Turbines in Invenergy's California Ridge wind farm dot the landscape of Pilot Township in western Vermilion County, north of Oakwood. Invenergy is considering additional wind farm development in Vermilion, but some residents opposed to wind energy are lobbying the Vermilion County Board for ordinance changes that could stifle additional development.
HOPE — An airplane that never takes off. A diesel truck idling. An interstate highway. Tennis shoes thumping in a dryer.
That's how several Vermilion County residents who live in the California Ridge wind project describe the various noises from wind turbines near their homes.
"Sometimes it sounds like a plane that never leaves ... other times like a diesel pickup or semi parked outside, and it idles all night. It depends on the wind direction," said Jeremy Lomax, a Vermilion County resident in the California Ridge wind project who, along with several other wind farm residents, is lobbying the Vermilion County Board to increase the distance a wind turbine can be built from the foundation of a house.
According to Lomax, Ted and Jessica Hartke, Jean and David Miles and Gina Isabelli — who all live within the project area — noise and shadow flicker from the turbines ranges from tolerable some days and nights when the wind isn't blowing to severe other days and nights and affects either their quality of life, health or both in various ways.
It's a familiar scenario playing out in other parts of the state, the nation, even the world — people living near wind turbines are voicing concerns about noise, shadow flicker, declining property values and other issues with this clean renewable energy source that the U.S. Department of Energy claims can provide 20 percent of the nation's electricity by 2030 and reduce projected emissions of carbon dioxide, the leading greenhouse gas, by 25 percent.
Just last week, the Iroquois County Board significantly increased its setback of 1,500 feet between turbines and houses of nonparticipating landowners — those without a wind turbine lease.
The new setback is now 12 times the rotor diameter, which would be about 3,936 feet for a 1.7-megawatt GE turbine with a rotor diameter of 328 feet.
Iroquois also approved noise limits on the turbines, along with a requirement that wind farm developers prove shadow flicker would not affect nonparticipating landowners who live within one mile of a turbine.
Residents within the California Ridge wind project voiced their concerns along with Iroquois County residents facing the same issues.
Iroquois County Board member Charles Alt voted for the changes to the county's wind ordinance. He said they were prompted by complaints from residents in the county, not only about noise from the turbines but also other issues, like road agreements with the wind companies.
Alt said there were also residents who argued against the changes, including landowners in the Wellington and Hoopeston area who have the possibility of leasing their land for a future wind farm development. Alt said that development may not happen now due to the changes.
"It seems like the controversy on wind energy is growing," he said, explaining that those with wind turbines on their property generally are okay with them, but others who live nearby are having issues. "We have to think about everybody in the county."
Alt said the effects of wind turbine noise and other issues is still a big question. But there have been residents from within all the area wind farms voicing concerns with noise and other issues. He said not all their stories are the same, but they are close, and he can't believe they are all making it up. He said the wind companies disagree with such claims, and board members have challenged them for proof.
"And I can't see why we should OK something that draws this much controversy from people who don't have anything to do with them," Alt said.
In Vermilion County, the board increased its setbacks two years ago from 1,000 feet to 1,200 feet. A proposal earlier this year from county board member Chuck Nesbitt to increase the setbacks even more, to 1,320 feet from a property line and 2,640 feet from the closest primary structure, did not gain support from an independent panel, which recommended no action to the county board.
But residents from the California Ridge project have not given up and continue to lobby Vermilion County for increased setbacks. Vermilion County officials argue that a 1,320 feet setback would eliminate such large sections of land from consideration for a wind turbine that it would effectively be a ban on future wind farm development.
Online for about a year now, the California Ridge wind project was developed by Chicago-based Invenergy and is the first wind farm in Vermilion and Champaign counties, stretching from western Vermilion, where there are 104 turbines, mostly in Pilot Township, to eastern Champaign, where there are 30 turbines.
Mostly rural, Pilot has a population of about 580 people, and many homes have wind turbines nearby. But more than a dozen residents there, including Lomax, the Hartkes, the Mileses and Isabelli, have put up vinyl signs on their properties with messages railing against the wind farm. One says "Welcome to wind turbine hell; home of the noisy turbines."
Earlier this year in response to wind turbine complaints, the Vermilion County Board organized a public hearing. Residents packed the Potomac Grade School gym. About half of the comments championed the construction jobs and income for local business from the wind farm in addition to rebuilt local roads and local property tax revenue it will generate for schools and other taxing bodies. But the other half of the comments came from the Hartkes, Lomaxes and other residents upset about noise, shadow flicker and other issues.
The controversy doesn't seem to be swirling in Champaign County, where the setback requirement is the same as in Vermilion: a minimum 1,200 feet from a wind turbine to the house of a nonparticipating landowner. John Hall, Champaign County planning and zoning director, said no one has lobbied Champaign County to increase its setback. And, he said, he is anticipating an application for a new wind farm development in northwestern Champaign County.
And more wind farms are a possibility in Vermilion, too.
Apex Clean Energy Inc., based in Virginia, already has a permit for its planned development in northwestern Vermilion County, called the Hoopeston wind project. Dahvi Wilson with Apex Clean Energy said the company is in the final stages of development for Hoopeston Wind, but there are still some key milestones that need to be met before the company determines a construction date and the number and size of turbines.
And Invenergy has plans for a phase two development of its California Ridge farm and does not have a permit yet. Alissa Krinsky, director of communications with Invenergy, said increased setbacks could have a real impact on those plans.
With more than 40 wind projects online, Illinois ranks fourth in the nation in the number of utility-scale wind turbines and total megawatts installed, according to the American Wind Energy Association. In 2012 alone, Illinois added more than 823 megawatts, ranking it fifth in the nation in the most new capacity added last year, but the percentage of Illinois' electricity provided by wind in 2012 was still under 4 percent, according to the association.
Vermilion County officials argue that in the absence of county-wide zoning, the county cannot legally deny a property owner from leasing his land, which would be the case if setbacks were increased to the point that wind farm development would not be possible.
Vermilion County Board Chairman Gary Weinard said the county has no zoning, no land-use policy and is not a home rule county, so if the county increases setbacks significantly, effectively banning wind farm development, then the county has, in effect, created a zoning ordinance and has taken property rights away from property owners and could be legally liable.
But Lomax, the Hartkes, Isabelli and Rankin-area residents Darrell and Kim Cambron have continued to attend Vermilion County Board meetings and publicly make their case, reiterating their complaints about noise and other issues, and calling for increased setbacks from property lines, not the foundations of their houses.
"To have a current setback of 1,200 feet, I guess they think we don't play outside, swim in the pool, ride our horses, work in our shops — all closer to the turbine than the foundation of our home," Isabelli said.
Jean and David Miles said there is a whining or humming noise when the turbines are repositioning. Sometimes there's a clunking or thumping from the rotating of the turbines, according to the Miles family and Isabelli, in addition to the constant whooshing from the blades turning. Lomax said the severity depends on the direction of the wind and the position of the turbines. David and Jean Miles said depending on the blade position and revolution speed, the turbines can sound like a jet engine. The Hartkes also compare the noise to an airplane in their backyard or diesel engines running.
The Miles family and Hartkes said the noise affects their sleep, and causes anxiety and headaches.
The Hartkes said they notified their local Invenergy representative and began reporting all noise events to the company, documenting the noise and weather conditions, hoping it would help Invenergy determine what conditions were causing the problem. The Hartkes said that within the first four months of California Ridge going operational, Invenergy shut down multiple turbines at their request on more than 50 occasions. The Hartkes said it was an improvement, but it still left the family sleep deprived and suffering from a number of health issues. Invenergy told them, the Hartkes said, to find contractors and get quotes for soundproofing their house. But the Hartkes did some research, and they said they learned it's not possible to soundproof against the type of "noise/infrasound produced by wind turbines." In May, the Hartkes said, Invenergy stopped turning off turbines at their request.
"The negative impact of this project on our family life, health and ability to live in our home and enjoy our property has been severe," the Hartkes said in a statement. Ted Hartke said the family has retained an attorney.
In 2009, Dr. Nina Pierpont's book, "Wind Turbine Syndrome: A Report on a Natural Experiment," was released. According to Pierpont, wind turbines emit infrasound — sound waves with frequencies below the lower limit of human hearing — and low-frequency noises that cause various symptoms in people, which she called Wind Turbine Syndrome.
There is conflicting research and medical opinions on whether wind turbines cause health issues, and in 2009, the American Wind Energy Association, the wind industry's main trade group, and the Canadian Wind Energy Association brought together a panel of professionals from a variety of disciplines to review current literature on the perceived health effects of wind turbines and concluded that subaudible, low-frequency sound and infrasound from wind turbines do not present a risk to human health.
According to the wind energy association, noise generated by wind turbines are in the range of 35 to 45 decibels at a distance of 1,148 feet, almost the same distance as Vermilion County's setback of 1,200 feet. According to the U.S. Department of Health and Human Services, a noise level of 40 decibels is equal to a running stream or refrigerator humming. The Hartkes, Miles family and Isabelli said the noise from the turbines is well beyond the noise their refrigerators make.
Krinsky said an independent noise study is in the process of being completed at the California Ridge wind project, but did not elaborate on the study.
"We are committed to compliance with all applicable laws and regulations and look forward to continuing to produce clean, renewable energy as a key economic participant in the community," she said.
Vermilion County is not the only county, and Illinois is not the only state, where Invenergy and other wind companies are clashing with residents over noise and other complaints. Just last month, a man in Oregon filed a $5 million lawsuit against Invenergy over noisy wind turbines claiming problems with sleep, headaches and other health issues.
Apparently the Iroquois County board cares about the health, safety, and quality of life of their residents. Money comes in second instead of first. Vermilion County would do well to look to the north!
Wind turbine setbacks of 350-400 meters are completely safe in all but the tiniest fraction of cases. The World Health Organization sets 50 dB of regular and prolonged night time noise that cannot be mitigated via closed windows and white noise generators as the level at which sleep loss becomes a concern. 40 dB, the level that Ontario's Regulation 359/09 regulations make the norm 99% of the time for 99% of properties, is a good cautionary level. 35 dB, the level set in a couple of Australian states, is even more conservative.
Mr Barnard spinning again. The last place anyone should go is to your site for information about adverse effects of wind. Try talking to some people who are affected instead of callously dismissing them. The only people who say there is no problem are those in the wind industry and who stand to gain massively from the subsidy payments. When there is finally justified recognition of people's suffering there will be lawsuits - I hope those who are to blame are well insured.
As for property prices get out of your office and go and knock on doors and ask people what is happening to them. Estate agents know it is a problem and so do those who try and sell. You may have been a master spinner once but I'm afraid the game is up. Wind is over - it's just a question of when.
How close are the turbines you live near, how high and how many?
I mean you must live really close to be such an authority on their effects.
Mike Barnard lives in Singapore. He's not even in North America, yet he feels compelled to tell people who live on the other side of the world, that the negative health effects they're feeling are all in their heads.
When you have lived next door (within half a mile) of an industrial electrical generation wind facility for at least 6 months Mike Barnard, then you can comment. Until that time, you should take your biased wind opinions and shove them. I've seen your comments all over the world promoting the greatness of wind. Thankfully everywhere you go you are trounced by the people who actually live next to these industrial blights. Someone who has that much time on his hands clearly is making money in some way off of this scam. I don't believe for a second you defend these things from the goodness of your heart -- if you have one.
Wind Turbines Harm Human Health Due To Sleep Deprivation: Mike, it is quite interesting that documentation from the World Health Organization was presented as evidence at a public hearing in Iroquois County before the revised wind farm ordinance became law stating that nighttime noise levels should be 30 db, and likely less than that if the noise is very low frequency. Also studies were entered as evidence from Europe and Australia showing the damage to ones health from sleep deprivation caused by wind farm noise. And nothing in the studies showed that it was permissible to create more noise (white noise) to drowned out the low frequency wind turbine noise. But testimony was presented at the public hearing by the wind farm advocates that was later decisively proven false. Anyone can state claims that are untrue and this public forum gives you that opportunity. I suppose you believe if you state false information often enough people will accept it as truth.
Since another myth has reared it's ugly head, another debunking is required.
Regarding torture, I was waterboarded once as part of a training exercise with the Canadian military. I find it offensive that someone would compare wind turbine noise to torture. Perhaps I'm oversensitive. Perhaps all of the people who really have been tortured as opposed to merely annoyed are just too sensitive too.
You're becoming boring now Mr Barnard.
Once again how close do you live to industrial turbines, how many and how big?
If you are such an authority on the effects of living with them tell us about your situation......otherwise I suggest you keep your offensive comments out of these forums. People ARE suffering whether you like it or not and to suggest otherwise is insensitive, callous and arrogant. You are behaving like many in the wind industry. People don't matter, the environment doesn't matter, wildlife doesn't matter - as long as they can get their grubby hands on the extortionate subsidies on offer for an unreliable and inefficient energy source. More and more are seeing it for what is and are not prepared to put up with the scam any more.
I am puzzled why you are so pro wind if you are not, as you imply, involved in the industry. Perhaps you just like looking at turbines! In which case offer to buy someone's property, that is near turbines, for market value so they can move away and live in peace and you can live happily ever after with your spinning friends - or not as they are the majority of the time.
I find it interesting that if there is no noise pollution low frequency or otherwise -- why--- people living in Ontario Canada, Australia, and parts of the US who are neighbors to wind projects have chosen to abandon their homes.
I find it interesting that property owners - who may "lease their land" are opposed to the increased setback. Have they read the leases - The wind company will put the IWT where they want it -- sure the sales guy will tell you we'll put it over there by your neighbor - but don't be too sure; and do you wonder why it includes stuff about noise and shadow flicker etc. and you can't say a thing about it. In our area there were people getting lease options that never would end up with a wind turbine & lease money - they end up being the "neighbor", but fought hard during the controversy.
I don't know what annoys me most about the wind industry - the affect on rural areas, their asking for the 30 year kill permit of eagles or the fact my taxpayer money props up the inefficiency of this industry.
Regarding comments being posted by Mike Barnard -- people need to be aware that Mr. Barnard is apparently working to carpet the USA and Canada with 450 - 500 foot-tall giant "Bird Cuisinarts" from his post in Singapore - where he works for IBM.
I have directly asked Mr. Barnard if he would buy and move his family into a home within a sprawling industrial wind factory - with its' 500-foot-tall machines and their 7-ton blades spinning overhead, only hundreds of feet from that home. He has refused to answer the question. Pretty much says it all, don't you think?!?
Wind energy is a horrendous waste of time and money. The federal tax subsidies alone cost America (that's you, me and future generations who bear this debt load) TWELVE BILLION DOLLARS A YEAR. That doesn't include the cost of new high voltage transmission lines needed to deal with the big surges from fickle wind and the need to transmit it to get it to population centers. For this princely sum, which mostly goes to big corporations like GE and foreign investors, America now gets 4% of its electricity from wind. That equals about 1% of our total energy portfolio. Wind doesn't reduce our usage of oil at all, because we generate electricity with gas, coal and nuclear, not oil. It would be FAR cheaper, greener, cleaner and nicer to residents to simply install natural gas-fired generators. We are sitting on an ocean of natural gas, and such generators would take up a tiny footprint, require virtually no extra high power lines, do not chop up raptors and kill endangered bats, and let residents sleep at night. Wind is so unreliable that we will have to build the gas generators anyway, so wind is a TOTAL waste. The government should not be spending money we don't have on a technology that doesn't work and in fact destroys the habitat for both humans and wildlife.
Obviously, Mr. Barnard did not actually study the latest LBNL report, or he would have noticed that table 7 proves there is a 28% value loss within 1 mile of turbines.
1,198 sales in that range compared to a larger group of sales 3-10 miles from turbines is pretty hard to deny, if one is concerned about the truth.
I have lived with wind turbines around my house for the last 3 years. There are 7 in our section and 3 1500 feet from the foundation of my house. We estimate that we have lost $100,000 in the value of our house. My husband was transferred out of state and I did not follow him because we would now owe more on our house than it was worth. I no longer spend time outside. I have hired people to mow our 2 1/2 acres. I did not bother to plant flowers this year. The turbines are noisy and create shadow flicker. When they first went up, we thought we could live with them because we had to. We tried sitting on our patio for a picnic and our guests ended up leaving because we were shouting at each other over the sound of the turbines. We decorated our Christmas tree at three in the afternoon one day to the pulsing strobe light created by the turbine blades. This whole process has been disillusioning and depressing. I thought local officials cared about their citizens, they don't. They care about THE MONEY. I thought our neighbors cared about each other, they don't. They care about THE MONEY. Wind turbines are an expensive hoax. They are the equivalent of a "Vegan in Leather Shoes". They are socially unjust, creating electricity that is 5 times the cost of convential forms of electric generation. What they do generate is MONEY for WIND COMPANIES that is paid for by taxpayers. Walk into Whole Foods and spend your Whole Paycheck so you can feel good about yourself. Wind Farms are just the same. They are a hoax, exploited by people who are out to make a buck, so FEEL GOOD looking at a Wind Farm. Fool yourself into thinking your are saving the planet, be a "Vegan in Leather Shoes", enjoy the land of the PC.... until someday, you are me.
We have lived in a wind complex now since 2010. If we only knew how much it would affect our daily lives we probably would have sold our home and farm prior to it being built. Unless you live in one and experience it on a daily basis, you have no right to make any conclusions or assumptions based on inexperience. What we experience daily, changes daily, and no two days are ever the same. Not only do we have shadow flicker almost 5 months out of the year but the noise is so bad in our home with the house closed up at that you cannot sleep at night. Even when you are sleeping at night, you can be awaken by the moon crossing in front of the blades for what is know as moon flicker. I am sure there are not too many people that would enjoy be awaken by this as nothing keeps it out of the rooms just like shadow flicker. We live with "jet rumbling" noises inside and outside our home daily. All the studies in the world will not conclude the same results as if the turbine was put up. We were told by the company from their study that we only had 20 hours of flicker a year. Well, with video, I was able to show them we had that amount of flicker in just two weeks in the Fall. So, they redid their study, came back with 43 hours of flicker. To make a long story short after being able to prove them wrong, they said I could have 80+ hours a year. Because the county did nothing to protect the residents prior them the wind turbines being put up, we have no recourse against the developer. So sad. We have health issues within our home that the noise and flicker have made worse. Does the county want to start being responsable for health bills? You do not know what health issues people already have let alone know what they will experience down the road from these complexes. These companies are only about money and not helping the residents that live in them. It is your responsibility as a board to protect the health, safety and well being of the residents in your county. They also stated they do not throw ice. But when they were asked to come out due to ice throws, then all of a sudden signs went up. So, now when you travel down the county road, you have to be aware of flying ice? In my opinion, even if the set back was a mile, it would not be far enough away. We have shadow flicker from two turbines that are just over a mile from our home. We currently get flicker from 6 turbines and we have 86 around our home.


