This is the main bulk of my EPQ which gained an A grade at A Level. There are other bits, such as a log, reflection and several appendices, but this is the main bulk of it
3. In the Next Decade, will Invisibility be a Feasible Military or Commercial use?
Abstract:
The purpose of this paper is to delve into the complexity of the processes of
making waves invisible. In respect of feasibility, I will discuss whether it is possible
to do it as the technology already exists and explore the question of social and
economic impacts on both military and commercially public environments.
Introduction:
The key to being invisible is to be undetectable to human senses and modern day
technology. We sense things by seeing, hearing, tasting, touching or smelling and
the senses are now heightened by technology developed primarily for military use,
but also found in commercially viable areas and vice versa. For example, GPS
using trilateration was created in 1972 by the US Navy to track targets[1]. This was
adopted by other military sectors and is now found in many mobile telephones or
cars. SONAR was created by Lewis Nixon in 1906 to detect icebergs floating in
the way of ships. By1915 Paul Langévin adapted this technology to detect
submarines in World War I[2].
My research has shown that, invisibility requires the diversion of a wave from its
natural path in order to make a receiver, like an eye, think that it is receiving an
uninterrupted feed of that wave. We see objects when light is reflected off the
3
[3]
[4]
Simple diagram showing how SONAR works
How GPS systems use a minimum of 3 satellites
to locate itself by finding the area covered by all
the satellites
4. object and directed towards the eye so, for invisibility, any reflection must be
eliminated. By creating a cloak to provide an uninterrupted feed, instead of a void
in place of the object, produces the effect of being able to see through the object.
This is the same idea for all the other types of invisibility.
The most relevant senses are seeing and hearing as they are the most
conceptualised as being able to be made invisible.
The ability to see and hear requires a source to emit particular waves, hit a receiver
and reflect off anything in its path because waves can only travel in straight lines.
Or they used to!
4
5. Cloaking of the Electromagnetic Spectrum
John Pendry and Ulf Leonhardt were the pioneers of modern day optical meta-
materials.
John Pendry, Professor of Optics at Imperial College London, invented the first
designs for meta-materials in the 1990s when he was asked to give advice to
Marconi Materials Technology, a stealth technology company. He was asked to
explain how radiation absorbing carbon materials used for warships worked. He
discovered that the way in which the carbon was structured affected its properties,
as previously discussed.
He then hypothesised about a composite meta-material, where small loops of
copper wire would be embedded within fibreglass and could mimic the way
electrons affected an atom, but on a larger scale. These are called spit-ring
resonators. He extended his idea to state that it could act like a slab of iron in
terms of weight when a large enough current was run through the wires. This
would give a magnetic response due to the resonating effect caused by the
structural design of the loops. He likened this to “pushing a swing, letting it swing
back and forth naturally and then pushing the ropes faster so that when you push
forwards the swing actually goes backwards.”[10]
This effectively causes it to swing at a different resonant frequency. He developed
and adapted his idea and eventually cut the wire loops. This became the first
magnetic resonator and allowed him to change the magnetic properties of a
material by choice, just as if it were a switch. He then combined this with the
radiation absorbing material that he started with and, without realising, created a
method of manipulating electromagnetic radiation, which is one of the fundamental
building blocks of optical cloaking, as visible light is in the electromagnetic
spectrum. Although it only works in the microwave length of the radiation,
everything else has developed from this technology.[10],[13]
East German Ulf Leonhardt, Chair of Theoretical Physics at St Andrews
University, started work in 1999 developing a visual cloak. In 2009 he thought his
work would be complete by 2011, but many other people had also made
5
6. breakthroughs - to be discussed. He published a paper, “Optical Conformal
Mapping”, in many notable journals, some of which I have already referred to. [12]
The first lines are quoted - “an invisibility device should guide light around an object as if
nothing were there.”
His work was similar to Pendry’s, causing concern about whether his paper should
be published in the Physical Review Letters Journal. Eventually it was seen as
separate but very similar work.
In 2006 David Schurig and David Smith, research engineers of Duke University,
developed a fully working device based on Pendry’s theories[13] and early
developments, proving the theory to many of the world’s sceptics.
The evolution of cloaking technologies soon followed and the ideas will be
expanded later.
6
[46]
Diagram showing the electromagnetic spectrum and also relating the size of the wavelengths in comparison to everyday objects
7. Current Developed Methods for Visual Cloaking:
Using Cameras-
Some early mechanisms, involving the use of multiple cameras, created a live
stream of what they were looking at, which would then be projected onto a screen
or special material made with reflective glass beads, only a few microns in size, on
the opposite side of the object. Researchers at Tokyo University[20] were the first to
publish results of such a device that had any potential. This makes the science
fiction seen in films like “James Bond: Die Another Day”[15], seem realistic and is
particularly evident in the scenes where the Aston Martin Vanquish car becomes
invisible using a more “advanced method” of the same process. This particular
method is simple to describe but there are still ways to improve the technology as
graphics processors, resolutions of screens and projectors improve, making it
appear seamless.
7
[17]
[18]
[20]
Top Left: LED screen when off, on the Mercedes
F-Cellcar
Top Right: LED screen on Mercedes F-Cell in
action
Bottom Right: Tokyo University’s initial
developement
8. Military
Pros Cons
Can be added to most objects and vehicles It is still fairly obvious that something is there
due to the screen resolution being different
to what we naturally see
Fairly simple mechanism so will keep costs
down
Not very ergonomic. Looks bulky and does
not follow the shape of the item being
cloaked
Can always be improved glitches will be obvious to enemy
Glass Beads - relatively cheap to make and
replace at about $37 per 20 grams (enough
for 5 metres squared)[48]
Glass beads likely to break due to fragility in
combat situations
Commercial
Pros Cons
Can be added to most objects and vehicles People may not like the look of the
technology when not in use
Useful to temporarily keep eyesores or
events (such as CSI or construction) hidden
from the public, without them walking into
them
Cannot be used with smaller objects, such
as computer cables, due to the size of it
Does not completely hide criminals so will
not aid them
Consumes a lot of power in order to achieve
its purpose
Can be use in vehicle interiors to remove
blind spots or to act as a navigation aid
Glass beads likely to break due to fragility in
combat situations
Works in most light levels Vast amount of energy required to power
LED screens as expected as the screens are
existing technology that already accounts for
most peoples electricity bills at home
Glass Beads - relatively cheap to make and
replace at about $37 per 20 grams (enough
for 5 metres squared)[48]
Vast amount of energy required to power
LED screens as expected as the screens are
existing technology that already accounts for
most peoples electricity bills at home
Calcite experiments-
Two significant studies have demonstrated how to hide an object completely. The
experiments were carried out by researchers at both Birmingham and Zhejiang
Universities and, although both methods are the same, each uses the science in two
different ways. [20],[21],[22],[23]
Calcite is an anisotropic material, where the physical and mechanical properties of
the material differ depending on its direction when measured. The best example in
relation to calcite is that when it is tilted at the correct angle it is like looking at a
3D IMAX screen without wearing 3D glasses. However, when tilted at a different
angle it is like looking through any other transparent material. This specific
property is called “birefringence” or “double refraction.”[22]
8
9. A common experiment carried out in a school Physics class uses a semicircular
block of glass or transparent Perspex. When a light source is focussed on the flat
side of the block directly through its centre, the light continues to travel in a
constant direction. However, if the light source is moved so that the concentrated
beam of light hits the semi-circular block from an angle, the path of the light will
change in relation to the refractive index of the material. This is caused by the light
accelerating and slowing down on one side before the other, changing its direction.
