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1
Making waves: the geometric derivation of Huygens' Principle for
wave propagation, and the problem of the wake
Forrest L. Anderson
Abstract
Huygens' Principle states: 'every point on a wave front serves as a source of secondary
wavelets. The new wave front is the tangential surface to all the secondary wavelets in the
direction of propagation'. But how do the wavelets vanish outside of new wave front?
Wavelets that exist outside of the new wave front combine to form a wake and the wave
does not propagate cleanly. Then Huygens' Principle can fail to properly describe wave
propagation. This problem has not previously been resolved by using a clear direct
geometrical theory and impulsive wavelets that are in harmony with Huygens' description.
An interpretation, developed here, of Huygens' geometrical construction shows Huygens'
Principle to be correct as depicted in his figures. Having waves propagate without having a
wake is achieved using only geometry and the theory of distributions: The Huygens'
wavelets disappear when combined and differentiated except where they contact their
common tangent surfaces, the new propagating wave fronts.
The following shows that whether or not there is a wake is determined solely by geometric
properties of the source of the propagating waves, and that planar and spherical sources
have those required properties. It also shows the form that the wavelets must take to have
the wake disappear.
Introduction
Figure 1 | Huygens' Principle as a geometrical construction. (a) A replica of Huygens' original
figure
1
(a spherical wave). (b) Huygens' geometrical construction presented as a plane wave.
Huygens' Principle states: every point on a wave front serves as a source of secondary wavelets.
The new wave front is the tangential surface to all the secondary wavelets in the direction of
propagation. ( Figure 1 presents this as a geometric construction. )
2
An abundance of literature
refers to Huygens' Principle
1-3
. Refs 4-32 are a very small sample
which show the tremendous variation in the application of Huygens' Principle. But there has been
insufficient theory developed which addresses the underlying Huygens' basic geometrical
construction of how a wave propagates via Huygens' wavelets. This is despite the common use of
Huygens' construction to introduce wave theory, reflection, refraction, and diffraction. In his
construction two problems arise: The portions of the wavelets not contacting the position of the
new propagating wave must disappear ('no wake')
2,3
. Also, the portions of the wavelets propagating
in the opposite direction to the new propagating wave must vanish ('no backward wave'). There
has been no clear direct geometrical theory which shows how this can occur. The following
focuses on the first problem, the problem of the wake. The phrase 'satisfies Huygens Principle' in
the literature most often refers to this.
Huygens' Principle's problems of the backward wave and the wake have been approached before
by relying on physics (for example, suitable initial values of the velocity potential and
condensation
3
), or by involving doublets, dipoles, obliquity factors, sinusoidal waves, and also by
using complex mathematics.
2,3
"..but the intuitive appeal of Huygens' simple principle is lost. .."
4
along with a clear direct connection to his geometrical construction.
Wave propagation is linear so superposition holds: It should be possible to decompose an
impulsive propagating wave into its constituent points, then consider the impulsive wavelets
radiating from each of those points at a future time, and combine those wavelets in a simple direct
geometric manner to obtain the progressing wave at that future time.
The following shows that Huygens' geometrical construction is correct as depicted in his figures
when the wavelets are interpreted as Dirac Delta distributions (Delta functions)
33-37
. The
development is geometric and does not involve any particular type of field. The symbol
ϕ
is used
for the initial propagating wave fields and f is used for wave fronts derived from
ϕ
. The symbol
δ
is
used for the Dirac delta distribution and
δ
f
is used for finite approximations to it. The symbol
θ
is
used for the Heaviside step function
35,36
. The following relationships are used:
δ
(ax) = (1/ |a|)
δ
(x);
δ
(-x) =
δ
(x); d
θ
(x)/dx =
δ
(x), and all results are in the sense of distributions.
Results and discussion
The ensuing analyzes the wave fields and wave fronts from infinite planar sources and spherical
sources including the nature of the wake.