text:
\section{Introduction}

Boundary conditions are fundamental building blocks in problems of mathematical physics, including electromagnetics. For given sources in a given space, Maxwell equations require additional constraints on the boundaries of the domain that force uniqueness and existence of the solution. These conditions between the electric and magnetic fields and fluxes at the boundary can in principle be rather general \cite{GBC}; however, in finite real-world problems, the boundaries often are characterized by approximate surface conditions \cite{Hoppe,Senior}.

Examples of such boundary conditions are perfectly electrically conducting (PEC) surfaces, artificial magnetic conductor and high-impedance surfaces \cite{Sievenpiper1,Sievenpiper2}, and anisotropic soft-and-hard surfaces \cite{Kildal}. But  the possibilities to engineer the electromagnetic response of complex boundaries have a great variety. Thin two-dimensional structures which are capable of modifying the reflection and transmission of the incident wave in multiple manners have been recently designed and fabricated, and the label {\it metasurface} has appeared in the electromagnetics literature to cover such advanced surfaces \cite{Holloway}. Generalizing the traditional frequency-selective surfaces and transmit/reflect arrays \cite{Munk,Maci}, these metasurfaces can control the amplitude and phase of the wavefront in various ways \cite{Grbic}, the effects can be nonreciprocal \cite{Caloz}, non-local, or even active \cite{Kusti}. In the optical range of the electromagnetic spectrum, these developments within the metasurface paradigm are known by labels like planar photonics \cite{Kildishev} and flat optics \cite{Yu}.

It is the purpose of the present paper to investigate still uncharted classes of complex electromagnetic boundaries and surfaces. The treatment is purely theoretical, concentrating on certain aspects of electromagnetic boundary conditions beyond those in the book \cite{GBC} by these authors. Theoretical predictions of unconventional boundaries, like, as example,  the perfect electromagnetic conductor PEMC \cite{PEMC} have found experimental realizations \cite{Shahvarpour}, and found practical applications in the manipulation of properties of electromagnetic waves.  The novel class of boundary conditions introduced in the present paper may be of use in the future design of  novel metaboundaries.

\section{General Boundary Conditions}

A typical electromagnetic problem involves solving the Maxwell equations in a region bounded by a surface with imposed boundary conditions. The most general (linear and local) boundary conditions (GBC) can be shown to take the form \cite{GBC,PIER2016,AP2017},
\ea \alpha_1 c\#n\.\#B + \frac{\beta_1}{\epsilon_o}\#n\.\#D + \#a_{1t}\.\#E + \eta_o\#b_{1t}\.\#H &=& 0, \l{BC1}\\
\alpha_2 c\#n\.\#B + \frac{\beta_2}{\epsilon_o}\#n\.\#D + \#a_{2t}\.\#E + \eta_o\#b_{2t}\.\#H &=& 0, \l{BC2}\fa
with
\e c = 1/\sqrt{\mu_o\epsilon_o},\ \ \eta_o=\sqrt{\eta_o/\epsilon_o}. \f
The unit vector $\#n$ is normal, and the vectors $\#a_{1t}\cdots \#b_{2t}$ are tangential, to the boundary surface. They are dimensionless and so are the scalars $\alpha_1,\alpha_2,\beta_1,\beta_2$. Because one of the scalar coefficients in each equation can be chosen, the number of free parameters defining the general boundary conditions is 10. Realization of a boundary defined by general conditions of the form \r{BC1} and \r{BC2}, in terms of the interface of a bianisotropic medium, has been discussed in \cite{GBC}, Section 5.14.