Like refraction, the wave changes direction. The difference is that instead of the
light being in a single cluster, changing direction, birefringence is where the
internal atomic structure of anisotropic materials sorts the beam into two by
reducing its speed into two different velocities, according to each of two mutually
perpendicular vibration directions. By slowing it down, the momentum is not lost
but it changes direction symmetrically and is called negative refraction. This is
another benefit of light’s wave particle duality. Light can behave like a particle
when it travels as a wave, the particles constantly vibrating in specific directions.
Therefore, materials with the birefringence property can sort perpendicular
vibrating particles into two.
However, this will not cloak anything only cause a blurred image where the
resultant paths of the light meet the retina of a person’s eye at different angles
simultaneously. This led the scientists at Birmingham to create an experiment,
improving on previous attempts, which would not need high powered electron
microscopes or special detectors to view any significant result, just using the
normal eye. They developed the experiment using Maxwell’s Equations as a basis.
The equations that contributed to this experiment are the fundamentals of
9
Images of the effect of birefringence in Calcite crystals
[41] [40]
10. electrodynamics and the way that this helped design the cloak was similar to any
sort of equation or function.
Setting the output to be a specific result (for example: 0 for not able to be seen)
the equation can be re-arranged to produce the necessary fields that need to be
produced to create the desired outcome. In this case, it was to produce the
necessary internal nano-properties of the calcite by artificially engineering crystals
with particular angles of the birefringence property and was perfectly transparent
and undamaged to sight (to name only but a few of the adjustments). [23]
The optical engineers then produced a successful experiment, based on the past
work of other scientists across the world, in relation to carpet cloaks. These are
devices that turn bulging reflective surfaces into flat ones, removing any suspicion
of an object being hidden underneath it. This uses the birefringence[23] of the
calcite to create a single flat plane for the source light to be reflected off rather than
a bumpy surface, which would cause the source light to be reflected in multiple
directions. This prevents the surface from looking like a bulge, as the light needs to
reflect in multiple directions in order to appear three dimensional. The main
breakthrough was that it worked on a much larger scale and in the visual light
spectrum, compared to the nano-scale of previous attempts. There was no
distortion or image loss from all viewable angles, which showed that the cloaking
technique worked as if only a flat mirror was present.
10
[20]
The refractive index of the surrounding media is assumed to be 1.532. (a, b) Light incident on a cloaked
bump at incident angles of (a) 45° and (b) 15°, showing almost no scattering. (c, d) Without cloaking, the
bump strongly scatters the light into different directions. Although the cloak is designed at 590 nm, in
accordance with the experiment, the ray-tracing calculation is performed at 532 nm.
11. Another interesting experiment is to hide a section of a paperclip and this is where
polarisation filters come into play. A paperclip was placed in between artificially
engineered calcite crystals and the source of light was a green laser polarised into a
transverse mode of polarisation. This put the photons of light into a single plane of
travel preventing the photons being scattered in multiple directions, similarly to the
carpet cloak experiment (see the fence diagram). After travelling through the
calcite crystal medium, the source light was polarised again. The resulting effect
was as if nothing was there in the first place unlike the other experiment, which
uses the mirror-like property of the bulging material to trick the viewer into
thinking that they are looking at a single flat object.
11
Computer generated simulation of how a
light ray is able to pass around the object
when the cloak is operational
Computer simulation of Maxwell’s Fish
Eye lens showing how light rays interact
with the lens to form a cloak
[21]
Left: Diagram of experimental setup of second experiment carried out by the researchers
at Zheijiang University. It uses the same method as that of the Birmingham researchers
Right: The calcite in action
[14]
[14]
12. The process is again related to the birefringence property of the calcite. As the
source light enters, it has been polarised to only exist in one plane, splitting into
two directions around the internal object. When the photons penetrate the second
crystal, birefringence causes them to converge into one path, making it appear as if
there was nothing in the way to block the light. Therefore, the process makes the
light waves circumvent the main object.[20],[23]
Military
Pros Cons
Completely cloaks the object or vehicle
allowing extreme tactical assaults,
ambushes and spying without the enemy
knowing what is going on
It can also be used by the enemy as the
technology has been published
Defence heightened- enemy can’t see your
position
sand or distortions show outline, footprints
Good camouflage if you are injured It is currently difficult to use on large objects
due to the size of the crystals and the need
for external polarisation
High abundance of the materials due to
large finding in Mexico[49]
High cost at the moment due to young
technology at £11.95 per 15cm squared, but
this should come down as the technology
allows thinner calcite to be used or a new
smaller crystal is discovered with the same
properties[49]
Commercial
Pros Cons
Eyesores and crime scenes can be
completely hidden from the public
people could walk into them or planes fly
into them unless there was a warning, which
defeats the purpose of cloaking in the first
place
Personal security- valuables hidden, no
security safe needed
Criminals have an unknown identity
advantage as they can’t be seen when
stealing or harming
Good for hunters to prevent scaring the
animals
Crystals still too large (without being toxic)
to be used on clothing, for example
High abundance of the materials due to
large finding in Mexico[49]
High cost at the moment due to young
technology at £11.95 per 15cm squared, but this
should come down as the technology allows
thinner calcite to be used or a new smaller
crystal is discovered with the same properties[49]
12
13. Microwaves-
Microwaves are usually associated with cooking foodstuffs. The frequency of the
emissions makes water and fat molecules vibrate causing heat production and, over
time, this causes the food to heat up and to cook. Microwaves are also used in
mobile phone technology to send and receive texts or calls, but modern technology
has started to look at using other types of signals.
