturbation in surface height h(x,y) that this source can produce.
To solve the problem, we consider the two half-planes y>0 and y<0 separately, using the y=0 line along which the boat moves as a boundary condition.
In each half-plane, we
decompose the surface perturbations into plane waves, taking advantage of
the fact that, in the frame moving with the source, the resulting pattern should be stationary.
Each plane wave of wave vector {\bf k}\equiv(k_{x},k_{y}) corresponds to a perturbation that is proportional to (the real part of) \exp[i(k_{x}x+k_{y}y-\omega({\bf k})t)], with \omega given by the medium’s dispersion relation. For deep-water waves it reads[17]
\omega({\bf k})=\sqrt{g|{\bf k}|},with g the gravitational acceleration.
In order for the amplitude to be stationary in the frame of reference moving with the source, the phase
of the plane must be proportional to (x-vt). In other words, since the height h of the disturbance must satisfy h(x,y,t)=h(x-vt,y,0), we have the following relation:
\exp\left[i(k_{x}x+k_{y}y-\omega(\mathbf{k})t\right]=\exp\left[i(k_{x}(x-vt)+k_{y}y\right],which is satisfied if
k_{x}v=\omega=g^{1/2}(k_{x}^{2}+k_{y}^{2})^{1/4}.Then, k_{y} can be expressed in terms of k_{x}:
k_{y}=\pm k_{x}\sqrt{\frac{k_{x}^{2}v^{4}}{g^{2}}-1}.We remark that, from Eq. (19), k_{x} and v must have the same sign and we take both as positive.
Also, when considering the y>0 (y<0) half plane, we must have k_{y}>0 (k_{y}<0) in order to have
plane waves that move away from the boat.
From now on, we focus on the y>0 half-plane, measure x and y in units of v^{2}/g, and use k to refer to k_{x}v^{2}/g.
The most general form of a surface disturbance then is:
h=\int_{1}^{\infty}dkA_{k}e^{i\left(kx+k\sqrt{k^{2}-1}y\right)}.The lower limit of the integral is 1, because lower values of k produce waves that exponentially attenuate for |y|\to\infty, and we disregard them. The coefficients A_{k} and the exact value of the upper integration limit are important in determining the actual details of the wake for a boat of given shape and dimensions.
As we are interested in a point source we can take A_{k}\equiv 1, and the upper limit as infinite.
There is a strong formal similarity between Eq. (21) and Eq. (15) that reveal a striking analogy between the supernumerary arcs and the water wake pattern.
To uncover this similarity, we first evaluate Eq. (21) with the saddle point method.
For large values of x and y, the argument of the integral in Eq. (21) is rapidly varying, and Eq. (21) gets its main contribution from k-values where the phase is stationary: \partial({kx+k\sqrt{k^{2}-1}}y)/\partial k=0. These points are
k_{\pm}(x,y)=-{1\over\sqrt{8}}{x\over y}\sqrt{4+\left({y\over x}\right)^{2}\pm\sqrt{1-8\left({y\over x}\right)^{2}}}.Equation (22) provides either two or no values, depending on whether x/y is less or greater than -\sqrt{8}. This condition defines an angle \theta_{K}=\tan^{-1}(1/\sqrt{8})=19.5^{\circ}, which is the angle of the famous Kelvin’s cone observed as a boat moves. Note that the value of \theta_{K} is totally independent of the boat’s velocity.
The cone appears as a caustic, separating a region with two phase-stationary solutions from another “dark” region without any possible stationary phase solution.
Re-inserting the values of k from Eq. (22) back into the phase of Eq. (21), we obtain an oscillatory envelope \exp\left\{i\left(k_{\pm}x+k_{\pm}\sqrt{k_{\pm}^{2}-1}y\right)\right\} of h(x,y) in the stationary phase approximation.
The maxima of these perturbations are plotted in Fig. 5.
This result, first obtained by Kelvin, predicts that the Kelvin cone acts as a sharp boundary separating regions of zero and finite wave amplitude. However, this sharp boundary is only an artifact of the approximation; an exact evaluation of the integral in Eq. (21) reveals a smooth crossover between the two regions.