In the present paper we assume for simplicity that the boundary surface $\#n\.\#r=0$ is planar, as defined by a constant unit vector $\#n$. Also, we assume that the medium is isotropic with parameters $\epsilon_o$ and $\mu_o$, in which case the boundary conditions \r{BC1} and \r{BC2} can be rewritten in the simpler equivalent form \cite{GBC,AP2017}
\ea \#a_1\.\#E + \#b_1\.\eta_o\#H &=&0, \l{bc1} \\
     \#a_2\.\#E + \#b_2\.\eta_o\#H &=& 0, \l{bc2} \fa
where the four vectors $\#a_1\cdots\#b_2$ may contain components normal and tangential to the boundary,
\e \#a_i= a_{in}\#n + \#a_{it},\ \ \ \#b_i= b_{in}\#n + \#b_{it},\ \ \ i=1,2. \f
It is assumed that \r{bc1} and \r{bc2} represent two linearly independent conditions, because, otherwise, they do not define a unique boundary-value problem.

\subsection{Special Cases}

The boundary conditions \r{bc1} and \r{bc2} contain a lot of familiar special cases starting from the perfect electric conductor (PEC), defined by
\e \#a_{1t}\.\#E = \#a_{2t}\.\#E=0,\ \ \ \ \#a_{1t}\times\#a_{2t}\not=0, \l{PEC} \f
and the perfect magnetic conductor (PMC), defined by
\e \#b_{1t}\.\#H=\#b_{2t}\.\#H=0,\ \ \ \ \#b_{1t}\times\#b_{2t}\not=0. \l{PMC}\f
A straightforward generalization of these two conditions is that of the perfect electromagnetic conductor (PEMC) \cite{PEMC,AP05,GPEMC},
\e \#b_{1t}\.(\#H + M\#E)= \#b_{2t}\.(\#H+ M\#E)=0, \l{PEMC} \f
or, in vector form,
\e \#n\times(\#H+ M\#E)=0, \l{PEMC1}\f
where $M$ is the PEMC admittance.

As other special cases of the GBC conditions \r{BC1} and \r{BC2} we may add
\begin{itemize}
\item The DB conditions \cite{DB},
\e \#n\.\#D=\epsilon_o\#n\.\#E=0,\ \ \ \ \#n\.\#B=\mu_o\#n\.\#H=0. \l{DB} \f
\item The soft-and-hard (SH) conditions \cite{SHS1},
\e \#a_t\.\#E=0,\ \ \ \ \#a_t\.\#H=0. \l{SH} \f
\item The generalized soft-and-hard (GSH) conditions \cite{GSH},
\e \#a_t\.\#E=0,\ \ \ \ \#b_t\.\#H=0. \l{GSH}\f
\item In the EH boundary conditions \cite{GBC},
\e \#a\.\#E=0,\ \ \ \ \#b\.\#H=0, \l{EH} \f
the vectors $\#a$ and $\#b$ are not restricted, whence \r{EH} are more general than  \r{GSH}.
\item The soft-and hard/DB (SHDB) conditions \cite{SHDB},
\e \#a_t\.\#E+ \alpha \#n\.\eta_o\#H =0,\ \ \ \ \alpha \#n\.\#E - \#a_t\.\eta_o\#H=0, \l{SHDB}\f
contain SH and DB conditions \r{SH}, \r{DB} as special cases.
\item The generalized soft-and-hard/DB (GSHDB) conditions \cite{GSHDB},
\e \#a_t\.\#E+  \alpha \#n\.\eta_o\#H =0,\ \ \ \ \beta \#n\.\#E +\#b_t\.\eta_o\#H=0, \l{GSHDB}\f
represent a further generalization of \r{SHDB}.
\item The impedance conditions
\e \#a_{1t}\.\#E + \#b_{1t}\.\eta_o\#H =0,\ \ \ \ \#a_{2t}\.\#E+ \#b_{2t}\.\eta_o\#H=0. \l{imp}\f
\end{itemize}

\subsection{Purpose of this Study}

In a  previous study \cite{GSHDB}, it has been shown that a boundary defined by the GSHDB conditions \r{GSHDB} has the following property: any plane wave incident from an isotropic half space to the boundary can be decomposed in two eigen plane waves, one of which is reflected as from the PEC boundary, and, the other one, as from the PMC boundary. Since the GSHDB conditions \r{GSHDB} contain \r{DB} -- \r{GSH} and \r{SHDB} as special cases, the same property is shared by all the corresponding boundaries. The converse property was shown in \cite{PIER2016,GBC}: if the PEC/PMC decomposition property is required to be valid for the eigenwaves associated to the GBC conditions \r{bc1}, \r{bc2} (reflection coefficient either $+1$ or $-1$), the boundary must actually satisfy the GSHDB conditions \r{GSHDB}. Thus, the generalized soft-and-hard/DB boundary is the most general boundary with the PEC/PMC equivalence property.

It is the purpose of the present paper to extend this theory by requiring that the reflection coefficients $R_+,R_-$ corresponding to the two eigen plane waves satisfy $R_\pm=\pm R$, where $R$ is independent of the angle of incidence ($\#k$ vector of the incident wave). For the special case of the GSHDB boundary we have $R=1$.

\section{Reflection of Eigen Plane Waves}

Let us consider time-harmonic plane waves satisfying the plane-wave equations
\ea \#k\times\#E &=& k_o\eta_o\#H, \l{kxE}\\
\#k\times\eta_o\#H &=& -k_o\#E. \l{kxH}\fa
and denote fields incident to, and reflecting from, the boundary surface, respectively by
\e \#E^i(\#r)= \#E^i\exp(-j\#k^i\.\#r),\ \ \ \ \#E^r(\#r)= \#E^r\exp(-j\#k^r\.\#r), \f
with (see Figure \ref{fig:geometria})
\e \#k^i = \#k_t -k_n\#n,\ \ \ \ \#k^r=\#k_t + k_n\#n,\f
\e \#k^i\.\#k^i=\#k^r\.\#k^r=k_o^2=\omega^2\mu_o\epsilon_o. \f

\begin{figure}[htbp]
	\centering
	\begin{tikzpicture}[scale=1.0]
		\draw[thick,dashed][->] (0,0) -- (0,2);
		\draw[green!50!black, line width=3pt] (-4,0) -- (5,0);
		\node[black] at (4.3,.3) {\bf{GBC}};
		\node at (-1,1.1) {$\mathbf{k}^i$};
		\node at (1,1.1) {$\mathbf{k}^r$};
		\node at (-1,-.4) {$\mathbf{k}_t$};
		\node at (1,-.4) {$\mathbf{k}_t$};
		\node at (-2.8,.7) {$k_n$};
		\node at (2.8,.7) {$k_n$};
		\node[red!30!black] at (.4,2) {$\mathbf{n}$};
		\draw[dashed] (-2.5,.01)--(-2.5,1.65);
		\draw[dashed] (2.5,.01)--(2.5,1.6);
		\draw [->] (.01,-.1)--(2.5,-.1);
		\draw [->] (-2.5,-.1)--(-.01,-.1);
		\draw[blue!50!black,thick] [->] (-2.5,1.7)--(-.05,7/70);
		\draw[blue!50!black,thick] [->] (.05,7/70)--(2.5,1.7);
	\end{tikzpicture}
	\caption{Plane wave incident on a surface characterized by a general boundary condition GBC. The incident and reflected wave vectors $\mathbf{k}^i,\mathbf{k}^r$ are decomposed into their tangential and normal components.}
	\label{fig:geometria}
\end{figure}

Eigen plane waves are defined by requiring that the tangential components of the electric fields at the boundary satisfy a relation of the form
\e \#E_t^r=R\#E_t^i, \l{ERE}\f
i.e., that the incident and reflected fields tangential to the boundary have similar polarization and are related by of a scalar factor $R$, the reflection coefficient.

Polarizations and reflection coefficients of the eigen plane waves depend on the vector parameters of the GBC boundary conditions \r{bc1}, \r{bc2}. Applying a previous analysis \cite{GBC,PIER2016}, the reflection coefficient can be shown to satisfy the condition
\e \#n\.((1+R)\#a_{1t}' - (1-R)\#b_{1t}'\.\=J_t)\times((1+R)\#a_{2t}' -(1-R)\#b_{2t}'\.\=J_t)=0, \l{R+}\f
where we denote
\ea \#a_{1t}'&=&\#a_{1t}+ (b_{1n}/k_o)\#n\times\#k_t, \l{a1'} \\
\#a_{2t}'&=&\#a_{2t}+ (b_{2n}/k_o)\#n\times\#k_t, \\
\#b_{1t}'&=&\#b_{1t}- (a_{1n}/k_o)\#n\times\#k_t, \\
\#b_{2t}'&=&\#b_{2t}- (a_{2n}/k_o)\#n\times\#k_t, \l{b2'}\fa
and
\e \=J_t = \frac{1}{k_ok_n}((\#n\times\#k)\#k_t + k_n^2\#n\times\=I_t). \l{Jt}\f
Here we assume $k_n\not=0$, which excludes the possibility of lateral eigenwaves, $\#k=\#k_t$. The dyadic $\=J_t$ is known to satisfy the properties \cite{GBC}
\e \=J_t\.\=J_t = -\=I_t = -\=I + \#n\#n, \f
and
\e (\#c_t\.\=J_t)\times(\#d_t\.\=J_t) = \#c_t\times\#d_t, \f
valid for any two vectors $\#c_t,\#d_t$ tangential to the boundary. The rule
\e k_ok_n\#b'_{it}\.\=J_t = (\#b_{it}\.\#n\times\#k- k_oa_{in})\#k_t -k_n^2\#n\times\#b_{it}, \l{bitJt}\f
for $i=1,2$, appears useful in the subsequent analysis.

In the general case, the solutions $R$ of the equation \r{R+} depend on the boundary parameters $\#a_1\cdots\#b_2$ and the tangential component of the wave vector, $\#k_t$.  Requiring that the solutions $R$ be independent of the wave vector sets conditions for the boundary parameter vectors. For example, for the GSHDB boundary, \r{R+} has the solutions $R=+1$ and $R=-1$ for any $\#k_t$ \cite{PIER2016}.

Equation \r{R+} is of the quadratic form
\e C_2R^2+ C_1 R + C_0 =0, \l{CR}\f
with
\ea C_2 &=& \#n\.(\#a_{1t}'+\#b_{1t}'\.\=J_t)\times(\#a_{2t}'+ \#b_{2t}'\.\=J_t) , \l{CC2}\\
C_1 &=& \#n\.(\#a_{1t}'+\#b_{1t}'\.\=J_t)\times(\#a_{2t}'- \#b_{2t}'\.\=J_t) + \nonumber\\
&+&\#n\.(\#a_{1t}'-\#b_{1t}'\.\=J_t)\times(\#a_{2t}'+ \#b_{2t}'\.\=J_t), \l{CC1}\\
C_0 &=& \#n\.(\#a_{1t}'-\#b_{1t}'\.\=J_t)\times(\#a_{2t}'- \#b_{2t}'\.\=J_t). \l{CC0}\fa
For $C_2\not=0$ the two solutions are obtained from
\e R_\pm = -\frac{C_1}{2C_2}\pm \sqrt{\frac{C_1^2}{4C_2^2}- \frac{C_0}{C_2}}. \l{Rpm}\f

The coefficients $C_0$ and $C_2$ are closely related. From \r{CC2} and \r{CC0} we can write
\e k_ok_nC_0 = A-B,\ \ \ \ k_ok_nC_2=A+B, \f
with
\ea A &=& k_ok_n\#n\.(\#a_{1t}'\times\#a_{2t}'+ \#b_{1t}'\times\#b_{2t}'), \l{A}\\
    B &=&  k_ok_n\#n\.(\#a_{1t}'\times(\#b_{2t}'\.\=J_t)- \#a_{2t}'\times(\#b_{1t}'\.\=J_t)). \l{B}\fa
Expanding these expressions as shown in the Appendix, the three coefficients take the form
\ea C_0 &=& \frac{-1}{k_ok_n}(k_o(\#a_1\times\#a_2 + \#b_1\times\#b_2)\.\#k^i+(\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I-\#k^i\#k^i)), \l{C0'}\\
   C_1 &=& 2\#n\.(\#a_1\times\#a_2 - \#b_1\times\#b_2) -\frac{2}{k_o}(\#a_1\times\#b_2 + \#b_1\times\#a_2)\.(\#n\times\#k_t),  \l{koC1}\\
    C_2 &=&  \frac{1}{k_ok_n}(k_o(\#a_1\times\#a_2 + \#b_1\times\#b_2)\.\#k^r+(\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I-\#k^r\#k^r)). \l{C2'} \fa
As a special case, for the GSHDB boundary \r{GSHDB}, defined by $a_{1n}=b_{2n}=0$ and $\#a_{2t}=\#b_{1t}=0$, we have $C_1=0$ and $C_0=-C_2$, whence, from \r{Rpm}, $R_\pm=\pm 1$.

