The swedish physicist Anders Angstrom was the first person to study...
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Why
some
things
are
darker
when
wet
John
Lekner
and
Michael
C. Dorf
Angstrom
has proposed
that
rough
absorbing
materials
are darker
when
wet
because
their
diffuse
reflection
makes
possible
total
internal
reflection
in
the water
film
covering
them,
increasing
the likelihood
of the
absorption
of
light
by
the
surface.
His
model
is
extended
here
in
two
ways:
the
probability
of internal
reflection
is calculated
more
accurately,
and
the
effect
on
absorption
of
the
decrease
of the
relative
refractive
index
(liquid
to
material
instead
of air
to
material)
is estimated.
Both
extensions
decrease
the
albedo
of
the
wetted
surface,
bringing
the
model
into
good
agreement
with
experiment.
1.
Introduction
From
early
childhood
we
learn
to
distinguish
wet
from
dry,
not
just
by
touch,
but
also
by
sight.
Most
objects,
especially
those
with
rough
and
absorbing
sur-
faces,
are
darker
when
wet:
they
reflect
less
light.
In
a
model
for
this
phenomenon
proposed
by
Angstrom,1
the
surface
roughness
leads
to
diffuse
reflection,
and
thus
to
total
internal
reflection
at the
liquid-air
inter-
face
of
the
thin
film
of
liquid
covering
the
surface.
This
reflection
gives
an
increased
probability
of
the
absorption
of
light
by
the
surface,
and
thus
a
darker
surface.
Angstrom's
model
is
extended
in the
following
two
sections.
The
results
are
then
compared
with
experi-
ment.
In
the
final
section
we also
compare
Angstrom's
approach
to
that
of Bohren
2
and
Twomey
et
al.
3
(TBM),
in which
the
darkening
is taken
to be
due
to
the
increase
in
the
forward
scattering
on wetting.
II.
Diffuse
Scattering
and
Internal
Reflection
An
optically
smooth
surface
reflects
specularly,
whether
or not
the
surface
is covered
by
a
film
of liquid.
A ray
of
light
reflecting
at
a smooth
surface
will
not
be
totally
reflected
at the
liquid-air
interface.
If
the
surface
is
rough,
its diffuse
reflection
sends
some
light
out
obliquely
enough
to be
totally
internally
reflected
at
the
liquid-air
interface.
This
increases
the
chance
of
absorption
at the
surface.
The
authors
are with
Victoria
University
of Wellington,
Physics
Department,
Wellington,
New
Zealand.
Received
6 June
1987.
0003-6935/88/071278-03$02.00/0.
©
1988
Optical
Society
of America.
Consider
a
rough
absorbing
surface,
such
as
a
black-
board.
We
treat
the
surface
as
an
array
of
randomly
oriented
facets,
each
of
which
reflects
specularly.
When
wet,
a thin
liquid
layer
covers
the
surface.
Light
incident
on
this
layer
has
probability
1 -
R
1
of reaching
the
surface,
where
R
1
is
the
reflectance
at
the
air-liquid
interface.
Some
fraction
of
the
transmitted
light
is
absorbed
by
the
surface.
Call
this
fraction
a.
Of
the
light
which
is
reflected
by
the
surface,
let
p be
the
fraction
which
is then
reflected
back
at
the
liquid-air
interface,
so
that
it
is once
again
incident
on
the
sur-
face.
The
process
is
illustrated
in
Fig.
1.
Continuing
the
process
illustrated
ad
infinitum,
the
total
probability
of
absorption
by
the
rough
surface
is
A=
(1-Rl)[a
+
a(l-a)p
+
a(l-a)
2
p
2
+...]
(1
- R)a
1
- p(l-a)
(1)
Angstrom
evaluates
p
as follows:
all
the
light
with
an
angle
of
incidence
greater
than
the
critical
angle
,
(=
arcsinl/n)
onto
the
liquid-air
interface
will
be
to-
tally
reflected.
Thus
p
may
be
estimated
as
the
frac-
tion
of
diffusely
reflected
light
which
lies
outside
the
cone
generated
by
rays
whose
reflection
angle
is 60.
For
a
Lambertian
surface,
the
intensity
reflected
at
angle
is
proportional
to cos6.
Light
emerging
at
angles
to
+
d
subtends
a
solid
angle
27r
sinOdO.
Thus
i/2
27r
dO
sinG
cosO
P
p /2
= Cos
2
OC =
1-
1/n .