When describing microwave cloaking or invisibility you might think I mean hiding
these signals in terms of covert phone calls, say, or for hiding objects from
detection by radar. However, this is another example of development of visual light
cloaking. At the moment this method is strictly used for microwaves only but,
according to the researchers at Texas and Duke Universities[43], the same
techniques should theoretically work on visible light and even other parts of the
electromagnetic spectrum. This comes from the idea Pendry came up with initially
(as discussed).
It is now easier to explain with an understanding of how calcite effectively directs
light around an object. Although using split ring resonators to do the same thing, it
is not quite complete invisibility. The main reason for the introduction of this
technology was to enhance the ability of a B-2 Bomber to absorb and bypass radar
detection and to further reduce the ability to be able to detect a bomber.
13
Left: Diagram of how the split ring resonators
effect the microwaves
Right: The microwave cloaking device
[45] [42]
14. To do this, the researchers first had to reduce reflection and you can only see
something if the object reflects the source light back to your eye. This works with
all relevant waves, so the split ring resonator device actually fulfils its primary
purpose. David Schurig’s analogy is:
"One first imagines a distortion in space similar to what would occur when pushing a pointed
object through a piece of cloth, distorting, but not breaking, any threads." [43]
This demonstrates how other designs were forged from the idea of redirecting the
waves around an object without distorting them to create perfect invisibility.
Military
Pros Cons
Vehicles can be hidden from radar detection
at distances
Enemy can have same technology
Computer data can be secured further Not 100% effective
Has triggered the development of other
cloaks using meta-materials in the future
Commercial
Pros Cons
Could help remove potentially harmful
microwaves
May not have work on a large scale due to
some waves still able to reflect
Could help aid wireless privacy Cyber criminals could cover their tracks
Removes conflict e.g.protests against
mobile phone masts
14
[44]
Visual data of the cloak in action
15. Infrared-
Cloaking the infrared part of the electromagnetic spectrum is about hiding
someone’s or something’s heat signature emission. A heat signature is a person’s or
object’s specific picture of cold and hot spots, created all the time. A person
constantly functions and does so particularly because of the body’s enzymes being
able to work and produce vital chemicals, such as proteins.
They have an optimum temperature at which they can work otherwise they
become denatured. As a result, the body maintains an internal temperature of
around 37 degrees Celsius. This temperature varies slightly across the body due to
things like an area having more veins, allowing heat to be picked up with blood
cells and emitted with the larger surface area of the skin or simply being areas
covered up by clothing.[24] This already creates a noticeable heat signature as the
cooler spots will be blue, whereas the hotter areas will be yellow or even red, and
depends on infrared detector settings. When a person exercises, the muscles do
more work and the core temperature increases. This increase needs to be
countered to maintain a constant internal temperature so, through a process called
homeostasis, the veins dilate and move closer to the skin’s surface. The tiny
muscles attached to the hairs on the skin then contract and this causes them to rise.
15
[25]
Simulation of how homeostasis is controlled via the skin
16. This lets heat to be emitted and cool air captured, helping maintain the body’s
natural temperature.
Homeostasis is generally a good thing but, in some cases, it can be the cause of life
or death. For example, it can cause major safety issues for military personnel in the
field as running during covert missions to avoid danger could actually cause more
danger by revealing even a visually cloaked soldier.
Blücher Systems GmbH created Ghost camouflage[26],[27] - Thermal/IR (Infrared)
Camouflage/ Signature Management Fabric Technology. The detail is top secret,
but we do know that it is a mixture of materials surrounded by metalised fibres,
causing the fabric to act like a meta-material removing most of the wearer’s heat
signature. Another advantage is that it comes in a visual camouflage print enabling
it to continue to conceal the wearer.
The same company has developed a method of turning camouflage nets into cloaks,
allowing vehicles to be concealed from infrared detection. The material allows cool
air in and hot air out, creating a mini-cycle around the materials, keeping the
internal temperature similar to the external temperature.[30] It removes the
problem of stagnant heat from cooling vehicles creating a hot spot, which can be
detected. The down side at the moment is that the technology has only been able to
create a natural temperature drop of four degrees.
16
[26]
Infrared image of a normally equipped soldier
against a soldier equiped with the Ghost technology
17. On the positive side, there have been new developments to make the vehicles
appear completely invisible on the infrared spectrum. There are two similar
methods using optoelectronic devices to capture the infrared emissions of the
surroundings, which make the exterior of the vehicle appear to have the same heat
signature as the space around it, rendering it non-existent. This is similar to the
video cameras filming one side and projecting on the other creating a visual
invisibility effect.
In my view, the most advanced method is BAE Systems Adaptiv Technology.[29] It
produces the effect of having the surrounding infrared emission, using a method of
effectively having pixels all over the armour.
Above: The Exterior of the vehicle cloak
Below: Diagram showing how the vehicles work
17
[30]
[29]
Left: ADAPTIVE technology fitted to land air and sea militay vehicleds
Above: Pictures showing how ADAPTIV can be used to disguise a tank as a
car or other vehicle
18. The pixels are hexagonal tiles and can change temperature rapidly, allowing the
vehicle to cool down or heat up, depending upon the surrounding environment.
Another advantage is that they are individual, allowing one vehicle to take on the
infrared signature of another vehicle or object in order to mislead the enemy.
Notably, this method can already be used on vehicles such as planes and ships
because of the tiles’ formation. It is also extremely light in comparison to stand-
alone armour and is just as durable, which should save costs on normal armour. In
comparison to the Eltic Black Fox technology,[28] for example, it is a very low
power consumer, which is important in military survival where power is important
for many things, like communications and survival.