Let us now concentrate on finding the boundary conditions corresponding to the relation $R_+=-R_-$.

\section{Boundaries Defined by $R_+=-R_-$}

From \r{Rpm} we find that $R_+=-R_-$ is equivalent to the condition
\e C_1=0, \l{C10}\f
without any restriction on $C_0$ and $C_2$. From \r{koC1} we obtain the corresponding relation between the vectors $\#a_1\cdots\#b_2$,
\e  k_o\#n\.(\#a_1\times\#a_2 - \#b_1\times\#b_2) +(\#n\times(\#a_1\times\#b_2 + \#b_1\times\#a_2))\.\#k_t =0. \f
Requiring this to be valid for any angle of incidence, i.e., for any tangential vector $\#k_t$, leads to the two conditions
\ea \#n\.(\#a_1\times\#a_2 - \#b_1\times\#b_2)&=&0, \l{cond1a} \\
 \#n\times(\#a_1\times\#b_2- \#a_2\times\#b_1)&=&0. \l{cond2a}  \fa
They are respectively equivalent to
 \ea \#a_{1t}\times\#a_{2t}-\#b_{1t}\times\#b_{2t} &=&0, \l{cond1}\\
 b_{2n}\#a_{1t} - b_{1n}\#a_{2t} + a_{2n}\#b_{1t}- a_{1n}\#b_{2t}&=&0. \l{cond2}\fa
For the GSHDB boundary \r{GSHDB},  these are identically satisfied.

With \r{C10}, the condition \r{CR} becomes
\e C_2R^2+C_0=0, \l{R2}\f
whence
\e R_\pm = \pm\sqrt{-C_0/C_2}. \f

Requiring that the solution $R$ of \r{R2} be independent of $\#k_t$, \r{R2} can be split in the following set of equations relating the boundary parameters $\#a_1\cdots\#b_2$ and the reflection coefficient $R$:
\ea (R^2-1)(\#a_1\times\#a_2 + \#b_1\times\#b_2)\times\#n &=&0, \l{odd1x}\\
 (R^2+1)\#n\.(\#b_1\#a_2+ \#a_2\#b_1-\#b_2\#a_1-\#a_1\#b_2)\times\#n &=& 0, \l{odd2x}\\
 (R^2+1)\#n\.(\#a_1\times\#a_2)&=& 0, \l{even2x}\\
 (R^2-1)(b_{1n}a_{2n}-b_{2n}a_{1n})&=&0, \l{even32x}\\
 (R^2-1)(\#b_{1t}\#a_{2t} + \#a_{2t}\#b_{1t}-\#b_{2t}\#a_{1t}- \#a_{1t}\#b_{2t})&=&0. \l{even31'dx}\fa
Derivation of this set of equations is given in the Appendix.

Now we can separate three possible cases:
\begin{itemize}
\item $R^2=1$
\item $R^2=-1$
\item $R^2\not=1$ and $R^2\not=-1$.
\end{itemize}

In the last case, all expressions in \r{odd1x} -- \r{even31'dx} multiplying $(R^2-1)$ or $(R^2+1)$ must vanish. Actually, this corresponds to setting $C_0=C_1=C_2=0$ in \r{CR}, which means that $R$ may have any value. Ignoring this, we have only two possibilities, either $R^2=1$, or $R^2=-1$. Each of these defines a class of boundaries through certain conditions for the vector parameters $\#a_1\cdots\#b_2$ arising from  \r{odd1x} -- \r{even31'dx}. Since it has been previously shown \cite{GBC,PIER2016} that the case $R^2=1$ corresponds to the class of GSHDB boundaries defined by \r{GSHDB}, let us concentrate on the case $R^2=-1$, which yields a second possibility for the class of boundaries satisfying $R_+=-R_-$. In this case, the reflection coefficients for the two eigenwaves are $+j$ and $-j$.

\section{Boundaries Defined by $R_+=-R_-=j$}

From \r{odd1x}, \r{even32x} and \r{even31'dx} we obtain conditions for the boundary corresponding to the case $R^2=-1$:
\ea a_{1n}\#a_{2t} -a_{2n}\#a_{1t} + b_{1n}\#b_{2t}- b_{2n}\#b_{1t} &=&0, \l{odd1x'}\\
 b_{1n}a_{2n}-b_{2n}a_{1n}&=&0. \l{even32x'}\\
\#b_{1t}\#a_{2t} + \#a_{2t}\#b_{1t}-\#b_{2t}\#a_{1t}- \#a_{1t}\#b_{2t}&=&0, \l{even31'dx'}\fa
which, together with \r{cond1} and \r{cond2}, define the present class of boundaries.

It has been shown that the case $R^2=1$ corresponds to the class of GSHDB boundaries, for which the vector parameters satisfy $\#n\.\#a_1\times\#a_2=0$ and $\#n\.\#b_1\times\#b_2=0$, as is seen from \r{GSHDB}. Similarly, from \r{cond1} and \r{even2x} we can conclude that the case $R^2=-1$ corresponds to $\#n\.\#a_1\times\#a_2\not=0$ and $\#n\.\#b_1\times\#b_2\not=0$. Because of this, we can assume that the tangential vectors $\#b_{1t},\#b_{2t}$ and $\#a_{1t},\#a_{2t}$ form two linearly independent pairs, whence we can expand
\e \left(\begin{array}{c} \#b_{1t}\\ \#b_{2t}\end{array}\right) = {\cal B} \left(\begin{array}{c} \#a_{1t} \\ \#a_{2t}\end{array}\right), \l{b1tb2t}\f
for some matrix
\e {\cal B} = \left(\begin{array}{cc} B_{11} & B_{12}\\ B_{21} & B_{22}\end{array}\right), \f
defined by four scalars $B_{11}\cdots B_{22}$. The condition \r{cond1} requires
\e {\rm det}{\cal B} = B_{11}B_{22}- B_{12}B_{21}=1. \l{detB}\f
From \r{cond2} we obtain a relation between the normal components of the four vectors as
\e \left(\begin{array}{c} b_{1n}\\ b_{2n}\end{array}\right) = -\left(\begin{array}{cc} B_{22} & -B_{12}\\ -B_{21} & B_{11}\end{array}\right) \left(\begin{array}{c} a_{1n}\\ a_{2n}\end{array}\right) = -{\cal B}^{-1}\left(\begin{array}{c} a_{1n}\\ a_{2n}\end{array}\right). \l{b1nb2n}\f
Combining these, we can write
\ea \#b_1 &=& B_{11}\#a_1 + B_{12}\#a_2 -{\rm tr }{\cal B}\ a_{1n}\#n, \l{b11}\\
\#b_2 &=& B_{21}\#a_1 + B_{22}\#a_2 - {\rm tr }{\cal B}\ a_{2n}\#n, \l{b22}\fa
with
\e {\rm tr }{\cal B} = B_{11}+ B_{22}. \f
Applying \r{b11} and \r{b22},  the boundary conditions \r{bc1} and \r{bc2} take the form
\ea \#a_1\.\#E  + (B_{11}\#a_1 + B_{12}\#a_2- {\rm tr }{\cal B}\ a_{1n}\#n)\.\eta_o\#H &=& 0, \l{a1E}\\
\#a_2\.\#E + (B_{21}\#a_1 + B_{22}\#a_2 - {\rm tr }{\cal B}\ a_{2n}\#n)\.\eta_o\#H &=& 0, \l{a2E}\fa
for some vectors $\#a_1,\#a_2$ and a matrix ${\cal B}$ restricted by \r{detB}.

Applying \r{b1tb2t}, \r{even31'dx'} becomes
\e -2B_{21}\#a_{1t}\#a_{1t}+ (B_{11}-B_{22})(\#a_{1t}\#a_{2t}+ \#a_{2t}\#a_{1t}) + 2B_{12}\#a_{2t}\#a_{2t}=0, \f
which for $\#a_{1t}\times\#a_{2t}\not=0$ yields
\e B_{12}=B_{21}=0,\ \ \ B_{11}=B_{22}=B, \ \ \ {\rm tr }{\cal B}=2B,\f
for some scalar $B$. From \r{detB} we obtain
\e {\rm det}{\cal B}= B_{11}B_{22}-B_{12}B_{21} = B^2=1. \f
with two possible solutions $B=+1$ and $B=-1$.

From \r{b11} and \r{b22} we obtain the corresponding possibilities,
\e \#b_1=\#a_{1t}-\#n a_{1n},\ \ \ \#b_2 = \#a_{2t}- \#n a_{2n}, \l{b12B+}\f
and
\e \#b_1=-\#a_{1t}+\#n a_{1n},\ \ \ \#b_2 = -\#a_{2t}+ \#n a_{2n}. \l{b12B-}\f
\r{odd1x'} and \r{even32x'} are now satisfied for any given vectors $\#a_1$ and $\#a_2$. Substituting \r{b12B+} or \r{b12B-}, the conditions \r{cond1a}, \r{cond2a} and \r{odd1x} -- \r{even31'dx} are also satisfied.

Thus, the above analysis offers two possibilities for the boundary conditions corresponding to the case $R^2=-1$,
\ea \#a_{1t}\.(\#E +\eta_o\#H) + a_{1n}\#n\.(\#E-\eta_o\#H) &=& 0, \l{a1EB+}\\
\#a_{2t}\.(\#E +\eta_o\#H) + a_{2n}\#n\.(\#E-\eta_o\#H)  &=& 0, \l{a2EB+} \fa
and
\ea \#a_{1t}\.(\#E -\eta_o\#H) + a_{1n}\#n\.(\#E+\eta_o\#H) &=& 0, \l{a1EB-}\\
\#a_{2t}\.(\#E -\eta_o\#H) + a_{2n}\#n\.(\#E+\eta_o\#H)  &=& 0. \l{a2EB-} \fa
The conditions \r{a1EB+}, \r{a2EB+} can be expressed in vector form as
\e \#n\times(\#E+\eta_o\#H) + \#p_t\#n\.(\#E-\eta_o\#H)=0, \l{EPEMC+} \f
and the conditions \r{a1EB-}, \r{a2EB-} as
\e \#n\times(\#E-\eta_o\#H) + \#p_t\#n\.(\#E+\eta_o\#H)=0, \l{EPEMC-} \f
with
\ea \#p_t &=& \frac{1}{A_n}(\#a_1\times\#a_2)\times\#n,
= \frac{1}{A_n}(a_{1n}\#a_{2t}- a_{2n}\#a_{1t}),  \l{pt} \\
A_n &=& \#n\.\#a_1\times\#a_2. \l{An} \fa
In the present case $R^2=-1$ we have assumed $A_n\not=0$.