3
Planar source
Figure 2 | (a) An infinite plane with impulsive excitation, δ
f
(t). The excitation duration or temporal
pulse width is
ε
. The parameter z
0
is the distance from the plane to a point in space, P, at which the
wave field will be observed at positive time t. The propagation speed is c. Ra and Rb are the radii
of an annulus defined by the intersection of the plane with two spheres of radii ct and ct +
ε
. Each
point on the plane is a point source which radiates a spherical wavelet given by (4πr)
-1
δ(t - r/c). The
radius of the spherical wavelet is r. (b) The wave field ϕ propagated from the plane. (c) The wave
front f in the Z+ direction derived from the wave field which will arrive at P at t = z
0
/c.
It is only necessary to compute the wave field for each point on any one line perpendicular to the
source since the planar source is uniform with respect to the x and y coordinates. Only the points in
the planar source intersected by the sphere with radius ct centered at P
can contribute to the wave
field φ(z
0
, t) when computing the wave field at P at time t (Figure 2(a)).
Initially the duration ε of the excitation pulse δ
f
will be finite and the pulse will have an amplitude
equal to
α
. Then the area under this finite version of δ(t) is εc
α
which is required to be 1. Later,
where needed, the limit as ε tends toward zero will be taken while still requiring
α
to take on a
value such that εc
α
remains equal to 1 ( i.e.
α
= 1/(εc) ) so that δ
f
→ δ.
The locus of all radiating points on the planar source which can contribute to the field φ(z
0
, t)
during the time interval (t, t +
ε
) and at distance z
0
from the plane is an annular area. This annular
area is bounded by two concentric spherical shells of radius ct and c(t + ε) .
The area Ap of the annulus is:
2 2
b a
Ap R R
π π
= −
(1)
where
2 2 2 2
0
a
R c t z
= −
(2)
and
4
(
)
(
)
2
2 2 2 2 2 2 2 2 2 2 2 2 2
0 0 0
2 2
b
R c t z c t t z c t c t c z
ε ε ε ε ε
= + − = + + − = + + −
(3)
So
(
)
2 2
2Ap c t
π ε ε
= + (4)
which has the limit
2
2 as 0
Ap c t
π ε ε
= →
( and so δ
f
→ δ ). (5)
Note that for (4πr)
-1
δ(t - r/c), r = ct when
δ
is non zero. So (4πct)
-1
δ(t - r/c) can be used for the
wavelets. Then since spherical spreading attenuation of the form 1/(4πct) attenuates δ(t - r/c) and
therefore the radiation from the annular area. As a result the attenuated radiation from Ap is not a
function of the field point's distance, z
0
, from the planar source except for the step at ct = z
0
as will
be shown in the following:
Including the 1/(4πct) spreading and the
α
amplitude factors gives the amplitude of the wave field,
φ, after the impulsive spherical wavelets have all passed (i.e. for 0
≤
|z|
≤
ct) :
2
2 2
4 4 4 2
c t ct c
Ap c
ct ct ct
α π ε π εα
ϕ α εα
π π π
= = = = (6)
Because cε
α
= 1, the result simplifies to
1 for
2
z ct
ϕ
= ≤
, (7)
or using a Heaviside step function,
θ
,
( )
( )
1
,
2
z t ct z
ϕ θ
= −
(8)
which as a spatial function of z is a rectangular pulse expanding in both advancing (toward P) and
retreating (away from P) directions (Figure 2(b)). Note that the dimensions of cεα remain hidden
in the formulas henceforth.
In the advancing direction the wave field observed at P as a function of time is
( ) ( )
0 0
1
,
2
z t ct z
ϕ θ
= − , (9)
which has a positive transition at t = z
0
/c when the wave field has advanced to P.
The derivative of φ yields an advancing planar wave front, f, which observed at P as a function of
time is (Figure 2(c)):
5
( ) ( ) ( ) ( )
0 0
0 0 0 0
1 1 1
,2 2 2 2 2
z z
c c
f z t ct z ct z z ct t t
t c c
θ δ δ δ δ
+
∂
= − = − = − = − = −
∂
(10)
which holds for every line perpendicular to the source. Note that only the single point source at N
contributes to the wave front at P: The wave field
ϕ
is a step function and its derivative is zero
except at the step's location, and the step originated at N. Because the derivative is zero except at
the step there is no wake.