Military
Pros Cons
Enemy infiltration easy- spying, ambush (in
the night)
Nothing stopping technology seeing the
extremely feint outline that is produced
Defence heightened- enemy can’t see your
position (in the night)
Sand or distortions show outline, footprints
Heat seeking weapons will not detect you Enemy could have same technology
Can be mounted on most vehicles
Ghost- same as current military gore -tex
layers (£25)
Adaptiv - no official cost given as public
information, but one can assume it is
affordable to a military budget to cover
vehicles such as tanks
Commercial
Pros Cons
Could help control infrared emission in the
form of heat, increasing efficiency of things
such as boilers
When developed further, could be mounted
to cars, giving criminals an advantage in
night police chases
Not really relevant to a commercial society
18
19. Acoustic Cloaking
We already encounter noise cancelling abilities in our day to day lives. This could
be from noise cancelling headphones, which use microphones to record the
ambient noise and play it back along with the music, creating a region of
destructive interference. It appears as if there is no external noise. Another
example is double glazing in our windows, which traps a variety of waves,
including sound, in a vacuum and prevents it from propagating through the
medium of air. It is also found in the design of major roads, where high walls are
shaped specifically to reflect sound waves upwards into the air and away from the
inhabited areas. [31]
Sound can only be one type of wave, longitudinal, where the wave particles of a
medium vibrate in the same direction as the propagation. The way we hear things
only happens with longitudinal waves due to rarefaction. [32] When we speak our
vocal chords vibrate caused by the chords flexing in and out. When they flex one
way, air particles are pushed on that side and then collide in a chain effect with
other air particles in the way. This is compression. When the vocal chords flex in
the opposite direction they pull the air particles towards them, effectively
stretching out the air particles, creating a drop in pressure, pulling in more
particles, which again creates a chain effect and pulls the particles even further out.
The pressure therefore decreases. This overall process is called rarefaction and
cannot happen with transverse waves.[33]
19
Diagram of how a longitudinal wave travels through a medium
[32]
20. Like transverse waves in the electromagnetic spectrum, longitudinal waves can be
reflected. However there is no phase change. As they can be reflected, you can use
sound waves to detect things, which are directly in the way, by recording the time
taken for them to travel and return.
Sonic, Ultrasonic and Infrasonic waves have different wavelengths and are used to
detect objects.
They rely on the same technique, where a source sends out the wave and a
receiver detects it after reflection off an object. This reflection shows the receiver
that something is actually there and the most common form is SONAR.
In conclusion, in order for an object not to be detected by sound, the waves would
have to either be displaced unnoticeably like visual invisibility, or absorbed.
Unlike visual invisibility, if the waves are absorbed there won’t necessarily be a
blind spot in the received signal. Otherwise there would be a void of darkness
instead of nothing. This could be used above sea level for ships or planes but would
not necessarily be good for submarines, as it would have the same effect as a blind
spot with visual invisibility. It would render the cloak useless as someone trying to
find it would notice a SONAR signal having a patch where it has been absorbed
and not reflected back.
Three methods of making things impervious to sound waves used for detection,
and even from emitting sound themselves, are worth mentioning.
German/ American Acoustic Invisibility Cloak: [34],[39]
The first does not yet have a specific name. It is simply an object made from two
polymers, one hard and the other soft, formed into a plate only a few millimetres
thick, which resonates or vibrates at 100Hz. As the audible frequency range for
humans is 12 Hertz to 20 Kilohertz, it is not within our hearing range, but is in the
range for SONAR frequencies[35]. However, the plate is not a plane and has
various holes of different shapes and sizes (tenth smaller than millimetre), making
it a nano structured material.
20
21. This was developed at the Karlsruhe Institute of Technology (KIT) in Germany.
The scientists created the micro-structure to cause an effect similar to the way in
which the calcite bends light seamlessly around an object. However, instead of
double refraction or polarisation, the object works by guiding the sound waves
around the circle in the centre. This circle could be likened to a blind spot. In
addition, to make the cloaking as seamless as possible, and not to affect the
frequency of the waves from what is perceived from input to output, the speed of
the waves is increased as they travel around the blind spot, due to the
microstructure.
Doctor Nicolas Stenger, Institute of Applied Physics at KIT, said:
“the key to controlling waves is to specifically influence their local speed as a function of the
running direction (or propagation) of the wave.”
By doing this, the time it takes for the waves to travel straight through the centre
will be the same as the time it takes for them to completely bypass the blind spot.
Another leading scientist, Professor Martin Wegener also from the Institute of
Applied Physics at KIT said:
“Contrary to other known noise protection measures, the sound waves are neither absorbed nor
reflected. It is as if nothing was there.”
Professor Nick Fang, university of Illinois, created a similar device. The holes were
replaced with bumps, in the same sort of microstructure, as they caused the sound
waves to reflect off the underlying material of the object being cloaked. This
provided a practical use for the design as a method of removing detection for
underwater objects by SONAR.
21
[36][34]
Left: KIT’s version of the cloak
Right: Professor Fang’s Version of the
cloak
22. Military
Pros Cons
Can prevent detection by SONAR Can work for the enemy as well
Easy to mount onto existing submersibles No detail on resistance to corrosion or other
damage
Commercial
Pros Cons
N/A currently no commercial market for this
Improved Sound Proofing:
This is like the soundproofing material you can buy for walls. The difference is that
it is a developed structure, made from a meta-material, which has improved the old
designs by minimising the sound waves to nothing. As with normal soundproofing
found in music studios, the aim of the material is to prevent sound being heard by
an external receiver. It does this by having a three dimensional pattern of deep
grooves, allowing the sound waves to diffuse out as they lose the kinetic energy
needed to move as they continually bounce from side to side before reflecting back
out. The materials used for this are usually porous as they diffuse the energy of the
waves having an internal structure that acts in the same way as the external
structure as a whole. The most common internal design is a honeycomb structure
and can trap the sound waves without releasing them again. However, this material
works both ways and works very well until the reflected waves reach a receiver.
This couldn’t be prevented, unless another method of putting a soundproofed room
within another soundproofed room was developed. This worked in exactly the
same way, except the soundproofed layer doubled and a gap of air was formed to
contain the sound waves. This is more expensive as construction is difficult and it
would take up space. [31]
22
23. One of the Navy’s biggest issues is being detected by techniques like SONAR.
There was no way to prevent detection, as the waves would either reflect back or,
if they were absorbed, a blind spot would be created and they could still be found.
So a new material and structure needed to be developed to counter this. The result
also ended up working both ways, so that it could actually soundproofed the inside
of a ship to prevent audible detection.