In the special case of impedance boundaries, the boundary parameters satisfy $a_{1n}=a_{2n}=0$, which corresponds to $\#p_t=0$. In this case, the condition \r{EPEMC+} becomes
\e \#n\times(\#E +\eta_o\#H) = 0, \l{EPEMC++}\f
and, the condition \r{EPEMC-} becomes
\e \#n\times(\#E -\eta_o\#H) = 0. \l{EPEMC--}\f
\r{EPEMC++} and \r{EPEMC--} equal special cases of the PEMC boundary conditions \r{PEMC}, with admittances $M=1/\eta_o$  and $M=-1/\eta_o$, respectively. The reflection coefficients for the two eigenwaves at the general PEMC boundary are known to have the form \cite{GBC}, Eq. (2.38),
\e R_\pm = \frac{1\pm jM\eta_o}{1\mp jM\eta_o}, \f
which satisfies $R_\pm^2=-1$ for both $M\eta_o=1$ and $M\eta_o=-1$.

One can easily verify that, for both of the boundary conditions \r{EPEMC+} and \r{EPEMC-}, the two solutions for the reflection coefficient equation \r{CR} satisfy both $R_+=-R_-$ and $R_\pm^2=-1$. The details are given in the Appendix.

In conclusion, requiring that the reflection coefficients $R_+$ and $R_-$ of the two eigenwaves satisfy the condition $R_+=-R_-=R$, with $R^2=-1$ for any angle of incidence, the boundary is defined either by the conditions \r{EPEMC+} or \r{EPEMC-}.

\section{Extended PEMC Boundaries}

The conditions \r{EPEMC+} and  \r{EPEMC-} are special cases of the vector condition
\e \#n\times(M\#E + \#H) + \#p_t\#n\.(M\#E-\#H) =0, \l{EPEMC} \f
defining what can be called the class of Extended PEMC (EPEMC) boundaries. The class depends on two parameters, a scalar $M$ and a vector $\#p_t$. The boundary vector parameters corresponding to \r{EPEMC} are
\e \#b_1=(\#a_{1t}-\#n a_{1n})/M\eta_o,\ \ \ \#b_2=(\#a_{2t}- \#n a_{2n})/M\eta_o. \l{b1b2}\f
For $\#p_t=0$, the condition of the EPEMC boundary \r{EPEMC} is reduced to that of the PEMC boundary \r{PEMC}, in which case the parameter $M$ equals the admittance of the PEMC boundary.

Because \r{EPEMC+} and \r{EPEMC-} are special cases of \r{EPEMC}, respectively corresponding to $M=1/\eta_o$ and $M=-1/\eta_o$, they can be called conditions of the EPEMC$_+$ and EPEMC$_-$ boundaries, respectively.
For $\#p_t=0$ both of them are reduced to the corresponding special PEMC boundaries.

It has been previously shown that a plane wave incident normally to the PEMC boundary is reflected cross polarized, i.e., satisfying $\#E_t^r\.\#E_t^i=0$, exactly when the PEMC admittance satisfies either $M\eta_o=+1$ or $=-1$ \cite{GBC}. It will turn out that the same property is valid for the EPEMC boundary.

\subsection{Matched Waves}

By definition, a plane wave is matched to a boundary when the boundary conditions are identically satisfied for the single plane wave, i.e., there is no reflected wave \cite{GBC}. Applying the plane-wave equations \r{kxE} and \r{kxH}, the conditions of the  EPEMC$_+$ and EPEMC$_-$ boundaries, \r{EPEMC+} and \r{EPEMC-}, can be respectively  expanded as
\ea k_o\#n\times(\#E+\eta_o\#H) &=& - \#p_t\#n\.k_o(\#E-\eta_o\#H)\nonumber\\
&=& -\#p_t\#k_t\.(\#n\times(\#E+\eta_o\#H)), \fa
\ea k_o\#n\times(\#E-\eta_o\#H) &=& - \#p_t\#n\.k_o(\#E+\eta_o\#H)\nonumber\\
&=&\#p_t\#k_t\.(\#n\times(\#E-\eta_o\#H)). \fa
Thus, the conditions for the matched waves at the respective EPEMC$_+$ and EPEMC$_-$ boundaries become
\ea (k_o\=I_t + \#p_t\#k_t)\.(\#n\times(\#E+ \eta_o\#H)) &=& 0, \l{match+}\\
(k_o\=I_t - \#p_t\#k_t)\.(\#n\times(\#E- \eta_o\#H)) &=& 0. \l{match-}\fa
Let us consider these two cases separately.

\subsubsection {EPEMC$_+$ Boundary}

From \r{match+} we have two possibilities: either
\e \#n\times(\#E+ \eta_o\#H)=0, \l{match+1} \f
or the two-dimensional dyadic in \r{match+} has no inverse.

In the latter case, the two-dimensional determinant of the dyadic must be zero \cite{GBC,Methods}, whence
\e \frac{1}{k_o}{\rm tr }(k_o\=I_t + \#p_t\#k_t)^{(2)} = k_o + \#p_t\.\#k_t =0, \l{disp1} \f
which is the dispersion equation of the matched wave. Because the dyadic can be expressed as
\e k_o\=I_t + \#p_t\#k_t = \#k_t\times(\#p_t\times\=I_t)= -(\#n\times\#k_t)(\#n\times\#p_t), \f
from \r{match+} the fields of the matched wave must satisfy
\e (\#n\times\#p_t)\.(\#n\times(\#E+\eta_o\#H) = \#p_t\.(\#E+\eta_o\#H)=0. \f
Applying the orthogonality $\#k\.(\#E+\eta_o\#H)=0$, the polarization of the matched wave corresponding to the dispersion equation \r{disp1} is
\e \#E+\eta_o\#H\sim \#k\times\#p_t. \f

Considering the case \r{match+1}, from \r{EPEMC+} we obtain
\e \#n\.(\#E-\eta_o\#H)=0. \l{nE-hH}\f
Applying \r{kxE} and \r{kxH} we can expand
\ea k_o\#n\times(\#E+ \eta_o\#H)&=& \#n\times(-\#k\times\eta_o\#H + \#k\times\#E) \nonumber\\
&=& \#k\#n\.(\#E -\eta_o\#H)- (\#n\.\#k)(\#E-\eta_o\#H)\nonumber\\
&=& - (\#n\.\#k)(\#E-\eta_o\#H)=0. \l{disp2}\fa
Thus, the second possible matched wave is governed by the dispersion equation
\e \#n\.\#k=0, \l{nk0}\f
which corresponds to a lateral wave. From \r{nE-hH} we obtain the polarization condition
\e \#E -\eta_o\#H \sim \#n\times\#k. \f

\subsubsection{EPEMC$_-$ Boundary}

This case can be handled similarly. From \r{match-}, for $\#n\times(\#E-\eta_o\#H)\not=0$, the dispersion equation becomes
\e k_o-\#p_t\.\#k_t=0. \f
Because the fields must satisfy
\e \#p_t\.(\#E-\eta_o\#H)=0, \f
the polarization condition becomes
\e \#E-\eta_o\#H \sim \#k\times\#p_t. \f
For the other possibility  $\#n\times(\#E-\eta_o\#H)=0$, the matched wave is again a lateral wave with dispersion equation \r{nk0}. From \r{EPEMC-} we now obtain
\e \#n\.(\#E+\eta_o\#H)=0, \f
whence the polarization of the wave is
\e \#E+\eta_o\#H\sim \#n\times\#k. \f

\subsection{Reflection of Plane Waves}

Let us finally consider plane-wave reflection from the EPEMC$_+$ and EPEMC$_-$ boundaries, defined by \r{EPEMC+} and \r{EPEMC-}.

Applying \r{kxE} and \r{kxH}, the sum of incident and reflected fields, at the  EPEMC$_+$ boundary can be shown to satisfy

\ea && k_o\#n\times(\#E^i+\eta_o\#H^i +\#E^r+\eta_o\#H^r) \nonumber\\
&=& - \#p_t\#n\.k_o(\#E^i-\eta_o\#H^i+\#E^r-\eta_o\#H^r)) \nonumber\\
&=& \#p_t\#n\.(\#k^i\times(\#E^i+\eta_o\#H^i)+\#k^r\times(\#E^r+\eta_o\#H^r))\nonumber\\
&=& -\#p_t\#k_t\.\#n\times(\#E^i+\eta_o\#H^i + \#E^r+\eta_o\#H^r).  \fa
The result can be written as
\e (k_o\=I_t + \#p_t\#k_t)\.\#n\times(\#E^i+\eta_o\#H^i+\#E^r+\eta_o\#H^r) =0. \l{EiEr+}\f
Omitting the case when $\#k$ satisfies the dispersion equation \r{disp1} of the matched wave, the dyadic in \r{EiEr+} has a two-dimensional inverse. In this case, the fields at the boundary must satisfy
\e \#n\times(\#E^i+\#E^r+\eta_o(\#H^i+\#H^r)) =0, \l{condPEMC+}\f
for any vector $\#p_t$. Actually, \r{condPEMC+} equals the boundary condition of the PEMC boundary \r{PEMC1}, for $M=1/\eta_o$.

The EPEMC$_-$ boundary can be handled similarly by starting from the condition \r{EPEMC-}, leading to
\e (k_o\=I_t - \#p_t\#k_t)\.\#n\times(\#E^i-\eta_o\#H^i+\#E^r-\eta_o\#H^r) =0, \l{EiEr-}\f
whence the result \r{condPEMC+} will be replaced by
\e \#n\times(\#E^i+\#E^r-\eta_o(\#H^i+\#H^r)) =0, \l{condPEMC-}\f
for any vector $\#p_t$. Again, this equals the boundary condition of the PEMC boundary \r{PEMC1}, for $M=-1/\eta_o$.

To summarize, the class of boundaries defined by the property $R^2=-1$ consists of two subclasses, EPEMC$_+$ and EPEMC$_-$, which are equivalent in reflection to two subclasses of PEMC boundaries, respectively defined by $M=1/\eta_o$ and $M=-1/\eta_o$, for any vector $\#p_t$.

As a example, let us consider a plane wave with normal incidence, $\#k^r=-\#k^i=\#n k_o$, reflecting from an EPEMC boundary. The fields are tangential to the boundary:
$\#n\.\#E^i=\#n\.\#E^r=0$, $\#n\.\#H^i=\#n\.\#H^r=0$. From \r{EPEMC}, the reflected field can be expressed as
\e \#E^r = -\frac{1}{M^2\eta_o^2+1}\left( (M^2\eta_o^2-1)\=I_t + 2M\eta_o\#n\times\=I\right)\.\#E^i, \l{ErC}\f
whence
\e \#E^r\.\#E^i = C(\#E^i\.\#E^i), \f
where $C$, defined by
\e C = -\frac{M^2\eta_o^2-1}{M^2\eta_o^2+1}, \l{C}\f
is a measure of copolarization in the reflected field. This function is depicted in Figure \ref{fig:copol.png}.