2,r I
dO
sinG
cosO
Equations
(1) and
(2) are
together
equivalent
to the
last
equation
in
Angstrom's
Sec.
4(ii),
except
that
he
omits
the
1 -
R
factor.
1278
APPLIED
OPTICS
/
Vol.
27,
No. 7
/ 1
April
1988
(2)
Fig.
1. Liquid
layer over
a
rough
surface.
The
coefficients
repre-
sent
the fraction
of the
incident
light
intensity
which
is transmitted
along
each
path.
Equation
(2) underestimates
p: the
reflectivities
for
both
polarizations
are
generally
small
but
not
zero
for
0
< 0,;
see,
for
example,
Ref.
4, Sec.
1.5,
or
Ref.
5,
Figs.
1-3 and
1-4.
For
a surface
which
reflects
diffuse-
ly according
to Lambert's
law,
the
probability
reflec-
tion at
the
liquid-air
interface
is
J r/2
| dO
sinG
cosOR(x,1/nj)
/2
X
dO
sinG
cosO
=
J dxR(x,
/ni),
where
x =
sin
2
o and
R(x,n)
is the
reflectance
at
an
interface
between
media
1 and
2, with
n =
n
2
/ni.
Here
n
= 1/ni <
1 and R
is unity
for x >
n
2
so
n2
p
1 - n
2
+
dxR(x,n).
(4)
For both
polarizations,
the reflectances
have
the
prop-
erty that
R(x,n)
= R(x/n
2
, i/n)
(5)
(this
was
noted
by
Stern
6
for
the
transmittances).
On
changing
to the
variable
y = x/n
2
,
the integral
in
Eq. (4)
may
be written as
n
2
j
dyR(n
2
yn)
=
n
2
J
dyR(y,i/n)
=
n
2
(/n).
(6)
The last
equality
expressed
the
integral
in terms
of
the
average
reflectance
of an
isotropically
illuminated
sur-
face,
r-l2
J
dO sinO cosOR(x,n)
I
R(n)
= °=
I
dxR(x,n)
(7,
j dO sinG cosO °
(in
correspondence
with
the
average
transmittance
de-
fined
by
Stern
6
). For
unpolarized
light,
Stern's
for-
mulas
(9a) and
(9b) lead
to
- 3n
2
+2n+i
2n
3
(n
2
+
2n-1)
n
2
(n2
+ 1)
3(n
+ 1)2
(n
2
+
i)
2
(n
2
- 1)
(n
2
- 1)2
n2- 1)2
n(n +
1)
(n
> 1). (8
(n
2
+ i)3
l o g
From
Eqs.
(4) and
(6) we
have
the result
p =
1- I [1-
R(nj)].
n
(9)
For
water,
this
formula
gives
a p
larger
by
-9%
than
the Angstrom
estimate
(0.475
instead
of 0.437).
Ill. Probability
of Absorption
by
a Wetted
Surface
The
parameter
a
in Eq. (1)
is the
fraction
of the
light
incident
on the
surface
which
is
absorbed.
(The above
refers
to a
single
interaction:
the
total
probability
of
absorption,
allowing
for
reflections
at
the
liquid-air
interface,
is A.)
Angstrom
takes
a to
have the
same
value
for
the wet
as
for the
dry surface.
We
expect
a,
(the
value
when
wet),
to
be greater
than
ad
(the
value
when
dry),
since
the absorbing
medium
will normally
have
the real
part
of its
refractive
index
greater
than
unity.
Since
reflection
is caused
by
wave
vector mis-
match,
and
since wave
vectors
are
determined
by
re-
fractive
indices,
covering
the surface
with
a layer
of
liquid
(with
nj
> 1) results
in less
reflection.
The
value
of
ad is
in principle
determined
by the
complex
refractive
index nr
+ ins of
the material,
and
by the roughness
of the surface
(which
influences
the
average
angle of
incidence
on its
randomly
oriented
facets,
and the probability
of multiple
interactions,
as
in a
crevice).
The value
of a,
is in addition
a
function
of nj,
the refractive
index
of the
liquid film
covering
it.
For
the purpose
of comparing
the albedos
1 - ad
and 1
-
A of the
dry
and wetted
surfaces,
we estimate
a, in
terms
of ad, nl,
and nr as
follows.