Figure 5: For a boat at the right tip, moving towards the right, we show with continuous lines lines the positions of the maxima of perturbations (crests) as obtained within the stationary phase approximation. There are no solutions outside the Kelvin cone (dashed line), with half angle 19.5∘ (dashed lines).
This diagram contains some of the essential features exhibited by the observed wake. The angle of the cone (the caustic) matches that observed in real boat wakes. In addition, inside the cone there are arcs corresponding to a slightly curved plane wave whose phase velocity matches the boat velocity v. This is clearly observed in Fig. 7. Using Eq. (17), we see that the velocity dependent wavelength \lambda(v) of this portion of the pattern is \lambda(v)=2\pi v^{2}/g. Once diffraction effects near the caustic are included, we obtain a more realistic pattern near the edge of the wake that includes a finite wave amplitude beyond the cone, in close analogy to what is observed in the case of the rainbow (See Fig. 6).
To evaluate Eq. (21) analytically near the Kelvin cone,
one should note that the phase in the integral in Equation (21) has an inflexion point at k_{0}=\sqrt{3/2},
at which its second derivative vanishes.
Expanding around this point, we obtain
kx+\left(k\sqrt{k^{2}-1}\right)y\simeq\left(\sqrt{\frac{3}{2}}x+\frac{\sqrt{3}}{2}y\right)+u(x+\sqrt{8}y)+\sqrt{8}yu^{3},where u=k-k_{0}. In this way, near u\sim 0
h=Ce^{i\left(\sqrt{\frac{3}{2}}x+\frac{\sqrt{3}}{2}y\right)}\int due^{i\left(u(x+\sqrt{8}y)+\sqrt{8}yu^{3}\right)},with C an overall constant.
After integrating over u
we obtain (taking the real part of the final expression)
h(x,y)\simeq 2\pi Cy^{-1/3}\cos\left(\sqrt{\frac{3}{2}}x+\frac{\sqrt{3}}{2}y\right)\times{\rm{Ai}}\left(\frac{x+\sqrt{8}y}{(3{\sqrt{8}}y)^{1/3}}\right).This expression is the equivalent to Eq. 16 for the rainbow.
Since the pattern is moving in the x direction with velocity v, the above equation shows that an observer at rest, at a distance y from the path of the moving boat
will observe an oscillation of the water level in time that can be described by an Airy function superimposed with an harmonic factor due to the cosine function in Eq. (25). The overall amplitude is proportional to y^{-1/3}.
This result, valid near the Kelvin cone, is plotted in Fig. 6, together with the exact result obtained by numerical integration of equation (21).
Note how the Airy function produces fringes of maximum amplitude, which are analogous to the supernumerary arcs in the rainbow problem.
In between these maxima,
there are lines where the perturbation amplitude vanishes, which correspond to the zeros of the Airy function.
Given a zero z_{0} of the Airy function ({\rm{Ai}}(z_{0})=0), the perturbation vanishes along the curve
x+\sqrt{8}y=z_{0}(3\sqrt{8}y)^{1/3}.These lines of “destructive interference” can sometimes be seen and are one of the most delicate and beautiful details of boat wakes (Fig. 7).
Figure 6: (Color online) Surface perturbation h produced by point source at (0,0) moving to the right. (a) is the exact result of Eq. (21), obtained through numerical integration (A_{k}\equiv 1). (b) is the analytical result of Eq. (25) (C\equiv 1), which is expected to provide accurate results near the Kelvin cone, and away from the source. In the lower part of both panels we plot as white lines the positions of zero amplitude from Eq. (26), corresponding to the first three zeros of the Airy function.
V (Quasi) non-dispersive quantum wave packets
The mathematical structure of Eq. (21) suggests its application to yet another physical problem.
Consider a quantum particle moving in one dimension in the presence of a periodic potential V(x) generated by identical atoms placed at positions x=na, with n being an integer. The eigenfunctions of this problem are characterized by a crystal momentum p with -\pi\hbar/a$
\Psi(an,t)=\int_{-\pi\hbar/a}^{\pi\hbar/a}dpA_{p}e^{{i\over\hbar}\left[anp-\varepsilon(p)t\right]}.
If a wave packet constructed at $t=0$ has its values concentrated around a well-defined spatial position, then the velocity of this packet is given by the group velocity $v_{g}=\partial\varepsilon(p)/\partial p$.