The reflected field appears cross polarized when $\#E^r\.\#E^i=0$, or $C=0$. This happens for $M\eta_o=\pm1$, i.e., when the EPEMC boundary is either EPEMC$_+$ or EPEMC$_-$. The same property is known to exist for PEMC boundaries with $M=1/\eta_o$ or $M=-1/\eta_o$ (\cite{GBC}, Section 2.4).  For $M=0$ (PMC, $C=1$) and $M\rightarrow\pm\infty$ (PEC, $C=-1$), the reflected field is totally copolarized. Note that, for a circularly polarized incident wave, satisfying $\#E^i\.\#E^i=0$, from \r{ErC} we have $\#E^r\.\#E^i=0$ for any $M\not= \pm j/\eta_o$.

\begin{figure}
\begin{center}
{
\includegraphics[width=11cm]{copol.png}
}
\caption{\label{fig:copol.png}
Visualization of the co-polarization quantity $C$ of \r{C}, as a function of the parameter $M\eta_o$, for a normally incident plane wave reflecting from an EPEMC boundary defined by \r{EPEMC}. The reflected field is co-polarized for $M\eta_o=0$ (PMC) and $M\eta_o\rightarrow\pm\infty$ (PEC) while, for $M\eta_o=1$ (EPEMC$_+$) and $M\eta_o=-1$ (EPEMC$_-$),  the reflected field is cross polarized, $C=0$.}
\end{center}
\end{figure}

\section{Conclusion}The problem considered in this paper has been to generalize the property of the class of electromagnetic boundaries known as that of generalized soft-and-hard/DB (GSHDB) boundaries. The class of boundaries is known to possess the unique property that the two eigen plane-waves associated to the boundary are reflected with reflection coefficients $R_+=1$ and $R_-=-1$ for any angle of incidence. Such a property has been called PEC/PMC equivalence, because the two eigenwaves are reflected as from a PEC or a PMC boundary. In the present case, the possibility of a class of boundaries with a more general property was studied by requesting that the reflection coefficients of the two eigenwaves satisfy a condition of the form $R_\pm=\pm R$, where $R$ is independent of the angle of incidence.

It was shown that, under this condition, $R$ must be either $1$ or $j$. While the former case corresponds to the class of GSHDB boundaries, the latter case was shown to correspond to another class of boundaries defined by two possible sets of boundary conditions, \r{EPEMC+} and \r{EPEMC-}. Both of these were shown to be special cases of a more general class of Extended Perfect Electromagnetic Conductor (EPEMC) boundaries defined by conditions of the form \r{EPEMC}, depending on a scalar $M$ and a vector $\#p_t$ tangential to the boundary surface. The two special cases of \r{EPEMC} were respectively called EPEMC$_+$ and EPEMC$_-$ boundaries as they correspond to the special parameter values $M=1/\eta_o$ and $M=-1/\eta_o$, respectively. For $\#p_t=0$, the EPEMC boundary is reduced to the PEMC boundary with $M$ equal to the PEMC admittance. Matched waves and plane-wave reflection properties were studied for both EPEMC$_+$ and EPEMC$_-$ classes of boundaries. While the matched waves depend on the $\#p_t$ vector, reflection of the eigenwaves does not. Actually, in reflection, the EPEMC$_+$ and EPEMC$_-$ classes of boundaries are equivalent to the corresponding classes of PEMC boundaries defined by the respective admittance values $M=1/\eta_o$ and $M=-1/\eta_o$, for any vector $\#p_t$.

\section{Appendix}

\subsection{Expanding $A$ and $B$}

Applying \r{bitJt}, we can expand the expressions \r{A} and \r{B} as
\ea A &=& k_ok_n\#n\.(\#a_1\times\#a_2 + \#b_1\times\#b_2)+ k_n\#n\.(\#b_2\#a_1-\#b_1\#a_2 + \#a_1\#b_2-\#a_2\#b_1)\.\#k_t \nonumber\\
&=& k_ok_n\#n\.(\#a_1\times\#a_2 + \#b_1\times\#b_2)- (\#b_1\#a_2-\#b_2\#a_1):k_n(\#n\#k_t + \#k_t\#n), \\
 B &=& (\#b_{1t}'\#a_{2t}'-\#b_{2t}'\#a_{1t}'):(\#n\times\#k)(\#n\times\#k)+ k_n^2(\#a_{2t}'\.\#b_{1t}'-\#a_{1t}'\.\#b_{2t}') \nonumber\\
&=& (\#b_{1t}'\#a_{2t}'-\#b_{2t}'\#a_{1t}'):(k_t^2\=I_t- \#k_t\#k_t + k_n^2\=I_t) \nonumber\\
&=& (\#b_{1t}'\#a_{2t}'-\#b_{2t}'\#a_{1t}'):(k_o^2\=I_t- \#k_t\#k_t).  \fa
Substituting
\ea \#b_{1t}'\#a_{2t}'-\#b_{2t}'\#a_{1t} &=& \#b_{1t}\#a_{2t}-\#b_{2t}\#a_{1t}\nonumber\\
&+& \frac{1}{k_o}\bigg((b_{2n}\#b_{1t}-b_{1n}\#b_{2t})\#n\times\#k -\#n\times\#k(a_{1n}\#a_{2t}-a_{2n}\#a_{1t})\bigg)\nonumber\\
&-& \frac{1}{k_o^2}(a_{1n}b_{2n}-a_{2n}b_{1n})(\#n\times\#k)(\#n\times\#k), \fa
we obtain
\ea B &=& (\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I_t- \#k_t\#k_t)\nonumber\\
&+& k_o(b_{2n}\#b_1-b_{1n}\#b_2 -a_{1n}\#a_2+a_{2n}\#a_1)\.(\#n\times\#k)- (a_{1n}b_{2n}-a_{2n}b_{1n})k_t^2, \nonumber\\
&=& (\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I- \#k_t\#k_t- k_n^2\#n\#n)+ k_o(\#a_1\times\#a_2+\#b_1\times\#b_2)\.\#k_t. \fa
These can be combined as
\ea A\pm B &=& \pm k_o(\#a_1\times\#a_2+\#b_1\times\#b_2)\.(\#k_t \pm k_n\#n)\nonumber\\
&&\pm (\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I- \#k_t\#k_t- k_n^2\#n\#n\mp k_n(\#n\#k_t + \#k_t\#n))\nonumber\\
&=& \pm k_o(\#a_1\times\#a_2+\#b_1\times\#b_2)\.(\#k_t \pm k_n\#n)\nonumber\\
&&\pm (\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I- (\#k_t \pm k_n\#n)(\#k_t \pm k_n\#n))  \fa
Thus, we can finally write
\e k_ok_nC_0 = -k_o(\#a_1\times\#a_2 + \#b_1\times\#b_2)\.\#k^i-(\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I-\#k^i\#k^i), \l{koknC0'}\f
and
\e    k_ok_nC_2 =  k_o(\#a_1\times\#a_2 + \#b_1\times\#b_2)\.\#k^r+(\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I-\#k^r\#k^r). \l{koknC2'} \f

\subsection{Expanding Condition \r{R2} }

Applying \r{C0'} and \r{C2'}, the condition \r{R2} can be expressed in compact form as
\e (R^2-1)(K_o+K_e) + (R^2+1)(L_o+L_e)=0, \l{KoKeLoLe}\f
with terms odd in $\#k_t$,
\ea K_o &=& k_o(\#a_1\times\#a_2 + \#b_1\times\#b_2)\.\#k_t \l{Ko}\\
L_o &=& -k_n\#n\.(\#b_1\#a_2+ \#a_2\#b_1-\#b_2\#a_1-\#a_1\#b_2)\.\#k_t, \l{Lo}\fa
and, terms even in $\#k_t$
\ea K_e&=& (\#b_1\#a_2-\#b_2\#a_1):(k_o^2\=I_t -\#k_t\#k_t +(\#k_t\.\#k_t)\#n\#n)\l{Ke} \\
L_e&=& 2k_n k_o\#n\.(\#a_1\times\#a_2) . \l{Le}\fa
For \r{KoKeLoLe} to be satisfied for any $\#k_t$, parts odd and even in $\#k_t$ must vanish separately. This leads to the respective conditions
\ea (R^2-1)K_o + (R^2+1)L_o &=&0, \l{odd}\\
 (R^2-1) K_e+ (R^2+1)L_e&=&0. \l{even} \fa

As a special case, for the GSHDB boundary \r{GSHDB} with $\#a_1=\#a_t$, $\#b_1=\alpha\#n$, $\#a_2=\beta\#n$ and $\#b_2=\#b_t$, we find $L_o=0$ and $L_e=0$. Since $\#a_t$ and $\#b_t$ may be any tangential vectors and $\alpha,\beta$ any scalars, $K_o$ and $K_e$ are nonzero. Thus, both \r{odd} and \r{even} yield $R^2=1$, or $R_\pm=\pm 1$, a result known from \cite{GBC,GSHDB}.

Let us consider the general case. For \r{odd} to be valid for any polarization of $\#k_t$, we must have
\ea (R^2-1)k_o(\#a_1\times\#a_2 + \#b_1\times\#b_2)\times\#n &+& \nonumber\\
- (R^2+1)k_n\#n\.(\#b_1\#a_2+ \#a_2\#b_1-\#b_2\#a_1-\#a_1\#b_2))\times\#n &=& 0. \fa
Since only the second term depends on $\#k_t$ (through $k_n$), the two terms must vanish independently, whence we can write
\ea (R^2-1)(a_{1n}\#a_{2t} -a_{2n}\#a_{1t} + b_{1n}\#b_{2t}- b_{2n}\#b_{1t}) &=&0, \l{odd1}\\
 (R^2+1)(b_{1n}\#a_{2t}+ a_{2n}\#b_{1t}- b_{2n}\#a_{1t} -a_{1n}\#b_{2t}) &=& 0. \l{odd2}\fa
These two conditions correspond to the equation \r{odd}. Similarly, \r{even} can be split in three conditions, based on  different orders of magnitude in $\#k_t$, as
\ea (R^2-1)(\#b_1\#a_2-\#b_2\#a_1):\=I_t &=&0, \l{even1} \\
 (R^2+1)\#n\.(\#a_1\times\#a_2)&=& 0, \l{even2}\\
 (R^2-1)(\#b_1\#a_2-\#b_2\#a_1):(\#k_t\#k_t -(\#k_t\.\#k_t)\#n\#n)&=&0. \l{even3}\fa
The condition \r{even3} can be further split in two parts because one of the terms does not depend on the polarization of $\#k_t$,
\ea (R^2-1)(\#b_1\#a_2-\#b_2\#a_1):\#k_t\#k_t&=&0, \l{even31}\\
 (R^2-1)(b_{1n}a_{2n}-b_{2n}a_{1n})&=&0. \l{even32}\fa
Setting $\#k_t=\#k_t'+\#k_t''$ in \r{even31}, we obtain
\e (R^2-1)(\#b_1\#a_2-\#b_2\#a_1):(\#k_t'\#k_t''+\#k_t''\#k_t')=0, \l{even31's}\f
which must be valid for any $\#k_t'$ and $\#k_t''$, whence \r{even31's} equals the dyadic condition
\e (R^2-1)(\#b_{1t}\#a_{2t} + \#a_{2t}\#b_{1t}-\#b_{2t}\#a_{1t}- \#a_{1t}\#b_{2t})=0. \l{even31'd}\f
Since the trace of \r{even31'd} covers the condition \r{even1}, the latter can be omitted.