For
small
absorption
(ni <<
nr),
ad
1 - R(n)
where
R(n) is
the average
reflectance
of an isotropically
illu-
minated
surface,
defined
in Eq.
(7). The
assumption
made
here is
that the
angle of
incidence
on facets
of the
rough
surface
(for,
say, normal
illumination)
has the
same
distribution
as would
be obtained
for
a plane
surface illuminated
isotropically.
Similarly,
a, - 1 -
(nr/nl)
when
ni <<
n,.
Thus
when
the
absorption
is
small,
a,
ad[1 - R(nr/nl)]/[l
-
R(n_)] = a).
(10)
When
the
absorption
is
large, on
the other
hand,
and
ad
1,
we expect
aw ad
a(".
We
will
use an
interpola-
tion
formula
which
incorporates
these
limiting
forms
by giving
a(°) and
a(') the
weights
1 - ad
and ad:
aw
(1 - ad)a(°)
+ ada(l),
or
(11)
ad
~
~~ 1-R(n,1n
1
)
(I-d
-
+ ad-
ad
(1
ad)
-R(nl)
Because
R(n)
is
a monotonically
increasing
function
of
n (for
n > 1),
R(nr/nli)
<
R(nr)
for ni
< nr, and
so
aw > ad,
with the
greatest
percentage
increase
occurring
at
low
absorption.
For small
ad and
ni = 4/3
(water),
the
ratio
of aw/ad
takes
the values
1.07,
1.08, and
1.10 for
nr
= 1.5, 2,
and 2.5
(most minerals
have
refractive
indices
within
this range).
When we
put a,,
as defined
by
(11),
for a in
Eq. (1),
we can
find the
ratio of
the total
absorption
by a wetted
surface
to the
absorption
by the
dry surface,
A/ad. This
provides
one
measure
of how
much
darker
a given
surface
becomes
when
wetted.
1 April
1988 / Vol.
27, No.
7 / APPLIED
OPTICS
1279
The
ratio
A/ad
is
plotted
as a
function
of
nj (for
two
values
of
ad) in
Fig.
2.
We
note
from
Fig.
2
that
the
darkening
effect
is
strongest
when
the
absorption
is
weak.
When
the
absorption
is strong
the
contribution
of
internal
reflec-
tions
is less
important,
since
a larger
fraction
of
the
light
is
absorbed
on
first
contact
with
the
surface.
The
interpolation
formula
(11)
gives
a
ad.
We
expect
this
to
be true
for all
cases
where
the
refractive
index
of
the
liquid
film
lies
between
that
of
air and
the
real
part
of
the
index
of
the
surface.
Now
from
Eq.
(1)
it
follows
that
dA
(1
- p)(1
- R
1
) >
da
[1 -p(i
- a)]2
(12)
for
all
physical
values
of a,
p, and
R,.
Thus
taking
an =
ad
provides
a lower
bound
for
A.
This
is
shown
by the
dashed
curves
in
Fig.
2,
which
neglect
the
increase
in
absorption
due
to the
change
in relative
refractive
index
at
the surface
on
wetting.
As such,
they
isolate
the
effect
of
internal
reflection,
which
is
seen
to
be
dominant
for
a weakly
absorbing
rough
surface.
IV.
Comparison
with
Experimental
Data
and
Discussion
Albedo
is
defined
and
measured
as the
ratio
of the
light
intensity
incident
on
a surface
to
that
diffusely
reflected
back
by
the
surface.
For
example,
Angstrom
measured
albedo
by
taking
the
reading
(a)
of his
up-
ward
facing
pyranometer,
then
the
reading
() of
the
same
pyranometer
facing
the
ground.
The
albedo
is
/
a.
His
Table
III
lists
albedo
comparisons
for
dry
and
wet
sand
(0.182
and
0.091)
and
for
dry
and
wet
black
mold
(0.141
and
0.084).
The
variability
is indicated
by
the
last
value
being
an
average
of
0.091,
0.081,
and
0.081.
These
are
plotted
as
circles
in Fig.
3,
on
which
we
compare
theoretical
albedos
(1
- ad
for
the
dry
surface,
1
- A
for
the
wet).
Other
data
available
are
those
listed
by
Sellers,
7
Table
IV, for
dry
and
wet
sand
dunes,
and
dry
and
wet
savanna
(grassland).
These
appear,
respectively,
as the
large
and
small
crosses
in
Fig.
3,
the
size
of
the
crosses
indicating
the
range
of
values
observed.