The spreading of the wave packet is controlled by the second derivative $\partial^{2}\varepsilon(p)/\partial p^{2}$.
In fact, by constructing a wave packet with different p values, slightly different group velocities are incorporated, and therefore
different parts of the packet move at different velocities, producing an overall spreading.
It is natural to ask what happens if a wave packet is constructed around a $\tilde{p}$ value
at which $\left.\partial^{2}\varepsilon/\partial p^{2}\right|_{\tilde{p}}=0$.
Although this is not possible for a free particle ($V(x)=0$) with
$\varepsilon=p^{2}/2m$, it arises in the case a non-zero potential $V(x)$.
In fact, in this case, the dispersion relation $\varepsilon(p)$ is periodic with periodicity ${}2\pi\hbar/a$,
and this implies that there are at least two points where $\partial^{2}\varepsilon/\partial p^{2}=0$ (see Figure 8).
For the particular case of the tight binding model[20] with all-equivalent atoms, the dispersion relation becomes
\varepsilon(p)=-2W\cos(pa/\hbar),
with W being an energy parameter associated with the jump probability of a particle between adjacent sites, which is taken as a constant.
We will analyze the spreading of a wave packet in such a situation.
Figure 8: The energy-momentum relation $\varepsilon(p)$ of the tight binding model of particles in a one-dimensional discrete lattice has two inflection points, indicated by the small circles.
Figure 9: (Color online) Space-time probability map ($|\Psi(n,t)|^{2}$) (where n represents space, $x=na$) for a particle located at site 0, at $t=0$, of a one-dimensional tight-binding
lattice. Compare with the result for surface perturbations in boat wakes (Fig. 6). Note that
the additional oscillations seen in Fig. 6 originate in the cosine factor in
Eq. 25 and do not appear in the present case.
Consider the particular case of a particle that is in the site 0 at the initial time $t=0$.
The wave function of this particle at a time $t>0$ is obtained directly from (29), as
\Psi(n,t)=\frac{a}{2\pi\hbar}\int_{-\pi\hbar/a}^{+\pi\hbar/a}dpe^{{i\over\hbar}\left[pna+Wt\cos(pa/\hbar)\right]},
where we have set all $A_{p}=a/2\pi\hbar$ to fit the initial condition
$\Psi(n,t=0)=\delta_{n,0}$.
Eq. (29) is an integral similar to Eq. (15) for the rainbow and Eq. (24) for the boat wake. Specifically, the function $\epsilon(p)$, expanded around $p_{0}=\pm\pi\hbar/2a$, is of the form $\epsilon(p)\simeq\mp 2W\left[(p-p_{0})+{1\over 6}(p-p_{0})^{3}\right]$, which is analogous to the cubic wavefront introduced in section II.
Fig. 9 shows the numerical solution of this equation. We clearly observe a maximum velocity in both directions, corresponding to the points of maximum group velocity, occurring at $p=p_{0}$, at which
$\partial^{2}\varepsilon/\partial p^{2}=0$. These are the caustic lines of this problem: $n\simeq\pm 2Wt/\hbar$, and correspond to the Kelvin cone in the boat wake problem.
The analogy extends further: in the quantum case, the “wake” becomes a cone in space-time, with the caustic line tracing its boundary at $n/t=\pm 2W/\hbar$. The wave packet, initially localized at the origin, spreads due to a uniform jump probability, propagating in both directions along the one-dimensional chain. Waves corresponding
to other values of p have lower velocity and interfere to form the Airy pattern near the caustic lines—closely resembling the interference seen in phenomena such as rainbows and Kelvin wakes.
The Airy pattern can be analytically worked out along the same lines as in the previous sections.
The result is
\Psi(n,t)=2\pi\left({\frac{\hbar}{tW}}\right)^{1/3}\exp{(i\pi n/2)}{\rm{Ai}}[(n-2Wt/\hbar)/(tW/\hbar)^{1/3}].
This is a wave packet that is only slightly dispersive. In fact, its width increases as
$t^{1/3},$ which is a slower rate than the typical increase as $\sim t^{1/2}$ when $\partial^{2}\varepsilon/\partial p^{2}\neq 0$[21]. To our knowledge, this “space-time wake” has not been observed experimentally, but it may be observable in quantum cold atom systems.