There will also be a retreating plane wave front f
-
(z
, t) = ½ δ(t + z
/c) propagating in the opposite
direction, away from P, which can be derived similarly. This also propagates without a wake.
(Here z is negative, and f
-
will not be seen at z
0
, the location of P.)
Therefore an infinite planar distribution of point sources radiating impulsive spherical wavelets
creates advancing and retreating impulsive plane wave fronts that propagates cleanly without a
wake. This shows that the Huygens' wavelets cancel each other when summed together and
differentiated except on their tangent surfaces (Figure 1(b)).
Spherical source
h
R0
R0- h
ct
a
z0
Impulsive spherical source
Field point P
Spherical
cap
Z+
Z-
Impulsive spherical source
h(t+ε)
ct
Field point P
h(t)
c[t+ε]
R0
Z+
Z-
z0
ab
Figure 3 |
(a)
A spherical cap on an impulsive spherical shell. ( The shell to be referred to as a
'sphere' hereafter. ) The radius is R
0
, and the sphere is centered at the coordinate origin and has
impulsive excitation δ(t). Each point on the sphere’s surface represents a point source which
radiates a spherical wavelet given by (4πr)
-1
δ(t - r/c). The radius of the spherical wavelet is r. The
6
parameter z
0
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 at time t. The propagation speed is c. The distance ct from
P to the sphere defines a spherical cap of height h with radius a. 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 R
0
which can contribute to the field at P during the time interval (t, t +
ε
)
and at distance z
0
from the sphere. The zone on the spherical source is bounded by its intersection
with two concentric spheres of radius ct and c(t - ε) centered at the field point P. The intersections
defines spherical caps of heights h(t) and h(t + ε).
Since the spherical source is symmetrical, it is only necessary to compute the wave field for each
point on any one radial line emanating from its center. The wave field will be the same on any
other radial line. As before, initially it is assumed that the temporal duration ε of the excitation
pulse δ
f
is finite and that it has an amplitude equal to
α
and area equal to εc
α
, where
α
= 1/(
ε
c) to
maintain an area equal to 1 as
ε
→
0 ( and δ
f
→ δ ).
The surface area of a spherical cap is 2πR
0
h (Figure 3(a)), and the surface area of the spherical zone
is As = 2πR
0
(h(t + ε) - h(t)) (Figure 3(b)); Note surprisingly that the surface area of the zone
depends only on its height, h(t + ε) - h(t), not on its vertical position within the sphere.
To find As the heights h(t) and h(t + ε) must be found. Two right triangles can be formed (Figure
3(a)):
T1:
( )
2
2 2 2
0
c t a h z
= + + , and (11)
T2:
(
)
2
0
22
0
hRaR −+=
.
(12)
Solving for h gets
T1 - T2:
( ) ( )
2 2
2 2 2
0 0 0
c t R h z R h
− = + − − , (13)
2 2 2 2 2 2 2
0 0 0 0 0
2 2
c t R h hz z R R h h
− = + + − + −
, (14)
2 2 2 2 2
0 0 0 0 0
2 2
c t R hz z R R h
− = + − + , (15)
2 2 2
0 0 0
2 2
c t hz z R h
= + + , (16)
(
)
2 2 2
0 0 0
2
h R z c t z
+ = −
, (17)
( )
2 2 2
0
0 0
( ) 2
c t z
h t
R z
−
=+ , (18)
7
and also (by substituting t + ε for t)
( )
2 2 2
0
0 0
( )
( ) 2
c t z
h t R z
ε
ε
+ −
+ = + (19)
where both equations are for time t where z
0
/c
≤
t
≤
(2R
0
+ z
0
)/c .