The
agreement
of Angstrom's
theory
(as extended
here)
with
the
data
is similar
to
that
obtained
by
Two-
mey
et
al.
3
(TBM),
as
shown
in their
Fig.
5.
Ang-
strom's
original
theory,
incorporating
only
the
possi-
bility
of internal
reflection
(and
that
approximately)
is
seen
to
show
the
same
trend,
but
agreement
with
the
data
is not
quite
so
good.
[However,
the
variation
in
the
darkening
of
sand
with
the
refractive
index
of
the
wetting
liquid
is
not
so strong
in
the
Angstrom
model
as
in
the experimental
data
of
TBM
(Ref.
3,
Fig.
7).]
The
fact
that
the two
theories,
based
on such
differ-
ent
models,
both
agree
with
the
(very
limited)
wet
and
dry
albedo
data
is
surprising.
Angstrom's
theory
would
seem
to
apply
best
to rough
solid
surfaces,
such
as
blackboards,
asphalt,
or
concrete.
The
TBM
the-
ory,
based
on
the
idea
of
enhanced
forward
scattering
when
the
interstitial
space
in the
medium
is
filled
with
liquid,
would
seem
to
apply
best
to finely
divided
media,
such
as
sand.
An intermediate
case
is
that
of
3
Aa
ad
1
0
1
2
Fig.
2. Ratio
of average
absorption
by
a wet
surface
to
that
of the
dry
surface,
for
varying
refractive
index
of the
wetting
liquid.
For
the
solid
curves
a,,,
is
given
by
(11),
with
nr =
2.
The
dashed
curves
are drawn
for a,,
= ad
(Angstrom).
Inall
cases
we
have
setRl
= (nl
-1)2/
(ni
+ 1)
2
(normal
illumination).
wet
albedo
0
0
dry
albedo
1
Fig.
3.
Wet
albedo
as
a function
of
dry
albedo,
for
a layer
of water
over
a rough
surface.
Normal
illumination
is assumed.
Dry
albedo
is given
by
1 -
ad and
wet
albedo
by
1 - A,
the
latter
calculated
using
nr =
2. A
is given
by
Eq.
(i);
the solid
line has
a
= a,
as given
by
(ii),
and
p
by
Eq. (9).
The
dashed
line
has
a
= ad
and
p given
by (2)
(Angstrom's
approximations).
The
experimental
data
are
as de-
scribed
in the
text.
clothing
fabric.
Further
work,
both
experimental
and
theoretical,
is
surely
needed
on
this
fascinating
every-
day
phenomenon.
We
wish
to
thank
Malcolm
Wright
and
Joe
Trodahl
for
stimulating
discussions
and
Craig
F.
Bohren
for
critical
comments
on
the
original
version
of
this
paper.
References
1.
A. Angstrom,
"The
Albedo
of Various
Surfaces
of Ground,"
Geogr.
Ann.
7, 323
(1925).
2. C.
F.
Bohren,
"Multiple
Scattering
at the
Beach,"
Weatherwise
(Aug.
1983).
3.
S.
A. Twomey,
C.
F.
Bohren,
and
J. L.
Mergenthaler,
"Reflectance
and
Albedo
Differences
Between
Wet
and
Dry
Surfaces,"
Appl.
Opt.
25,
431
(1986).
4.
M.
Born
and
E.
Wolf,
Principles
of Optics
(Pergamon,
Oxford,
1965).
5. J.
Lekner,
Theory
of Reflection
of Electromagnetic
and
Particle
Waves
(Martinus
Nijhoff,
Dordrecht,
1987).
6. F.
Stern,
"Transmission
of
Isotropic
Radiation
Across
an
Inter-
face
Between
Two
Dielectrics,"
Appl.
Opt.
3,
i (1964).
7. W.
D.
Sellers,
Physical
Climatology
(U.
Chicago
Press,
1965).
1280
APPLIED
OPTICS
/ Vol.
27,
No.