## VI Conclusions
This paper has delved into the remarkable similarities between seemingly disparate wave phenomena: rainbows, boat wakes, and quantum mechanical wave packets. By emphasizing the underlying principles of ray folding, caustics, and Airy interference, we have highlighted the unifying concepts that bridge these diverse phenomena, as highlighted in Table 1.
| Phenomenon | Function involved | Coordinates |
| --- | --- | --- |
| Rainbow | $$
e^{ik\rho}{\rm{Ai}}\left(-(kd)^{2/3}\alpha\right)
$$ | α is the angle measured from the main rainbow. Fold caustic at the rainbow angle $\alpha=0$ . |
| Boat Wakes | $$
{\tiny\cos\left(\sqrt{\frac{3}{2}}x+\frac{\sqrt{3}}{2}y\right)\times{\rm{Ai}}\left[\frac{x+\sqrt{8}y}{(3{\sqrt{8}}y)^{1/3}}\right]}
$$ | x and y are the distances from the boat, located at $(0,0)$ . The caustic is at the Kelvin cone $x=-\sqrt{8}y$ . |
| Quantum Packets | $$
{\rm{Ai}}\left[\frac{n-2Wt/\hbar}{(tW/\hbar)^{1/3}}\right]
$$ | The caustic appears at $n=2Wt/\hbar$ defining a “Kelvin cone” in space-time. |
Table 1: Comparison of amplitude modulation given by the Airy function in the three phenomena described in this paper.
Through this comparative study, we hope to have provided a deeper appreciation of the interconnectedness of physical principles. By demonstrating how concepts from one domain can elucidate phenomena in another, we aim to enrich the pedagogical value of these topics and inspire further exploration into the beautiful and complex patterns observed in the natural world.
## References
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Antonio Machado. “Fields of Castile/Campos de Castilla: A Dual-Language Book”. Stanley Appelbaum (Editor, Translator). Dover Dual Language Spanish, (2007), p. 148.
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Paul Williams and Kenny Ascher, “The Rainbow Connection”, Fuzzy Muppets Songs, (1979).
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Frank S. Crawford
“Elementary derivation of the wake pattern of a boat”
Am. J. Phys. 52, 782–785 (1984).
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Carl Benjamin Boyer
“The Rainbow: From Myth to Mathematics”,
Princeton University Press (1987).
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See Ref [4], p .287
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G. B. Airy, “On the intensity of light in the neighborhood of a caustic”, Transactions of the Cambridge Philosophical Society, 6, 3, pp. 397-402 (1838).
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Jean-Pierre Eckmann,
“Broken pencils and moving rulers: After an unpublished book by Mitchell Feigenbaum”. Am. J. Phys. 89, 955–962 (2021).
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A. Mark Smith, “Optics to the Time of Kepler,” Encyclopedia of the History of Science (November 2022). https://doi.org/10.34758/9482-n985.
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See for example Section 27.2 of Raymond A. Serway; John W. Jewett, “Principles of Physics: A Calculus-Based Text”, 5th Edition. Cengage Learning (2012).
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Thomas Young, “The Bakerian Lecture: Experiments and Calculations Relative to Physical Optics” Philosophical Transactions of the Royal Society of London, 94, pp. 1-16, (1804).
- [11]
Our description here is the basics of the “stationary phase approximation”. An elementary explanation of the method can be seen in
C. Goutis and G. Casella, “Explaining the Saddle Point Approximation”. The American Statistician, 53 (3), 216.
(1999). doi: ${}10.2307/2686100$
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Philip Laven, “Supernumerary arcs of rainbows:
Young’s theory of interference”, Applied Optics, 56, 19, G104-G111 (2017).
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William Thomson, “On ship waves”, Institution of Mechanical Engineers, Proceedings 38 (1887), pp. 409–34; illustrations, pp. 641–49. Available in Google Books.
