Solutions To Fresnel Diffraction Problems (Part 2)

  1) Consider a circular aperture with center at 0, and n zones comprised of essentially equal areas a _i .  

a) If the n zones are even estimate the total amplitude. 

b) If the n zones are odd, estimate the total amplitude.   

Solns.

a) Let n be even  then we have for the sum of amplitudes:

a  =  a ₁ /2  -  ( a ₁ /2  +   a ₂  -  a ₃ /2)  -  ( a ₃ /2  +   a ₄  -   a ₅ /2)  - .....-    a _n /2

= a ₁ +   a  ₂ /2  +  ( a ₂ /2  + a ₃ -  a ₄ /2) +  (a ₄/2  +  a ₅ -  a ₆ /2)

 + ......+  a _n-1 /2 +  a _n

Because the amplitudes for any two adjacent zones are very nearly equal we can write:   a ₁   »    a ₂    and:  a_n-1  »    a _n

Then we have:

a  =  a ₁ /2  -   a_n /2

b)  Let n be odd then we have for the sum of amplitudes:

a  =  a ₁ /2  +  ( a ₁ /2  -  a ₂  +  a ₃ /2)  +  ( a ₃ /2  -  a ₄  +  a ₅ /2)  + .....+    a _n /2

= a ₁ -  a  ₂ /2  -  ( a ₂ /2  - a ₃ +  a ₄ /2) -  (a ₄/2  -  a ₅ +  a ₆ /2)

 - ......- a _n-1 /2 +  a _n

Because the amplitudes for any two adjacent zones are very nearly equal we can write:   a ₁   »    a ₂    and:  a_n-1  »    a _n

Then we have:

a  =  a ₁ /2  +   a_n /2

2) Show the obliquity factor: 

cos (n, r) - cos (n, r')  =  cos (n, r) + 1

And explain why.

Soln.

We assume again in conformance with the presentation that the diagram below applies:

Since from this diagram:    r' is constant and cos (n, r') = -1

Then: cos (n, r) - cos (n, r') =  cos (n, r) - (-1) =  

cos (n, r) + 1

3) Let the first 10 even zones of a zone plate be covered. Find: a) The whole wave front amplitude,  b) the whole wave front disturbance magnitude, c) the relative amplitude at point P, and d) the intensity at point P.

Solns.:

a)  If the first ten even zones are exposed, then this leaves the amplitudes:  A ₂, A ₄ ,  A ₆,   ... A ₂₀

The sum of which »   10 A ₂  »   10 A ₁ 

The whole wave front amplitude is then:  ½ A₁ 

b) The whole wave front disturbance magnitude is:  ½ U₁

c)  The relative amplitude at point P is twenty times as great as when the plate is removed.

d)  The intensity at point P, from the inverse square law for light, is then   400 times as great as when the plate is removed.

4) The average distance of P from a zone is 15 mm.  If the obliquity angle q  is p/15  and the applicable path difference  D =  0. 15 mm, find the amplitude of the 13th zone.

Soln.:

The amplitude of the mth zone can be expressed: 

A _m  = k (s _m  / d _m  ) (1   +   cos q )

Where k = const., d _m is the average distance to P,   and q  is the angle at which light leaves the zone. 

We have then: d _m  =   15mm

m = 13,    s _{m     ~}   Ö m    » Ö 13  ,    q  =  p/15

Then:

A ₁₃  »   k (Ö 13/ 15 mm ) (1   +   cos p/15 ) 

But the applicable path difference  D =  0. 15 mm, which means  this must be factored into the computations. This means we recast d _m  = b + D

And now: (s _m  /  d _m )  = (  b + D )/   ( b + D )  =  1

(s ₁₃  /  d ₁₃ )  =  1

Then:  A ₁₃  = k (s ₁₃  / d ₁₃  ) (1   +   cos  p/15 )

Or:

A ₁₃  = k  (1   +   cos  p/15 )

