Huygens' Principle geometric derivation and elimination of the wake and backward wave

Abstract

Huygens' Principle (1678) implies that every point on a wave front serves as a source of secondary wavelets, and the new wave front is the tangential surface to all the secondary wavelets. But two problems arise: portions of wavelets that exist outside of the new wave front combine to form a wake. Also there are two tangential surfaces so wave fronts are propagated in both the forward and backward directions. These problems have not previously been resolved by using a geometrical theory with impulsive wavelets that are in harmony with Huygens' geometrical description. Doing so would provide deeper understanding of and greater intuition into wave propagation, in addition to providing a new model for wave propagation analysis. The interpretation, developed here, of Huygens' geometrical construction shows Huygens' Principle to be correct: as for the wake, the Huygens' wavelets disappear when combined except where they contact their common tangent surfaces, the new propagating wave fronts. As for the backward wave, a source propagates both a forward wave and a backward wave when it is stationary, but it propagates only the forward wave front when it is advancing with a speed equal to the propagation speed of the wave fronts.

Figure 1

Huygens' Principle as a geometrical construction. (a) A replica of Huygens' original Figure1 (a spherical wave). (b) Huygens' geometrical construction presented as a plane wave.

Figure 2

Planar source. (a) An infinite plane with impulsive excitation, δ_f(t). The excitation duration or temporal pulse width is ε. The parameter z₀ is the distance from the plane to a point in space, P, at which the wave field will be observed as a function of time. The propagation speed is c, where c > 0. R1 and R2 are the radii of an annulus defined by the intersection of the plane with two spheres of radii ct and c[t + ε]. The point N is the nearest point on the plane to the field point P. Each point on the plane is a point source which radiates an impulsive spherical wavelet. (b) The wave field ϕ that is propagated from the plane. (c) The wave front f in the Z + direction that is derived from the wave field which will arrive at P at t = z₀/c.

Figure 3

The geometry of an infinite planar impulsive source moving toward P with speed v. (a speed away from P would be given by − v.) R1 is the radius of the circle created by the intersection at time t of a sphere of radius ct with the plane. R2 is the radius of the circle created by the intersection at time t + ε of a sphere of radius c[t + ε] with the plane.

Figure 4

(a–c) Source motion displacement for the forward wave: (a) the original forward wave front pulse compared with the additional forward wave front pulse due to the source motion. (b) The original forward wave front pulse which is the same as the wave front pulse from the stationary source. (c) The additional forward wave front pulse due to the source motion. (d–f) Source motion displacement for the backward wave: (d) the original backward wave front pulse compared with the additional backward wave front pulse due to the source motion. (e) The original backward wave front pulse which is the same as the wave front pulse from the stationary source. (f) The additional backward wave front pulse due to the source motion. (g–i) The original and additional wave fronts with v = c. (g) Forward and backward wave front pulses that would propagate from a stationary source. (h) Additional wave front pulses originating from source motion. (i) Forward wave front pulse created by the sum of the other two wave front pulses (g) and (h).

Figure 5

Non expanding spherical shell impulsive source. (a) A spherical cap on an impulsive spherical shell source. (The spherical shell may be referred to as a 'sphere' hereafter.) The radius is R₀, and the sphere is centered at the coordinate origin and has impulsive excitation δ_f (t). Each point on the sphere’s surface represents a point source which radiates an impulsive spherical wavelet. The parameter z₀ is the radial distance from the sphere's surface to the point in space, P, at which the resultant wave field φ will be observed as a function of time. The propagation speed is c. The distance ct from P to a point on the sphere defines a spherical cap of height h(t) with radius R1. The sphere is transparent to radiation so that the whole sphere will contribute to the wave field at any one point in space. (b) The locus of all contributing radiating points is a spherical zone. These are the only points on the spherical source of radius R₀ which can contribute to the field at P during the time interval [t, t + ε] and at distance z₀ from the sphere. The zone on the spherical source is bounded by its intersection with two concentric spheres of radius ct and radius c[t − ε] centered at the field point P. The intersections defines spherical caps of heights h(t) and h(t + ε), also radii R1 and R2.

Figure 6

Radiated wave fronts and individual terms in the equations. The factor R₀/[2[R₀ + z₀]] is not included in the figures' annotations. (a) A spherical source of radius R₀ with expanding and converging wave fronts. The distance z₀ is from the surface of the spherical source to the field point P at which the wave fronts will be observed. The two points N and F are the nearest and furthest points from P on the near and far surfaces of the spherical source. (b) Shows the impulses from the equation for the wave front f(z₀,t): the impulse δ(t − z₀/c) observed at P, at distance z₀, is caused by an expanding impulsive spherical wave. This is the forward wave front for the spherical source from its near surface. The impulse − δ(t − [z₀ + 2R₀]/c) is caused by an impulsive spherical wave front, observed at P, that had initially converged with decreasing radius inward toward the spherical source's center, passed through the center while changing sign, and then expanded with increasing radius toward P. This is the backward wave front for the spherical source from the far surface. (c) The wave field corresponding to the forward wave front. (d) The wave field corresponding to the backward wave front. (e) The combined wave field resulting from the summation of the forward and backward wave fields.

Figure 7

The impulsive spherical shell source (the 'sphere') with initial radius R₀ expanding with a positive relative radial speed v where as with the planar source, v will be positive for forward wave fronts (in the direction of the expansion) and will be proceeded by a negative sign for backward wave fronts. R1 is the radius of the circle created by the intersection at time t of a sphere centered at P of radius ct with the spherical source. R2 is the radius of the circle created by the intersection at time t + ε of a sphere centered at P of radius c[t + ε] with the spherical source.