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See for example Marc Rabaud and Frédéric Moisy (2013) “Ship wakes: Kelvin or Mach angle?,” Phys. Rev. Lett. 110 (21) : 214503, (2013); Alexandre Darmon and Michael Benzaquen and Elie Raphaël , “Kelvin wake pattern at large Froude numbers,” Journal of Fluid Mechanics, 738: R3-1–R3-8, (2014).
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J. Lighthill, “Waves in Fluids”, Cambridge University Press, (2001).
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Gouy, L. G. 1891.
Sur la propagation anomale des ondess.
Annales de chimie et de physique
24 6e série,
145–213.
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For an elementary derivation, see F. S. Crawford, “Speed of gravity waves in deep water:
Another elementary derivation,” Am. J. Phys. 60, 751–752, (1992).
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Stokes, G.G., “On the numerical calculation of a class of definite integrals and
infinite series”, Trans. Camb. Phil. Soc. , 9, 166-187, (1850).
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See for example Chapter 19 of John A. Adam, “Rays, Waves, and Scattering”,
Princeton University Press, (2017).
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For a discussion of spreading of wave packets in the tight binding model see K. Schönhammer, “Unusual broadening of wave packets on lattices”
Am. J. Phys. 87, 186–193 (2019).
- [21]
See for example, Claude Cohen-Tannoudji , Bernard Diu, et al., “Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications” Wiley-VCH; 2nd edition (2019),
p.59.
## VII Appendix I. The Stationary Phase Approximation
Even though the stationary phase method, also called the “saddle point approximation” is commonly discussed in textbooks, we provide a brief overview here to ensure that our presentation is self-contained.
The approximation is relevant in cases where we have integrals of the form
I=\int_{-\infty}^{\infty}dxe^{if(x)},
and $f(x)$ is a rapidly varying function of x. Rapid variations of $f(x)$ over most of the integration range mean that the integral averages almost to zero, except close to values of x where $f(x)$ has an extremum. The approximation consists of expanding $f(x)$ around an extremum $x_{0}$, and evaluating the integral close to $x_{0}$.
The Taylor expansion of $f(x)$ around $x_{0}$, where $f^{\prime}(x_{0})=0$ is
f(x)\simeq f(x_{0})+{1\over 2}f^{\prime\prime}(x_{0})(x-x_{0})^{2},
which, substituted in the above integral gives
I\simeq e^{if(x_{0})}\int_{-\infty}^{\infty}dxe^{if^{\prime\prime}(x_{0})(x-x_{0})^{2}}=e^{if(x_{0})}\sqrt{\pi\over if^{\prime\prime}(x_{0})}.
Notice that the integral can be evaluated in the range $(-\infty,\infty)$ using the quadratic expansion of f because the contribution from large values of $x-x_{0}$ to the integral is negligible.
### Application to the Airy integral
For $x\gg 1$ the integral
\displaystyle{\rm{Ai}}(x) \ \displaystyle= \ \displaystyle\frac{1}{\pi}\int_{0}^{\infty}du~{\cos(ux-u^{3}/3)}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}du~e^{i(ux-u^{3}/3)}
has a rapidly varying argument of the exponential which has two extrema at $u=\pm\sqrt{x}\equiv\pm u_{0}$.
Expanding the phase to quadratic order around these extrema, we obtain:
\displaystyle 2\pi{\rm{Ai}}(x) \ \displaystyle= \ \displaystyle\int_{-\infty}^{+\infty}du~e^{i(ux-u^{3}/3)} \ \displaystyle\simeq \ \displaystyle e^{-i|u_{0}^{3}|}\int_{-\infty}^{+\infty}du~e^{iu_{0}(u-u_{0})^{2}}+e^{i|u_{0}^{3}|}\int_{-\infty}^{+\infty}du~e^{iu_{0}(u+u_{0})^{2}},
and the results is
\displaystyle 2\pi{\rm{Ai(x)}}\simeq e^{i|u_{0}|^{3}}\sqrt{\frac{\pi}{{i|u_{0}|}}}+e^{-i|u_{0}|^{3}}\sqrt{\frac{\pi}{{-i|u_{0}|}}} \ \displaystyle=\pi^{-1/2}x^{-1/4}\cos\left(\frac{2}{3}x^{3/2}-\frac{\pi}{4}\right)
Applying this result to Eq. (15), noting that $x=\alpha(kd)^{1/3}$ we obtain
U(\rho,\alpha)\simeq{1\over\alpha^{1/4}}{1\over\pi^{1/2}(kd)^{2/3}}\cos\left(\frac{2}{3}(kd)\alpha^{3/2}-\frac{\pi}{4}\right).