Google Maps Capitulating To Dotard's Illegitimate Geography Has Kyle Kulinski Unloading On All Those Who 'Bent the Knee'

 

Kyle Kulinski rips Dotard a new one

                                         The senile freak tyrant shows his coloring book

I have The Donald Trump Memorial Lake right in my bathroom… and I always flush twice because it’s a long way to Mar-a-Lago. – NY Times comment

Someone needs to clue in Trump that the term, America, refers to the giant landmass of North and South America. -– NY Times comment

I propose that Canada rename the United States “South Canada”, except for Alaska (“North Canada” and Hawaii (“West Canada”). -– NY Times comment

Only in Trump's fevered mind is Lake Ontario "Lake America" and the Gulf of Mexico the "Gulf of America". Of course his sycophants in the Administration and business rush to support this, but it's laughable. -– NY Times comment

Incredibly, Google has updated its maps application to rename Lake Ontario as “Lake America” ​​for users in the United States, reflecting a recent  bogus “executive order” by Trump that speciously changed the name of the lake (given 400 years ago or before the Declaration of Independence.  

The announcement was too much for Patreon podcaster Kyle Kulinski who in Tuesday night's Youtube video exploded:

"Google Maps has now capitulated to Trump, renaming Lake Ontario. People will look back on this and be fucking disgusted at everybody who bent the knee to this pedophile, racist fascist. All who bent the knee because they got some arm twisting and’ didn’t want to lose out on a bid, a subsidy or contract."

Kulinski was right to explode, and 1, 055 subsequent comments backed him up, also referring to the trade war with Canada - and what a sorry pariah nation the U.S. had become under Trump. 

 Where Google had the power and heft to hold the line instead of giving in to Dotard's imbecilic ego trip, they caved. In other words, Google – a supposed redoubt of intelligent techies – has ‘bent the knee’ to a grifting orange extortionist traitor who doesn’t even know the history of the name.

Google tried to justify the move in a blog post Saturday asserting that "the decision was based on its practice of using names assigned by the U.S. Geographic Names Information System, also known as GNIS. "

Except that Bonespurs Trump does not control GNIS any more than he controls the Kennedy Center, or the Smithsonian. It’s all a sham, a put on, enabled because of a neutered Reepo congress that would rather suck Trump's balls than oppose him.  Besides, no executive order has the force of law. A Wikipedia entry is useful to shed light:

"There is no constitutional provision nor statute that explicitly permits executive orders. The term executive power in Article II, Section 1, Clause 1 of the Constitution refers to the office of President as the executive."

In other words, these glorified scrap sheets that so entrance Donny Dotard are not based on laws. Further, it is the Article I powers of congress that translate into force of law, not randomly issued executive orders- especially by a half-cocked, derelict fool occupying the White House.  An orange clown unfit to even be in charge of cleaning out Porto Potties.

Google Maps was therefore not obliged to conform to this witless BS, and indeed lost significant gravitas by knuckling under to a senile, fungus -ridden criminal grifter. They buckled under simply because they were following the bend the knee template of the other tech bros and companies. Can't miss out on a future AI deal after all.

The Google Maps update, coincidentally, comes days after Trump escalated a trade war with Canada which will now impose enormous costs on U.S. citizens in Michigan as well as Iowa. Michigan, in terms of the Trump tariffs increasing the cost of vehicles astronomically given auto assembly entails multiple back and forth movements of supply between Canada and the U.S. – with each move now incurring tariffs.

That has led both countries to impose billions of dollars in added tariffs on each other’s goods. It echoes what Google did in 2025 when Trump issued an order to rename the Gulf of Mexico as the Gulf of America.

Google actually posted that:

“People using Maps in the U.S. will see ‘Lake America,’ those in Canada will continue to see ‘Lake Ontario,’ and those outside of the U.S. and Canada will see both names,”

Sorry, no one ought to see more than ONE name – Lake Ontario – because that is the only REAL name. ‘Lake America’ is a fiction hatched from a Trump fecal brain fart expulsion like his renaming the Gulf of Mexico. As well as hundreds of other lawless, BS executive orders. None of which and merits any more respect or recognition than honoring one of his orange turds.