7 /
1 April
1988

Discussion

Can anyone please explain how it subtends such an angle? What I understand from this is something like this: https://prnt.sc/ruroj6. However, I can't understand how it is expressed mathematically. It was the grandson of the Ångström that the unit is named after. Huh -- didn't know Angstrom was so recent, seems like the unit would have been named long ago. The question of why a soaked fabric is more transparent is slightly different. The fabric is made of many fibers with air in between and this structure causes light to bounce around many times inside, making it hard for light to get through. When we soak the fabric, we are replacing the air with water, which has a closer index of refraction to the fibers, making the inside reflections less important and more light goes through. The swedish physicist Anders Angstrom was the first person to study this effect. He was intrigued by the darkening of soil when in contact with water and published the first proposed explanation of the effect in 1925. ![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Anders_%C3%85ngstr%C3%B6m_painting.jpg/225px-Anders_%C3%85ngstr%C3%B6m_painting.jpg) What we are doing here is calculating the ratio of radiance (power per unit of solid angle per unit of area) that is totally internally reflected (which corresponds in the figure to the blue area) ![](https://i.imgur.com/WD67C8B.png) $$ p = \frac{R_{blue}}{R_{total}} = \frac{\int^{\pi/2}_{\theta_c}\int^{2\pi}_{0}\frac{\phi_{source}}{4\pi r^2 }\cos \theta d\Omega}{\int^{\pi/2}_{0}\int^{2\pi}_{0}\frac{\phi_{source}}{4\pi r^2 }\cos \theta d\Omega} = \frac{2\pi \int^{\pi/2}_{\theta_c}\cos \theta \sin \theta d\theta}{ 2\pi \int^{\pi/2}_{0}\cos \theta \sin \theta d\theta} $$ where $d\Omega = \sin \theta d\theta d\psi$ is the differential solid angle. The integral is easy to solve since $\frac{d}{d\theta} \cos^2 \theta = -2 \cos \theta \sin \theta$ and so $p = \cos^2 \theta_c$. Note that if you isolate the term $a(1-R_l)$ we get $$ \frac{A}{a(1-R_l)} = 1+(1-a)p+(1-a)^2p^2+(1-a)^3p^3+ \ldots $$ which is the same as the geometric series $\sum^{\infty}_{n=0} x^n = \frac{1}{1-x}, |x|<1$ with $x = p(1-a)$. Thus $$ A = \frac{(1-R_l)a}{1-p(1-a)} $$ When light hits a surface we can decompose the propagating electric and magnetic fields in 2 components: the component perpendicular to the surface and the component parallel to the surface. The reflectivities for these 2 components are not the same and in particular, there's a small but non neglectable reflectivity for $\theta < \theta_c$ - which means that there's a percentage of light that is reflected in the liquid-air transition even at angles below the $\theta_c$. In the following to graphs we can see in blue the reflectivity for the liquid-air transition for the 2 polarizations: perpendicular and parallel. As we can see for both polarizations the reflectivity for $\theta<\theta_c$ is small but not zero, and so we also have to take that into account. ![](https://i.imgur.com/SjFCGbi.png) ![]() ![](https://i.imgur.com/Hzvd1c3.png) The first graph corresponds to the polarization where the electric field vector is perpendicular to the plane of incidence and the other with its electric field vector parallel to the plane of incidence. In the following image we can see the difference between *Diffuse reflection*, where light is reflected from a surface such that a ray incident on the surface is scattered at many angles rather than at just one angle as in the case of *specular reflection*. For ideal diffuse reflecting surfaces the intensity of emitted light (luminance) is the same regardless of the direction we look in the upper half plane. ![](https://i.imgur.com/OGGZciA.png) All the light rays diffused in the blue regions are going to be totally reflected as they hit the liquid-air transition making it impossible to escape to the air. ![](https://i.imgur.com/EPp839S.png) Note: If you don't know what total internal reflection is you can quickly learn [here](https://courses.lumenlearning.com/physics/chapter/25-4-total-internal-reflection/). Saying that in a Lambertian Surface the intensity reflected at angle $\theta$ is proportional to $\cos \theta$ (first derived by Johann Heinrich Lambert in 1760) is equivalent to saying that the surface has the same radiance (power transmitted per unit solid angle per unit projected area) when viewed from any angle. ![](https://i.imgur.com/hdCzIWP.png) To prove that, let's analyze the following diagram representing a Lambertian Surface. The radiance at A is $R_A = \frac{\phi_{source}}{4\pi x_0^2} $ where $\phi_{source}$ is the luminous flux of the source. At B, the radiance is $R_B= \frac{\phi_{source}}{4\pi x^2}\cos \theta$ since the plane of incidence is tilted at an angle $\theta$ from the normal direction. We just proved that for a degree $\theta$, $R\propto \cos \theta$!