The intensity $I\sim U^{2}$ coincides with Young’s expression (Eq. (13))
for $kd=({2\pi/\lambda})[R\sqrt{2/\theta^{\prime\prime}(x_{0})}]$
with an extra $\pi/4$ in the phase. This shift, the so-called Gouy’s phase, is a subtle diffraction effect that occurs close to singularities such as focal points or caustics (see Supplementary Material).
## Gouy’s phase distilled to its essence
Let us consider plane waves of wavelength λ that originate from a coherent source which is an arc of circle spanning an angle of ${}2\theta_{0}$ (Fig. 10). On moving from left to right, the waves are convergent up to the focal point f (where they arrive in phase), and then become divergent.
Let us evaluate the wave amplitude along the horizontal axis. Each wave forming a particular angle $\theta$ with the horizontal axis will produce along this axis a perturbation of an effective wavelength $\lambda_{eff}(\theta)\equiv\lambda/\cos(\theta)$, which is larger than λ. When summing up the contributions for all values of $\theta$ this “stretching” of wavelength produces the Gouy’s phase.
Figure 10: Plane waves of wavelength λ are emitted from a circular arc, converging to a focal point f. Any wave originated at an angle $\theta\neq 0$ has an effective wavelength–when observed along the horizontal–larger than λ. This wavelength stretching produces the Gouy’s phase of the beam when passing through f.
To calculate the effect, we first notice that
the contribution of a wave with a particular value of $\theta$ to the amplitude A along the x axis is
A(x,\theta)=A_{0}\cos(2\pi x/\lambda_{eff}(\theta))=A_{0}\cos(kx\cos(\theta))
with $k=2\pi/\lambda$, and where the coordinate x is measured from the focal point.
We expand for small angles:
\displaystyle\cos\left(kx\cos\theta\right) \ \displaystyle\simeq \ \displaystyle\cos\left(k\left[1-{\theta^{2}\over 2}\right]x\right) \ \displaystyle= \ \displaystyle\cos(kx)\cos\left({k\theta^{2}\over 2}x\right)+\sin(kx)\sin\left({k\theta^{2}\over 2}x\right),
and integrate over $\theta$ (using the additional approximation $kx\gg\theta_{0}$):
\displaystyle\int_{\theta_{0}}^{\theta_{0}}d\theta\cos\left({k\theta^{2}\over 2}x\right) \ \displaystyle\simeq\int_{\infty}^{\infty}d\theta\cos\left({k\theta^{2}\over 2}x\right)=\sqrt{2\pi\over k|x|} \ \displaystyle\int_{\theta_{0}}^{\theta_{0}}d\theta\sin\left({k\theta^{2}\over 2}x\right) \ \displaystyle\simeq\int_{\infty}^{\infty}d\theta\sin\left({k\theta^{2}\over 2}x\right)=\sqrt{2\pi\over k|x|}{\rm{sgn}}(x)
Then using elementary trigonometrical identities, we obtain the final result
\displaystyle A(x)=A_{0}\int_{\infty}^{\infty}d\theta\cos\left(\left[k\cos\theta\right]x\right) \ \displaystyle\simeq \ \displaystyle 2A_{0}\sqrt{2\pi\over k|x|}\left[\cos(kx)+|\sin(kx)|\right] \ \displaystyle= \ \displaystyle 4A_{0}\sqrt{\pi\over k|x|}\cos\left(kx-{\rm{sgn}}(x){\pi\over 4}\right)
In other words, compared to the $\sim\cos(kx)$ amplitude oa a single wake that originates in the center of the arc, the nodes of the bundle of waves from a finite angular range are shifted by $\pi/4$ before reaching the focus, and by $-\pi/4$ after passing through it.
The precise value $\pi/4$ results from the symmetry of the Fresnel integrals (Eq. (38)) involved.
The total shift of $\pi/2$ occurs for all waves when passing through a focal point or a caustic, and is the Gouy’s phase.
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