Although not named name yet, Researchers led by Steven Cummer of Duke
University, have developed a solution using stacked sheets of plastic, covered in
holes. These holes, like the German acoustic cloak described earlier, are arranged
in a specific fashion to enable sound waves to speed up and around it to make it
appear as if they have not hit anything and just travelled through it. Therefore, the
sound waves would neither be absorbed nor reflected, removing the issue of
SONAR detection. As a result, the design of the underneath of the structure acts
as a sort of super sound proofer because the underside forms a porous structure
that diffuses the sound wave energy. The difference is that the sound waves diffuse
as they leave and lose their energy extremely quickly due to the structure. This has
been proved by both Duke researchers and ratified by researchers in Hong Kong.
The main advantage of this method is that it is simple enough to be manufactured
cheaply on a mass scale. It can also be applied to any existing surface, just like
sound proofing a room. Many environments could benefit from this, ranging from
housing to concert halls, ships and many other types of places where sound causes
issues. [37]
23
Modern day warships are not only huge, but they also carry a large amount of
technology making them more susceptible to detection from large distances
[37]
24. Military
Pros Cons
Can be added to most objects and vehicles May be flimsy in comparison to adding more
metal armour
Fairly simple design If damaged cloak becomes useless
Inexpensive - can be made on a mass scale
using sustainable materials
If damaged then pieces could harm wildlife
Commercial
Pros Cons
Can be added to most objects and vehicles May be more bulky than other
soundproofing
Fairly simple design
inexpensive - can be made on a mass scale
using sustainable materials
Could be used to soundproof concert halls
from surrounding businesses and
residences
24
Recording studio with acoustic dampening tiles
[31]
[37]
Computer render of he form of a section of the cloaking system. Note how the plastic
is also layered in a sort of macrostructure along with their individual microstructures
25. Speech Cancelling Gun: [38]
This is completely different to anything you would expect for sound cloaking. This
method involves aiming the gun at someone or something making noise and makes
them mute!
This is very different to the previous methods described. Rather than physically
cancelling out or redirecting the sound waves produced by a person speaking, it
prevents them speaking altogether. It is more of a psychological blocker. This
could be very useful in situations where someone needs to keep quiet before they
“spill the beans” and it could keep people quiet where silence is required.
This simple method was invented by two Japanese Scientists, Kazutaka Kurihara
of the National Institute of Advanced Industrial Science and Technology and Koji
Tsukada of Ochanomizu University. It works by recording what the target is
saying and then “shoots” or plays it back to them with a 0.2 second delay. Now if
hearing your voice wasn’t bad enough, you hear what you are saying at nearly the
same time as you are actually saying it. We perceive this as happening at exactly
the same time because our cognitive brain processes allow us to live in terms of
understanding events going on around us and also to solve problems. By problems,
I mean the processes that go into our thoughts, which need to be turned into
actions, such as lifting an object.
By playing the voice of the person so soon after they have said it, causes the part of
the brain in charge of this cognitive function to go into a sort of mini meltdown. If
the cognitive function cannot work, the victim will start to stutter as they try to
work out the problem of speech. Eventually they will not be able to remember how
to speak and go completely silent. This has a range of thirty metres.
This is not acoustic cloaking as already described. However, it does prevent the
sound occurring in the first place. The main advantage of this is that, it does not
physically affect the subject of the “shot” but just confuses them completely.
The researchers discovered that this works extremely well for people who are
reading at pace or people who are speaking spontaneously. This would be useful in
libraries to remove disturbances of meaningless chatter or boring speakers.
I believe that the cognitive function does not have sufficient time to realise what is
happening as the mind is focussed on reading out what has been prepared. So
25
26. when the researchers said that saying sounds like “ahhhhhhhh,” could not be
silenced, my hypothesis seems correct as the “ahhhhhhh” is just a long meaningless
sound and requires much less of our cognitive function to be produced, so our
brain would still have time to work out that the sound is repeated but isn’t said at
the same time.
In terms of other practical uses, the researchers said:
“there are still many cases in which the negative aspects of speech become a barrier to the
peaceful resolution of conflicts.”
This suggests that they could develop this method for use in political situations to
help prevent world conflicts, removing things like insults and other statements.
26
[38]
The speech cancelling gun broken down into components
27. Military
Pros Cons
Can prevent political issues taking place by
stopping a figurehead insulting someone
(for example)
May not always work
Can be done from a distance of 30 metres
away
The laser aim gives it away, depending on
the reaction time of the target
Can always be improved May be a political issue in using it
No official pricing, but it has been noted to
cost a total of $225, well in a military style
budget
Commercial
Pros Cons
Could prevent misleading comments in
situations such as a court house
In a way against human right of free speech
Can stop people from talking loudly in a
library or any other quiet place
Quite bulky
Could calm people down in angered
situations by making them confused
May not always work
No official pricing, but it has been noted to
cost a total of $225, might be too much for
comercial use, but components may get
cheaper over the years and become more
affordable
27
28. Conclusion
The following are my ideas about the feasibility of methods of cloaking, expanding on the
advantages and disadvantages.
Visual and Infrared –
· Military:
!Would have the ability to spy and infiltrate without being detected. It would also
mean less equipment would be needed to be carried by an individual, as less
protection would be required, which otherwise would give them away. However, this
can work both ways.
One problem I do see is that, despite being undetectable to the human eye, there is
still the problem of being detected on other wave lengths, such as infrared when
cloaked for light. As technology develops I am sure the scale of each cloaking
method could be reduced to a less bulky technology. This can be seen with things
such as Moore’s Law where the number of transistors on an integrated circuit double
every 2 years. Although saying this I doubt it will be completely fine-tuned in the next
decade as there are a lot of issues finding non-toxic crystals small enough to act like
calcite, for example. Also, in order to achieve complete visual cloaking, a combination
of the technologies would be required and even more time needs to be spent in
research and development to make it a more manageable accessory.
Ideally, only one solution is required to achieve what all the current ideas do.
· Commercial uses:
I think this is best used to prevent eyesores affecting the public, like wind turbines or
power stations. Even damaged architecture could be made invisible using screens
integrated into the scenery showing images of what it should be like.
This would clear up many issues about new green energy producers being built in
open areas and make the places where people live feel better as it would always
appear to be maintained and in good order.
Another use would be in vehicles such as cars, where side pillars obstruct the
driver’s view. These could be made invisible resulting in the reduction of the number
of accidents with cyclists and pedestrians, for example.
Crime scenes could be covered up by the police to prevent public alarm. Although
there would still be the problem of having some sort of barrier to prevent people
walking into the crime scene tent. This would then render the cloaking useless as
people would still know something had happened albeit more discreet. This also
comes back to the use of invisibility for criminals. There would be no way of knowing
if a professional thief had stolen anything.