 Google then had the nerve to put this twaddle out in a brief statement:

“These updates follow our long-standing policy for bodies of water with names that vary from country to country, and are starting to roll out now.”

Wrong.  These updates are purely pretentious and performative horse shit to try to gain a fleeting business advantage (which the Dems won't give) from a grifting traitor and extortionist. Specifically, an executive order does not carry statutory authority to force private commercial entities (like Google, Apple, or MapQuest), or state governments to change what they call a geographical feature. The key aspect for people to bear in mind:

"Executive orders are subject to judicial review, and may be struck down if deemed by the courts to be unsupported by statute or the Constitution."

If this latest renaming Lake Ontario outrage stands, no one will respect Google Maps or the validity of its decision making. Many of us will also begin to question the quality of the Google search engine.

See Also:

Canada | Donald Trump

Trump’s Bonkers Bullying of Canada Is Backfiring

by Robert Reich | September 3, 2026 - 5:15am | permalink

— from Robert Reich's Substack

Friends,

What do bullies do when someone stands up to their bullying and punches them in the face? They look for someone weaker to bully who won’t punch back.

Presumably that’s why Donald Trump has now taken on Canada, after Iran punched back so hard in the cage match the U.S. president began more than six months ago as to leave him humiliated and baffled.

Trump wants to be the bully who humiliates others and forces them into submission. Needing to assert his dominance, Trump has now started a trade war with our northern neighbor, friend, second-largest trading partner (after Mexico), and, at least until now, reliable ally.

He’s hammering Canada on his Truth Social and in media appearances, saying Canada has “ripped off” the U.S. for years, and signing an absurd order to rename Lake Ontario “Lake America.”

» article continues...

And:

Lake Ontario Is One of Many Bodies of Water Caught in a Naming Fight - The New York Times

Excerpt:

Like Mr. Trump’s order to rename the Gulf of Mexico to the Gulf of America, last week’s order required only federal entities to recognize the change, but was not binding on other countries.

Lake Ontario thus joined an extensive group of bodies of water with disputed names. Though for some there is no dispute. Prime Minister Mark Carney of Canada spiked searches for the word “hydronym,” which means a body of water’s proper name, by more than 5,000 percent in a statement lamenting a changing America.

The United States’ latest rechristening is mostly about “rage-bating and trolling Canadians,” said Reuben Rose-Redwood, a professor of geography at the University of Victoria in Canada.

He said the tiff surrounding Lake Ontario lacks the deep symbolism and geopolitical stakes of some more longstanding naming disputes.

And:

Perhaps Hard To Believe: But Why Most Of Trump's Executive Orders Are Bogus As In Unconstitutional

And:

Culture | Donald Trump

Trump’s Cultural Vandalism: Bullshit ‘War on Woke’ Will Trash The Country’s Museums and Libraries

by Bill Berkowitz | March 23, 2025 - 4:55am | permalink

Trump’s Executive Order Defunding Museums and Libraries Across the Country is Another Authoritarian Attempt to Control Culture

“They were careless people … They smashed up things and creatures and then retreated back into their money or their vast carelessness, or whatever it was that kept them together, and let other people clean up the mess they had made.”
—F Scott Fitzgerald, The Great Gatsby

Historically, authoritarian regimes have used cultural appropriation to consolidate power and control cultural narratives. Suppressing artistic expression, attacking the media and academia, banning books, destroying historical documents, saturating the culture with groupthink, are all part of an authoritarian’s supercharged playbook. Donald Trump, in his rush to eradicate and/or rewrite history and reshape America’s cultural institutions has, among other acts, named himself Chairman of the John F. Kennedy Center for the Performing Arts and dismissed existing board members, appointing a new Trump-aligned board; rewritten guidelines for National Endowment for the Arts grants, emphasizing the need for so-called patriotic art; appointed a new head of the National Archives; and most recently signed an executive order to eliminate the agency that funds museums and libraries.