Then the zone's surface area is
( ) ( )
( )
(
)
( )
2
2 2 2 2 2
0 0
0 0
0 0
2 2 2 2 2 2 2 2 2
2 2 2
0 0
0 0
0 0 0 0
2 2 2
22
2 2( )
c t z c t z
As R h t h t R R z
c t c t c z c t z c t c
R R
R z R z
ε
π ε π
ε ε
ε ε
π π
+ − − +
= + − = +
+ + − − + +
= =
+ +
(20)
and as ε
→
0 ( so δ
f
→ δ )
2
0
0 0
2
R c t
As
R z
π ε
=+ (21)
This is the same as for the plane's annulus Ap when R
0
→
∞
or when z
0
→
0.
As before spherical spreading of the form 1/(4πct) attenuates the wavelets δ(t - r/c) and therefore the
radiation from the spherical zone. The amplitude of φ observed at P for z
0
/c
≤
t
≤
(2R
0
+ z
0
)/c is
produced by combining the 1/(4πct) attenuation and the
α
amplitude with As:
( ) ( )
2
0 0
0 0 0 0
2
4 4 2
R t c R c
As
ct ct R z R z
α π ε εα
α
ϕπ π
= = =
+ + (22)
Because cε
α
= 1,
( )
0
0 0
2
R
R z
ϕ
=+
which holds for z
0
≤
ct
≤
(2R
0
+ z
0
) (23)
For fixed z
0
, |φ| observed at P is a rectangular pulse in time of temporal width 2R
0
/c.
Note that as z
0
→
0 or R
0
→
∞
, |φ|
→
1/2, which is the same as for the planar source.
The wave field
ϕ
observed at P as a function of time is produced by using Heaviside step functions
to implement the time dependence, z
0
≤
ct
≤
(2R
0
+ z
0
):
( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
( )
0
0 0 0 0 0 0 0
0 0
, 2 2
2
R
z t ct z ct z R ct z c t z R
R z
ϕ ϕ θ θ θ θ
= − − − + = − − − +
+
(24)
8
The derivative of φ yields the wave front observed at P as a function of time (Figure 4(b)):
( ) ( ) ( ) ( )
( )
( )
( )
( )
0 0
0 0 0 0
0 0
0 0 0 0
0 0
0 0 0 0
0 0
0 0 0 0
0 0
( , )
, 2
2
2
( ) ( )
2
2
1 1
2
2
2
z t cR
f z t ct z ct z R
t R z
cR z z R
c t c t
R z c c
cR z z R
t t
R z c c c c
R z z R
t t
R z c c
ϕ
δ δ
δ δ
δ δ
δ δ
∂
= = − − − −
∂ +
+
= − − −
+
+
= − − −
+
+
= − − −
+
(25)
For finite z
0
and t in the equation for f(z
0
,t), as R
0
→
∞
the argument of the
δ
(ct - z
0
- 2R
0
) never
equals zero, so never 'activates' and
δ
(ct - z
0
- 2R
0
) may be ignored. Then
( ) ( )
0 0 0
0
0 0
1
,2 2
R z z
f z t t t
R z c c
δ δ
= − = −
+
as R
0
→
∞
(26)
which is the same as the equation for the planar source advancing wave.
R0
z0
Field point P
2R0
Far surface
Near surface
Converging
Expanding
Source
N
F
Z+
Z-
wave
wave
a
t
0
δ([z0+ 2R0] / c)
z0 / c
f(z0,t) observed at P
δ(t - z0 / c)
[z0+ 2R0] / c
b
c
0
θ(ct-z0)
ct
z0 z0+ 2R0
d
0
- θ(ct-z0-2R0)
ct
z0 z0+ 2R0
e
0
Φ(z0, t)
ct
z0 z0+ 2R0
Figure 4.
Radiated wave fronts and individual terms in the equations. The factor R
0
/(2(R
0
+ z
0
)) is
not included in the figures. (
a)
A spherical source of radius R
0
with expanding and converging
wave fronts. The distance z
0
is to the field point P at which the waves will be observed. The two
9
points N and F are the nearest and furthest points from P on the near and far surfaces of the
spherical source. (
b)
The terms in the equation for f(z
0
,t). The impulse
δ
(t - z
0
/c) observed at P, at
distance z
0
, is caused by an expanding impulsive spherical wave. This is the forward wave for the
spherical source from its near surface. The impulse -
δ
(t - (z
0
+ 2R
0
)/c) is caused by an impulsive
spherical wave, 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 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.