Sound Cloaking-
This is vital if you are already visually cloaked as there would be no point being
visually hidden if you could still be heard. For example, if you were in a bank vault
28
29. stealing money and someone heard you, the whole vault could be sealed off with you
in it.
· Military Uses:
The most obvious use would be in the Navy. When submarines discreetly move
about hundreds of leagues below the sea, they can’t really be seen unless they are
positioned near to the surface and can then be visually spotted or detected by a
ship’s radar. This was the case until SONAR was invented. It can also keep vehicles
and gunfire quiet so the enemy can’t locate one’s position.
· Commercial uses:
As someone who takes part in Sports, I personally prefer to go into my own sort of
bubble before matches. Using noise cancelling headphones is a method of blocking
out sound, but this only makes the world seem invisible to you. On a larger scale, it
could be used in places like examination halls to help students to concentrate and
prevent any potential cheating.
As described above, it could also be used to conceal publicly visual architecture,
including power stations, to help to keep places quieter, people happy and their living
areas comfortable whilst still having the energy source.
Plasmonic Time Cloaking:
This toy plays with the idea of personal security and privacy and the details are
described in appendix C.
· Military Uses:
There is a clear use for this method to encrypt data and transmission across military
networks. This would prevent security threats and also increase the safety of the
military and the public by protecting information and locations. It would also allow for
organised assaults to take place without the enemy knowing anything was going on.
· Commercial uses:
!Similar to the military uses, the technology could be used to protect personal
information being transferred illegally online, ranging from bank details to where
people live. Of course if someone was using this technology to “hack” someone’s
details they could do so without being detected at all. Censorship is another potential
use for the technology, be it by the censor or against the censor as neither party
would be known to breach each others’ security
29
30. Overall-
My research shows that invisibility does exist and potentially will be more widely
available in the near future, with developments in meta-materials and carbon nano-
tubes. However, despite all of the practical uses for the many different methods,
there is still the obstacle of it “being in the wrong hands.” This is really the one major
barrier to having this technology available and has often been a dilemma for many
other technological developments, from the mobile phone to the internet.
There will always be some debate about whether it is a good thing, but from past
experience the technology is still likely to be made public and used both
commercially and almost definitely in the military.
I am sure we won’t find the technology as individual elements but combined, as
invisibility must be covered from all angles. After all, it would be pointless hiding in
camouflage gear in the long grass if someone could see you through an infrared
detector or hear you.
Also, the cost of the materials used varies. Some materials, such as the LED
screens, are existing technologies so will cost a lot less than new ones (about 10% of
the cost). This affects the commercial market more than it does the military one as
invisibility is currently a concept and, in an economic crisis, the development of the
technology would not be a priority due to cost. The military finds itself in a similar
situation, but its budget is higher and they can develop these concepts into full
working technology in order to improve safety and defence. This would then
eventually turn the technology into a commercial use as evident with SONAR and
GPS.
30
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[40]: Olympus Digital - http://www.olympusmicro.com/primer/lightandcolor/
birefringence.html
[41]: All About Gemstones - Information Website for Gemstones - “Optical Effects in Gems:
Birefringence” - Viewed - http://www.allaboutgemstones.com/gemstone_birefringence.html
Author References:
1. John G. Delly, Michel-LŽvy Interference Color Chart . www.modernmicroscopy.com
2. Stanford University, Refraction in Gems . www.stanford.edu
3. Wuerzburg Uniaxial Minerals . www.geographie.uni-wuerzburg.de
4. Stephen A. Nelson Introduction to Uniaxial Minerals . www.tulane.edu
5. Douglas B. Murphy Optical Birefringence . www.micro.magnet.fsu.edu
[42]: Greg Gbur - Guest Blog - Created 11th January 2011 - “Invisibility: After Several
Years of Research, i’t’s Just Gotten Weirder” - Images Used - Viewed - http://
blogs.scientificamerican.com/guest-blog/2011/01/11/invisibility-after-several-years-of-
research-its-just-gotten-weirder/
[43]: Duke Today - News Releases for the University - Created 19th October 2006 - “First
Demonstration of a Working Invisibility Cloak” - Viewed - http://today.duke.edu/2006/10/
cloakdemo.html, http://today.duke.edu/cloak-invisibility-demonstration#video
[44]: Fahad Inc. - Technology Blog - Created 22nd October 2006 - “First Demonstration of
a Working Invisibility Cloak” - Images Used - Viewed - http://www.fahad.com/2006/10/first-
demonstration-of-working.html
[45]: ZeroPaid - Online Forum - Created 2006 - “A Real Invisibility Clokak? Wizard!” -
Images Used - http://www.zeropaid.com/bbs/threads/40057-A-real-invisibility-cloak-
Wizard!
[46]: NASA - Electromagnetic Spectrum Diagram - Created 26th October 2007 -
“Electromagnetic Spectrum” - http://en.wikipedia.org/wiki/Electromagnetic_spectrum
36
37. [47]: Dr Parnell - Manchester University News - Created 14th February 2012 - “Invisibility
Cloak could Protect Buildings from Earthquakes” - Viewed - http://www.manchester.ac.uk/
aboutus/news/display/?id=7968
[48]: Microspheres - Nanospheres - Retailer of retroreflective glass beads - “Glass
Nanospheres and Microspheres” - Viewed - http://www.microspheres-nanospheres.com/
Microspheres/Inorganic/Glass/Glass.htm
[49]: Crystal Reflection - Crystal Shop - Created 2011 - “Optical Calcite” - Viewed - http://
www.crystalreflection.co.uk/NSCO1656-optical-calcite.html
37
38. Appendix A
Meta-materials
What is a meta-material?:
A meta-material is specifically engineered for a purpose and is made to utilise the
properties of non-homogenous materials. Instead of acquiring their properties from
chemical composition, it is their combined structure which is the defining
characteristic.[5]
Differences between meta-materials and normal materials:
Naturally occurring materials exhibit behaviour based on the molecules they are
made of. The atomic structure of the elements the materials are composed of
determines what properties the product will have. This is similar to chemistry
where a lone electron in the outer shell of an atom of an element determines what it
will react with and how well. [5]
For example, take the relationship between wood and light; like all natural matter,
wood reflects and refracts light allowing us to see it.