In conclusion, the medium parameters $\#a_1\cdots\#b_2$ and the reflection coefficient $R$ must satisfy the conditions \r{odd1}, \r{odd2}, \r{even2}, \r{even32} and \r{even31'd}

\subsection{Expanding $C_1=0$}

Let us find the conditions for the boundary parameters $\#a_1\cdots\#b_2$ in the case when the coefficient $C_1$ vanishes for all wave vectors $\#k_t$. Because the second term of \r{koC1} changes sign in $\#k_t\rightarrow\#k_t$, from $C_1=0$ we obtain the two conditions
\ea \#n\.(\#a_1\times\#a_2 - \#b_1\times\#b_2) &=& 0, \l{C11}\\
(\#a_1\times\#b_2 + \#b_1\times\#a_2)\times\#n&=&0. \l{C12}\fa
Assuming $\#n\.\#a_1\times\#a_2\not=0$, the tangential vectors $\#a_{1t},\#a_{2t}$ make a planar basis and we can expand
\e \left(\begin{array}{c} \#b_{1t}\\ \#b_{2t}\end{array}\right) = \left(\begin{array}{cc} B_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right) \left(\begin{array}{c} \#a_{1t}\\ \#a_{2t}\end{array}\right). \l{b1tb2t''}\f
Inserting in \r{C11}, we obtain
\e B_{11}B_{22}-B_{12}B_{21}=1. \l{detB11}\f
Expanding \r{C12} as
\e a_{1n}\#b_{2t}- b_{2n}\#a_{1t} + b_{1n}\#a_{2t}- a_{2n}\#b_{1t}=0, \f
and applying \r{b1tb2t''}, with $\#a_{1t}\times\#a_{2t}\not=0$, after some steps we obtain the expansion
\e \left(\begin{array}{c} b_{1n}\\ b_{2n}\end{array}\right) = \left(\begin{array}{cc} -B_{22} & B_{12}\\ B_{21} & -B_{11}\end{array}\right) \left(\begin{array}{c} a_{1n}\\ a_{2n}\end{array}\right) = - \left(\begin{array}{cc} B_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right)^{-1}\left(\begin{array}{c} a_{1n}\\ a_{2n}\end{array}\right). \l{b1nb2n'} \f
In conclusion, vectors $\#b_1$ and $\#b_2$, depending through \r{b1tb2t''} and \r{b1nb2n'} on any vectors $\#a_1$ and $\#a_2$ whose tangential components are linearly independent, in terms of any parameters $B_{11}\cdots B_{22}$ satisfying \r{detB11}, yield $C_1=0$ for any $\#k_t$.

\subsection{Verifying EPEMC$_+$ and EPEMC$_-$ boundary conditions}

Let us verify that the EPEMC$_+$ boundary conditions \r{a1EB+} and \r{a2EB+}, and the EPEMC$_-$ boundary conditions \r{a1EB-} and \r{a2EB-}, depending on the two vectors $\#a_1$ and $\#a_2$, both correspond to the property $R_\pm^2=-1$.

Substituting \r{b12B+} and \r{b12B-} and expanding
\ea \#a_1\times\#a_2 &=& \#a_{1t}\times\#a_{2t} + \#n\times(a_{1n}\#a_{2t}- a_{2n}\#a_{1t}), \\
 \#b_1\times\#b_2 &=& \#a_{1t}\times\#a_{2t} -\#n\times(a_{1n}\#a_{2t}- a_{2n}\#a_{1t}), \fa
we obtain
\e \#n\.(\#a_1\times\#a_2- \#b_1\times\#b_2) = 0. \f
Similarly, we can expand
\e (\#a_1\times\#b_2+ \#b_1\times\#a_2)\times\#n = \pm(a_{1n}\#a_{2t}+a_{2n}\#a_{1t} -a_{1n}\#a_{2t}-a_{2n}\#a_{1t})=0. \f
Substituting these in \r{koC1} we obtain
\e C_1=0, \f
which, from \r{Rpm}, ensures the validity of the condition $R_+=-R_-$.

Further, we can expand
\ea \#a_1\times\#a_2+ \#b_1\times\#b_2 &=& 2A_n\#n, \\
\#b_1\#a_2-\#b_2\#a_1 &=& \mp A_n(\#n\times\=I + \#n\#p_t+ \#p_t\#n), \fa
with the upper sign corresponding to EPEMC$_+$, and the lower sign to EPEMC$_-$, boundary conditions.

Substituting these in \r{C0'} and \r{C2'} yields
\ea C_0 &=& \frac{-2\#n\.\#k^i}{k_ok_n}A_n(k_o \pm\#p_t\.\#k_t), \\
 C_2 &=& \frac{2\#n\.\#k^r}{k_ok_n}A_n(k_o \pm\#p_t\.\#k_t) = C_0, \l{C2=C0}\fa
whence, from \r{Rpm}, the reflection coefficients are $j$ and $-j$ for both of the EPEMC$_+$ and EPEMC$_-$ conditions.

redpajama_set_name: RedPajamaArXiv


text:
Blog › Strange and Dangerous things people experience with Letterboxes
Strange and Dangerous things people experience with Letterboxes
Tags: antiarsonletterbox | letterboxfire | highsecurityboxrange |
What is a letterbox designed for?
A simple question with an obvious answer you may say. The definition is…
"A box attached to an outside wall, or a slot in the door of a building, into which mail is delivered."
(Oxford English Dictionary)
Mmmm, some people don't seem to fully understand this concept. Many other bizarre objects appear through a letterbox, other than mail.
Take this strange but amusing example of a letterbox posting of a trout with a cigarette in it's mouth. A Bournemouth shop owner smelled something fishy after discovering a trout with a cigarette in its mouth had been posted through his letterbox.
Darren Burns, who owns 'Safercigs' in Blandford opened up shop on Sunday morning to be greeted by the strange sight and smell. "I had absolutely no idea what it was all about," he said. Darren reported this to the police about the incident. They decided that no further action would be taken as they had bigger fish to fry!
Trout with cigarette in its mouth posted through shop's letterbox (yes, you read that correctly)
A SHOP owner smelled something fishy after discovering a trout with a cigarette in its mouth had been posted through his letterbox.
www.bournemouthecho.co.uk
What about this strange tale of the mystery woman in Kirkby, near Liverpool, who cries and shouts through letterboxes in the middle of the night. Merseyside Police told the Liverpool Echo: "We received a call from a lady telling us that her mother who lives in Kirkby, had been disturbed in the early hours of the morning by a woman who knocked on her front door for over an hour. "The woman was shouting 'help me, let me in, I've got nowhere to go' but the lady didn't let the woman in." Residents suspect that the bizarre wailing is part of a ploy to burgle people's homes.
Mystery over woman 'crying through people's letterboxes in early hours of morning'
Local residents suspect bizarre wailing is a ploy to commit a burglary
www.independent.co.uk
Even more bizarre and frightening was when a resident of Buderim in Queensland Australia, got the shock of his life when he went out to collect his mail, only to discover a huge carpet python slumbering peacefully in his letterbox. Max from the…Sunshine Coast Snake Catchers' organisation was called to remove the sleeping intruder, who clearly just wanted a cool, dark place to crash out.
Giant python discovered in Queensland letterbox
WELCOME to Australia, where the wildlife is so dangerous you can no longer even check for your bloody mail.
www.news.com.au
A little more dangerous is a letterbox that bites! This story from the online website 'Mumsnet', is bizarre.
We have just received a letter from a firm of solicitors telling us that we are being held responsible for an accident that occurred at our house when someone was hand delivering a letter. Basically the top of the person's finger was amputated, the finger was fractured and there is 'possible nerve damage'. We had some warning around the time that the accident happened as her employer told my husband but that was the last we heard till now, almost a month later.
We saw no evidence of an accident - no blood, skin etc. There have never been any previous accidents and although the edge of the letterbox flap is quite keen, in proper use (as it has been used by numerous people for four years), it shouldn't present a hazard. We opened the letter too late to contact our insurers so not sure if we have cover but we have never been in this situation before and have no idea of what we might be liable for and how long this may go on for. Obviously we will be contacting our insurers as soon as possible.
We have just received a letter from a firm of solicitors telling us that we are being held responsible for an accident that occurred at our house whe
www.mumsnet.com
Finally, a terrifying tale of letterbox arson earlier this month. Fire crews were called to Kitchener Road in Rochester, just after 4.20am on Sunday, 6 th August, after the front door of the property was alight and had spread inside to the hallway carpet. The fire was out by 4.35am and there were no reported injuries.
A Kent Fire and Rescue Service spokesperson said: "The cause of the fire is thought to have been deliberate and is believed to have been caused by a flammable liquid being ignited. An investigation is underway." Kent Police was called minutes after the fire had been put out and is now appealing for witnesses to come forward.
A police spokesperson said: "Anyone who was in the area at the time or has information about the incident, is asked to contact Kent Police on 01843 222289 quoting ZY/36246/17…Alternatively, contact Kent Crimestoppers, anonymously, on 0800 555111."
404 - Resource not found
www.kentnews.co.uk
This collection of letterbox stories are strange, funny but dangerous. The final letterbox arson story is the most alarming. What can be done to safeguard your family against this silent and potentially deadly criminal act?
One solution is a purchase from our Postbox Shop anti arson letterbox range. To protect yourself or your business against arson attacks, you should install an anti arson letterbox. It is simply to fix behind a door and easy to maintain, as each box is manufactured from electro-galvanised steel with a white polyester powdered coating to improve durability, and ensure it is easy to clean. All our anti arson letterboxes include an anti theft baffle too, with secure locking systems, and 2 keys provided.
Take a look at this anti-arson letterbox for your home.
One of the larger fire risks in residential areas is arson. Usually, fireworks are dropped through the letter plate, or petrol is poured through the letter box and then set alight with a lit rag pushed through the letterbox. Arson endangers lives and property and in the case of an arson attack through the front door, the fire will block your main route of escape, increasing the risk to your family even more.
To protect yourself or your business against arson attacks, install our anti-arson letterboxes on the back of your entrance doors or letter openings. The letterboxes will catch the burning liquid and extinguish the fire. This mailbox is designed to fit neatly behind your door and letter flap. It is fitted with an extinguisher system that operates automatically within seconds of detecting excessive heat, flames or fireworks, protecting your home from arson attacks. This is also used in Sanctuary Scheme applications.
The letterbox is constructed from electro-galvanised steel and polyester powder coated in white for a tough durable easy to clean finish. Simple to install and easy to maintain our letterbox is fitted with an anti-theft baffle and quality lock with 2 keys for added security. There is free delivery for this postbox.
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text:
How the CTV Sector Is Bracing Itself for the Upcoming Privacy Overhaul
Pick up just about any survey or predictions paper for the advertising industry in 2020, and it can all feel like a case of whiplash: excitement over connected television and caution around privacy legislation.
As a measure of the technology's growth, the few remaining publicly traded ad-tech companies used their latest earnings calls to speak effusively of how CTV is spurring their fortunes and point to independent analyses to back this glowing outlook.
However, the rising tide of privacy has also ushered in a mentality of "caution first" among some in the media landscape, as legislation across the globe underlines the importance of data stewardship.
In the CTV space, U.S. federal authorities set the scene in 2017 by issuing a $1.5 million fine to TV manufacturer Vizio for recording audience data without consumer consent, and the Federal Trade Commission later issued guidance for responsible data collection.
Speaking with Adweek earlier in the year, Ameet Shah, senior director of global technology and data strategy at Prohaska Consulting, warned that further congressional inquiries into the TV industry's incursion upon citizen's living rooms were likely on the way.
As the industry continues to implement its digital transformation–OpenAP launched its marketplace offering earlier this week–multiple trade organizations are attempting to bridge the divide between the pillars of freewheeling opportunity and the more somber prospect of responsibility.
In this vein, the IAB Tech Lab is currently constructing a consent-signaling framework (primarily to facilitate the use of ad tech in compliance with data laws) ahead of the enactment of the California Consumer Privacy Act (CCPA) in January 2020. The challenge, and importance, of which was recently highlighted in a BritePool survey indicating that 87% of consumers would opt out of ad targeting under CCPA terms.
Jessica Lee, co-chair of privacy, security and innovation at law firm Loeb & Loeb, told Adweek how establishing (and transmitting) consumer consent in the CTV space represents an extra layer of complexity compared to desktop advertising, primarily because of the absence of cookies.
She advised that those looking to experiment with CTV need to examine the technology at work on their supply chain, including how well their partners can pseudonymize (i.e., encrypt) personally identifiable information (PII).
"The fact they are collecting personal information [including household-level data] means they [all players in the value chain] have to communicate this with consumers," she added, advising this applies under both CCPA and the EU's General Data Protection Regulation (GDPR).
Most privacy policies don't address addressable
For its part, Tru Optik, a data-management platform specializing in CTV, has introduced a privacy-compliance tool dubbed Privacy.TV, which it claims will help advertisers adhere to CCPA requirements.
This includes a framework for a consumer disclosure statement, opt-in and opt-out processes and a regularized audit of consumer permissions, according to Tru Optik CEO Andre Swanston.
"Privacy compliance across CTV is broken … Most publishers, data providers, ad-tech platforms and device manufacturers are ill-equipped to honor consumers' growing demand for transparency," he added.
A recent economic impact assessment of CCPA compliance estimated that the introduction of the legislation could cost U.S. businesses $55 billion, but Tru Optik is introducing the tool for free to encourage usage of its 80-million-household graph.
Swanston further explained the rationale for making the service available for free, adding that "most people out there don't have the proactive capabilities to address CTV in the right way."
He added, "We are the most distributed data marketplace among the CTV ecosystem … so we need to be able to show that we can support this in a safe and transparent way so that we can support our entire business."