Both propagating spherical waves (Figure 4(a)) are attenuated by the spherical spreading term R
0
/(R
0
+ z
0
) which supplies the additional attenuation due to the radius increasing from the initial R
0
to R
0
+ z
0
, ( It is implied that 1/(2πR
0
) attenuation is included in the initial amplitude of the
wavelets originating on the sphere's surface. ) For example, (1/R
0
) (R
0
/(R
0
+ z
0
)) = 1/(R
0
+ z
0
). In -
δ
(t - (z
0
+ 2R
0
)/c) the inverse spreading due to going from the far surface to the center is exactly
cancelled by the spreading resulting from going from the center to the near surface. So the same R
0
/(R
0
+ z
0
) spherical spreading term can ultimately be applied to both waves that are to be observed
at P.
The wave field θ(ct - z
0
-2R
0
) originated as a positive propagating field on the far surface. In the
limit as R0 → ∞ the wave fields close to the surface of the sphere must be the same as the wave
fields from the plane, and the wave fields on both sides of the plane are positive. As the initially
converging spherical wave passes through the center it changes sign ( Gouy phase shift )
38,39
(Figure 4(d)) and begins expanding. When it expands through the surface of the spherical source
the now negative wave field cancels the wave field that originated on the near surface,
θ
(ct - z
0
),
(Figure 4(c)). The combined wave field (Figure 4(e)) when observed in time at P is a square pulse
of duration 2R
0
/c .
Notice that only the two points N and F (Figure 4(a)), contribute as sources to the wave front f(z
0
,t).
This is because φ(z
0
, t) is a rectangular pulse of temporal width 2R
0
/ c for fixed z
0
and its
derivative is zero except on its leading and trailing edges (which originated at the two points N and
F). Because the derivative is zero there is no wake.
Therefore a spherical distribution of point sources radiating impulsive spherical wavelets creates
two impulsive spherical wave fronts (Figure 4(a) and (Figure 4(b)). Both propagate cleanly without
a wake. Again this shows that the Huygens' wavelets cancel each other when summed together and
differentiated, except on their two tangent surfaces.
Huygens' Principle as depicted in his figures
It was shown that for the spherical and planar source, when the excitation is
δ
(t), the resulting wave
field ϕ when differentiated disappears except at the locations of the propagating wave fronts f. The
following shows how the intermediate step of differentiation can be eliminated.
Let L represents the linear wave propagation operator which transforms the excitation on the plane
or sphere into the wave field ϕ, If
δ
(t) is the excitation, the resulting wave field due to the wave
propagation process is
10
(
)
(
)
(
)
,
z t L t
φ δ
= (27)
Because L is linear (LTI) the derivative can be taken of ϕ and of
δ
(t)
40
yielding
( ) ( ) ( )
( )
,
z t L t L t
t t
φ δ δ
∂ ∂
′
= =
∂ ∂
(28)
But the derivative of the wave field ϕ is the wave front f, so
( ) ( )
( )
( , ) ,
f z t z t L t
t
φ δ
∂
′
= =
∂ (29)
As a result, if
δ′
(t) is used as the initial excitation instead of
δ
(t) for the wavelets, the Huygens'
wavelets when combined disappear except where they contact their common tangent surfaces.
These tangent surfaces are the new propagating wave fronts, obtained without the intermediate
step of differentiation. In Huygens' figures the wavelets then would represent doublets (4πr)
-1
δ′
(t
- r/c). However, notice that the resulting wave fronts are of the form
δ
not
δ′
so they must be
differentiated before being used as the initial excitation in the next iteration of wave propagation.
Conclusion
Huygens' geometrical construction has been shown to be literally correct as depicted in his
figures (see Figure 1) when the wavelets represent the first derivative of Delta distributions. For
the sphere and the infinite plane there is no wake. What other geometric shapes would produce
similar results? The approach used here for conventional 3D (or 3D + 1D) space could be
applied to other spaces with different dimensions to investigate the existence of wakes.
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