However, how much light it reflects and refracts depends on how the
electro-magnetic light waves interact with the particles that make up the wood.
This is best seen when light reflects off a shiny surface in comparison to a matte
surface. [6]
As explained, meta-materials are not chemically combined but are structurally
positioned. Therefore, it is not the atomic particles of one material involved that
determine the overall properties but it is the sum of the parts, not the parts
themselves, that determines how the material behaves[5]. Researchers found that
by using certain materials, such as gold and copper arranged in particular patterns
and shapes, they can combine the properties of those materials. These are often
found in electrical circuits due to their highly conductive properties [5]. An
example of a chemical change rather than a positional or structural change, are the
properties of sulphur and iron after they are chemically combined. As a mixture,
the iron can be extracted by using a magnet leaving the sulphur behind. When the
mixture is heated with a bunsen burner, a chemical reaction occurs forming iron
(II) sulphide resulting in a chemical change to the overall structure and the
product is no longer magnetic[7].
38
39. So, unlike natural materials, the behaviour of meta-materials depends upon the
properties of the materials that they are made of and the distinct way they are put
together, not the chemical sub-atomic structure.
What are meta-materials used for?:
Meta-materials are currently used for six main purposes[10]
1. Super lenses
2. Antenna systems
3. Optical sensors
4. Solar Cells
5. Highly sensitive detectors
6. Invisibility cloaks
Before we get onto invisibility, the main uses of meta-materials as you can see have
been used for scientific and infrastructural purposes. Super lenses, for example,
work because the properties of the materials have created a negative refractive
index, which allows a spacial resolution (or image) below the light source’s
wavelength to be viewed. This shows that like naturally occurring materials, meta-
materials can be sub-divided into groups. In the case of invisibility we will be
focusing mainly on photonic meta-materials.
39
Chemically combined atoms of elements forming a more complex macro structure
[8]
[9]
A visual representation of different meta-material structures
40. Types of Meta-materials Used for the Relevant Cloaking Purposes:
- Photonic meta-materials,
The term photonic derives from the involvement of photons, an elementary
particle, which is the quantum of all forms of electromagnetic radiation. The part
of the electromagnetic spectrum affected by these is the visible, infrared (IR) and
Terahertz(THz) frequencies. These collectively can also be called optical
frequencies.
- Terahertz meta-materials,
Follow the same principals as other meta-materials used for electromagnetic
radiation. However, they only affect the terahertz frequencies.
- Acoustic meta-materials,
Manipulate sound. The three main sub-forms affected are sonic, infrasonic and
ultrasonic waves and can occur in all the basic states of gas, liquids and solids.
Sonic waves in particular can be extended to the negative refraction domain.
- Seismic meta-materials,
Are designed to counteract the adverse effects of seismic waves on man-made
structures, which exist on or near the surface of the earth.
- Plasmonic meta-materials,
Are special materials created by only a few scientists, in particular Professor
Martin McCall of Imperial College London. This topic will be developed later in
this paper and involves the cloaking of a small section of time!
What Form do we find most meta-materials?:
As previously discussed, meta-materials are made up of structured material
combinations. We now come to the fact that in order for them to have any
significant effect on a process outcome, they usually have to be designed on a nano
or micro scale, but this is not always the case. The materials usually involved are a
metals and dielectrics. An example of such a composite is shown below:
40
41. These are split ring resonators. This is the most common form that we can find
them in.
41
[11]
A diagram of the product of using split ring resonators as part of a meta-material, used to manipulate microwaves
42. Appendix B
Mr Savage - A Military Point of View
My interview with Mr Savage went extremely well. As a member of the Royal Air Force
(RAF) he knew exactly what I was looking for in relation to a military point of view into the
subject matter of invisibility.
I have already covered the aspects of various methods of actually cloaking vehicles and
potentially soldiers, but I have not really gone into much depth to how it has developed
good old fashioned espionage and camouflage.
Camouflage-
It goes without saying that camouflage has been a vital addition to the military whether it
be land, air or sea. It has saved the lives of many thousands of soldiers by disguising them
with the surrounding scenery. I do not think I have to go into too much detail, other than
the way camouflage works by being able to distort the human figure. However, this too has
its limitations. Other than being able to be detected by methods such as infrared detection,
on board cameras in surveillance aircraft, such as USAFs, can be used to recognise
outlines of human figures. As a result we now have digital camouflage which conceals
people from both a person’s eye and the cameras. This obviously has variations
depending on what the surrounding area is like.
Historic events-
There are countless historic events where invisibility could have been of use. Mr Savage
chose to briefly explain the potential effects that the technology could have had during the
Battle of Britain and the The Falklands War.
In 1940 during World War II we had the Battle of Britain. In brief this event was the result of
Hitler’s German Luftwaffe shifting focus from the British coastal defences to the mainland.
On August 24th the Luftwaffe tried to level RAF installations across the south of the
country. When this failed the Germans started the night blitzes of London and then other
cities. This also failed so the germans shifted their focus, again, on the coastal defences.
By now it was October. The Germans then gave up as they failed to overwhelm the British
and force a surrender. The Germans lost 2300 planes, whilst the British only lost 900. As
much as the RAF played such a considerable role in the victory, it would likely not have
been possible without the newly installed radar systems, which allowed the spitfires and
hurricanes to be scrambled into operation so quickly. Now one can imagine what might
have happened if the Germans had cloaking technology to counter the radar systems. This
is one of those things that puts a negative on the technology being used in a military
situation
42
43. On the other hand, the Falklands War in 1982 could have used the technology to save
lives. In particular the Battle for Goose Green, 2 Para was outnumbered 8 to 1. They may
have still won, but the loss of life and injuries could have been limited if they were not
detected.
Argentina United Kingdom
StrengthStrength
790 army
202 Air Force
10 coast guard personnel
690
Casualties and lossesCasualties and losses
47 killed
145 wounded
961 captured
17 killed
64 wounded
This was only a short meeting, but from this there is an immediate sense of controversy. At
the moment there is a lingering negative side to the use of invisibility in a military sense as
it can cause a huge loss of life. However, to counter this there is also the advantage of
saving lives by preventing people being detected by an enemy.
My interview with Mr Savage has helped a great deal with this side of the argument.
Transcript: Interview with Gary Clark-Savage of Lord Wandsworth College’s CCF. He
is the SSI.