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Digital Rights Weekly Update September 11 - 17, 2020
Check out or Digital Rights Weekly Update Report!
Digital Rights Weekly Update
-Leonardo DiCaprio, Kim Kardashian and more 'freeze' Instagram to Stop Hate For Profit - Page Six
Leonardo DiCaprio, Kim Kardashian, Sacha Baron Cohen, and dozens of other celebrities joined the Stop Hate For Profit initiative and froze their Instagram and Facebook accounts for 24 hours in order to hold Mark Zuckerberg's tech company accountable.
-A message to search engines and social media platforms - Put Palestine on the map - Middle East Monitor
Over a million people have signed a petition since July calling on Google to recognise Palestine on its maps, and to clearly designate and identify the Palestinian territories illegally occupied by Israel.
-The announcement of the emergence of the Number Portability application on the 20th of this month (Arabic) - Alhayat
The Palestinian Ministry of Communications and Information Technology, in cooperation with Jawwal and Ooredoo, announced the start of the commercial digital transmission service in Palestine as of 9/20/2020. This service is considered to be a great achievement because it constitutes an important turning point in the telecommunications and information technology sector. It will also lead to a fair competition among companies and preserving the citizens phone numbers when moving from one mobile operator to another, and allowing them the freedom to choose the operator that provides services with better quality and reasonable prices.
-Cell Phone Hacking and Millions in Gulf Deals: Inner Workings of Top Secret Israeli Cyberattack Firm Revealed - Haaretz
Leaked documents confirm that Candiru does not just operate spyware for computers but also has operational mobile spytech.
-Israeli Defense Contractor Rafael Announces New Facial Recognition Firm - Haaretz
Israeli state-owned defense contractor Rafael Advanced Defense Systems and the shareholders of Anyvision, a high-tech startup focused on facial recognition, are setting up a jointly-owned security company. The company will focus on developing machine vision and artificial intelligence for defense purposes.
This bi-weekly update is prepared by 7amleh - The Arab Center for the Advancement of Social Media. For further information please visit our website at www.7amleh.org.
7amleh program to put Al-Zarnouq on the digit...
Digital Rights Weekly Update September 11 - 1...
Update September


WINSTON-SALEM, North Carolina – African-American civil rights leader Franklin McCain, who as a college student helped launch the U.S. sit-in movement against segregation at whites-only lunch counters, has died, his alma mater said on Friday.
McCain and three classmates from North Carolina A&T State University began the first of many similar peaceful protests around the country when they tried to order lunch at the F.W. Woolworth store in Greensboro on February 1, 1960.
The store denied service to the four college freshmen, but they refused to leave until closing and returned in the following days with more students to sit at the lunch counter.
"We had a common outlook to change the unjust conditions in the society that was pushed forward by frustration," McCain said of the decision to begin the sit-in protests, according to a biography on the university's website.
To read this article in its entirety visit Reuters.


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Lets keep hope alive, Atiku urges Nigerians
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Former Vice President, Atiku Abubakar
Former Vice President and chieftain of All Progressives Congress (APC), Atiku Abubakar, says democracy, despite its limitations and challenges, remains the best form of government anywhere in the world.
In his message to mark Nigeria's Democracy Day, the former Vice President explains that the freedom of choice is one of the most important ingredients of the democratic system, stressing that the freedom to choose your leaders makes democracy fashionable.
While pointing out that any system that subverts the will of the people destroys the progress of the society, the Turaki Adamawa added that says as an active participant in the democratic struggle in Nigeria and an ardent proponent of the rule of law, rather than the rule of man, he recognises the sacrifices of gallant Nigerians whose struggles brought us to the current stage.
"As Nigerians celebrate democracy day, the memory of late Shehu Yar'Adua and MKO Abiola would resonate across the country," he noted, adding that the sacrifices of countless others who paid the supreme price for opposing dictatorship would also not be forgotten.
The former Vice President says no sane society would choose despotism over democracy because, under dictatorship, dissent is met with death or torture.
On the current challenges facing ordinary Nigerians, the former Vice President says no democratically elected government would intentionally hurt the voters that put it into office, remarking that the APC administration remains committed to the welfare of Nigerians, and that the current challenges are temporary.
APCAtikuDemocracy DayMKO Abiola


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DFW Black Arts
Color Purple: The Musical is Being Reimagined With a All Star Cast Movie
Director Blitz Bazawule is reimagining The Color Purple for a new generation. The 1985 historical drama based on the Alice Walker novel delved into the intimate suffering of Southern Black women in a way rarely portrayed in mainstream productions.
Bazawule co-directed Black Is King and distributed his feature debut The Burial Of Kojo on Netflix through Ava DuVernay's Array Films.
The original production was directed by Steven Spielberg and starred Whoopi Goldberg, Oprah Winfrey, and Danny Glover. It was then adapted into a Broadway musical in 2005, experiencing several star-studded revivals in locations including Chicago and London. The production earned Tony, Grammy and Emmy awards for its innovation.
At Gallery 12.26, David-Jeremiah Finds a New Vehicle To Explore Black Masculinity
The artist's Lamborghini-inspired 'I Drive Thee' exhibition is on display through March 5.
For more than a decade, David-Jeremiah has been one of the most provocative artists in Dallas. He first made a name for himself as an actor, then as a performance artist. As a visual artist, he's gone from zero to 60 in two years flat, a feat as impressive as the 2.2 seconds it takes a Huracán from Lamborghini, the car brand that inspired the work in Jeremiah's exhibition "I Drive Thee," now on display at Gallery 12.26.
Jeremiah references Lamborghini specifically because the company names its car models after bulls who have been victorious over matadors in the tradition of Spanish bullfighting. This series of eight circular paintings, or tondos, are painted in the slick bright red, black or yellow colors popular for sports cars. These steering wheel-shaped works, such as 2021′s El Cobarde, are aggressive in size but subtle in imagery, invoking elements of the human male form, or more particularly the corpse.
New art exhibit in Irving explores civil rights movement, 1960s American South
Photographs include striking sanitation workers, segregated lunch counters and Ku Klux Klan gatherings.
A new art exhibit in Irving explores the civil rights movement and life in the American South in the 1960s.
Called "I Am A Man," the exhibit gets it name from the iconic photographs of striking Memphis sanitation workers who carried signs with that phrase.
The BLK Experience Pop-Up Museum Returns for 2nd Immersive Installation
The BLK Experience Museum, an innovative pop-up museum dedicated to celebrating black lives and black excellence returns to Dallas' Urban Arts Center for the month of February 2022. The limited engagement exhibition will open to the public on February 5thand run throughout Black History Month.
Urban Arts Center, 807 Hutchins Road, Dallas, Texas starting February 4, 2021
Transport into the black experience at this fully immersive popup art gallery that spans black lives, black excellence, black history and black triumph. Step onto the stage of the world-famous Cotton Club. Dive into the black hair mosh pit. Journey through the History of Black Dallas. The BLK Experience Museum features iconic images and nostalgic commentary that will soon fill up your social media feed.
DFW Black Arts is your trusted source of news and events for the DFW metroplex
Photo used under Creative Commons from emerzon
Urban Arts Collective
IRMA P HALL BLACK THEATRE AWARDS


Silver Beaver Award is the council-level distinguished service award of the Boy Scouts of America. Recipients of this award are registered adult leaders who have made an impact on the lives of youth through service given to the council. The Silver Beaver is an award given to those who implement the Scouting program and perform community service through hard work, self sacrifice, dedication, and many years of service. It is given to those who do not actively seek it.
The Sequoia Council Boy Scouts of America, acting through the National Court of Honor, is given the opportunity to present the Silver Beaver Award for distinguished service to youth.
The award is presented to registered adults of exceptional character within the jurisdiction of the Sequoia Council for noteworthy service to youth. The award is limited to adults (over 21 years of age) who are currently registered volunteer Scouters. It cannot be presented posthumously.
Any volunteer can make nominations for this award. An appointed Silver Beaver Committee makes the selection of the Silver Beaver recipients from the nominations received. The Council is allowed to recognize a specific number of volunteers for this award, which is then presented at the Annual Council Recognition Dinner. It is a general policy that no public announcement shall be made by the Council in advance of action by the National Court of Honor with reference to the names presented for consideration. Nominations should be processed without the knowledge of the nominee.
All three parts of the nomination form must be completed.
Please give the full name and title (if any) of the nominee just as you want it to appear on the certificate (Part I ' NOMINATION FORM [Pg. 1 of 6]).
You may use the back of the form or additional sheets of paper, if necessary, to complete Sections A ' G of Part II, the BACKGROUND INFORMATION WORKSHEET. It is essential that you research your nominee and include as much information as is possible.
Include a minimum of three letters of support (Part III ' LETTERS OF RECOMMENDATION) from a variety of sources: Scouts or other individuals who can attest to the nominee's service to youth.


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