We talked about The Falklands war and how, if invisibility had been available in 1982, the
Task force could have been moved the 8000 miles to the south atlantic without being
detected. The battle for Goose Green, when 2 Para found themselves outnumbered 8 to 1,
would still have been won but without such great loss of life for the Paras. Going further
back to 1940, one of the main reasons that the Battle of Britain was won, apart from the
extreme heroism of the few, was our use of RADAR, which enabled us to scramble our
squadrons as soon as we knew that the enemy was airborne. Now imagine how much
more effective we would have been had our aircraft been invisible.
He then also did some background research for me. It was about the Apache helicopter
flown by the AAC, it has been designed so that the only part that is visible is the top
mounted Radome, this is the eyes for the pilot allowing him to see all the targets on the
battlefield, the rest of the aircraft has been designed to be invisible to enemy forces. So as
you can see the concept of invisibility is being used in modern warfare.
He also recommended I look the Apache up on the internet as I would find a lot more
details that could help me, in particular USAFs spy planes and bombers such as the B-2
Spirit , these have special paint and panels that are angled to avoid RADAR, so although
these aren't invisible as such they are un detectable to RADAR.
43
44. Appendix C
Plasmonic Time Cloaking
Transcript: Interview with Martin McCall of Imperial College London. He is a
professor of optical engineering.
Below is a brief list of questions that I asked Martin McCall via Skype and the sort of
answers he replied with.
Question Answer
What military applications
could there be?
Conceal communications, protect data, censorship, data
protection
What commercial
applications could there
be?
Protecting online information, such as bank details and
social media, data protection
Could the technology have
any affect on censorship in
areas across the world?
As much as any online data could be affected. The
technology would allow data transfer and
communications to occur without censors knowing at all.
On the other hand it could also be used by the censors
to censor data without being seen to do so
Teleportation idea- can
people still move across the
system whilst it is in use
without being effected by or
affecting the end result?
In a way yes, if they walk through a gap that has been
created it would be possible, but this gap will only occur
if designed to be there in the first place
Will it always need to be in
a closed system?
Yes. Not the area being affected, but the entire system
that the technology is being used in
Briefly explain the details of
the “chicken crossing the
road” analogy
Once the chicken steps onto the road cars must
stop to let it pass, but as soon as it leaves the other
side the cars would accelerate to catch up with the
traffic ahead.To an observer farther down the road,
the stream of passing cars would display no
evidence of having slowed down. Therefore, the
chicken must be quick and move forwards and
laterally to compensate.
44
45. This has helped me understand this method more thoroughly. In doing so I will add this to
part of my conclusion as it helps with the social aspects of personal security and privacy
as it deals very specifically with data movement on the internet.
45
46. Appendix D
Fermat’s Principle and Snell’s Law
The following was taken directly from Wikibooks. The purpose of this is to help show what
is being explained (more simply) at the beginning of the Electromagnetic section. I will not
go into too much detail as it quite complicated, but should be understandable to those who
do maths. Also, I have taken it from Wikibooks as it is still correct.
46
48. Appendix E
Future Uses and Developments
As I have gone through my research it keeps occurring to me that a lot of the technology
has being initially designed for military purposes. This is nothing new as this is the case for
many high tech things, such as SONAR which was discussed in the introduction.
So before I come to my final conclusions I am going to discus the use of several types of
invisibility employed by military projects still in their development stages.
B2 Stealth Bomber successor-
The B2 stealth bomber is renowned for its stealth ability during both surveillance and
assault missions. This is due to its shape and radar deflecting abilities. This only works for
a short time as detection technology becomes better at the likely hands of Moore’s Law
doubling the number of transistors in a chip every two years. The successor is very much
similar to its predecessor. However, more involved materials and attention to
aerodynamics have been employed.
48
Exterior Material:
The old B2 had a radar cloaking material. This
usually consisted of using iron laced paint. The
problem with this is it made the plane heavier
than it needed to be therefore reducing fuel
efficiency. Also due to the high speed and
altitude this material
Shape:
The angled shape of the
wings not only help with
fluid dynamics, they also
cause radar signals to
deflect away from the
detector at the source of
the signals
Carbon Nano-tubes and cenospheres:
The paint of the old bomber used to
contain suspended iron particles in the
paint. This managed to absorb a lot of
incoming radar signals, however not all
were absorbed. A new development,
called cenospheres, sees hollow ceramic
spheres being used with a carbon nano-
tube coating. This absorbs a much wider
range of radar signal wavelengths,
which makes the aircraft extremely well
hidden to detection. Particles of silver
are also employed in the censpheres to
make the bomber more invisible to being
49. Stealth Submarines of the Future-
Similarly to the B2 Bomber, stealth submarines have been highly successful in both
surveillance and assault mission, until detection technology catches up with it.
49
50. Earthquake Proofing -
This follows on from acoustic cloaking in the sense that an earthquake is essentially a
series of waves of pressure caused by the Earth’s plates slipping. These waves are also
longitudinal.
The difference with this method of cloaking is that at the moment it is pure mathematical
theory. In my view, like many other mathematic and scientific theories, this too will come
into fruition.
In short it involves having parts of structures covered in pressurised rubber. By doing this
the pressure waves would appear to travel straight through the structure (similarly to a few
other the other methods described in this paper). This is because the waves are partially
absorbed by the highly pressurised rubber. These are then resonated at a different
frequency so that they are harmless. This prevents any vibrations that could interact with
the building materials of a structure and removing the issue of these then collapsing or
being damaged.
At the moment the researchers at the University of Manchester have based this on pre
stressed natural materials, but as we know with meta-materials, there will be some
materials, which could be several times better in practice. All that needs to be done is
discover them.
The uses of this method of cloaking if successfully developed further are huge. There are
many regions in the world, such as Asia, that are constantly experiencing the effects of
earthquakes. For example in 2008 an earthquake measuring 7.9 on the Richter Scale hit
China’s Sichuan Province. The effects of this were huge with a death toll of 51000 and
29000 people missing. The main cause of these were a result of building collapsing on
groups of people and to make things worse bridges connecting Sichuan to the rest of
China were in ruins meaning aid services could not come in and help.
From what I understand about this method, there would also be an area for use in
protecting buildings from other forces that could impact on them. The main thing that
comes to mind here are the events of 9/11. Although I am sure that the top part of the
building would almost definitely still be in ruin, if this technology was available throughout
the structure the rest of it would have had the pressure waves from the impacts of the
planes would have been removed from the equation.
50