» article continues...

And:

Donald Trump Has No Idea What He's Doing | Steve Schmidt

Solution To Mensa 3D Geometry Circle Problem

 

Let O be the center of the cube after envisioning the two spheres centered at O (passing through the 2 circles) and sketched in two dimensions (as shown above).  Then the sphere that passes through the inscribed circle has radius:

s Ö (2)/ 2 

The larger sphere that passes through the inscribed circle has radius:

 s Ö (3)/ 2 

The  distance between the 2 circles cannot be smaller than the distance between the spheres. Thus, for any point P on the circumscribed circle and any point Q on the inscribed circle:  

PQ  >  (s/ 2) (Ö (3)  -  Ö (2) )

To show that this lower bound is attained, envisage a ray from O aimed toward the point where the inscribed circle is tangent to the face on which the other circle is circumscribed (as depicted in the revised 2D sketch).  Initially this ray passes through the interior of the circumscribed circle.

Now, visualize rotating the ray while keeping it tangent to the inscribed circle. Eventually, the ray will rotate far enough that it no longer passes through the interior of the circumscribed circle. By the principle of continuity, it follows that in some intermediate position the ray touches both circles.

Let P be the point where the ray touches the circumscribed circle and Q be the point where the ray touches the inscribed circle.  The points O, P and Q are co-linear and:

PQ = (OP - OQ) =  sÖ (3)/ 2)  -  sÖ (2)/ 2) 

= (s/ 2) {Ö (3)  -  Ö (2)}  

Which is the minimal distance between the two circles.

Reviewing Algebra II - Part Two: Exponents & Solving Exponential Equations

  Simplifying and solving exponential equations is one of the more important aspects given the wide array of applications in the physical sciences, such as radioactive decay, obtaining the half life of an isotope etc. So in this Part Two we focus on reviving the skills involved in solving these equations - which are  fundamental part of Algebra II- or learning them for the first time. Again, this is also based on the rules and properties of manipulating exponents which are generally part of Algebra I.   For example, we already saw in the previous post one of these rules applied to quotients, i.e.

a ^m /a ⁿ     =  a ^{m - n}

^{So for:} 

a ⁵ /a ³   we get:    a ^{5 - 3}    =  a 

Other rules for exponents:

(i)  a ^-n  =  1/a ⁿ

(ii)   (a ^m) ⁿ     =  a ^m n 

(iii)   a ^1/n    =   ⁿ Ö a

(iv)   a ^m/ n     =   (a ^1/n ) ^m     = (a ^m) ^1/n

=  ( ⁿ Ö a ) ^m     =   ⁿ Ö( a ^m )

There are generally 2 ways to solve exponential equations: i) where both sides of the equation have the same base, and (ii) when they have different bases.

For case (i) consider:   

3 ^x =  3 ^{2x - 2}

^{Here the same base (3) allows us to solve for x by equating the exponents:}

^{x =  2x - 2} 

^{so:  x = 2   which can be checked by plugging x = 2 back into the eqn.  Then:}

^{3 2 =  3 2(2) - 2}

3 ² = 3²

When different bases are involved, we have basically 2 choices: a) work to get both sides to the same base, then proceed as shown for case (i) or b) crank out the needed value using logs, say base 10 or natural. Since most scientific work is now with the latter we use that:  For example, take the equation:  4^x = 15

We begin by taking the natural logarithm of both sides:

ln 4^x   =   ln 15   

Then use the power rule: i.e. ln b^x   = x ln b, to bring the variable exponent (x) to the front, i.e.

x ln 4 =  ln 15

We then can solve by dividing both sides by ln 4:

x =    ln 15/  ln 4  =   2.708/ 1.386  = 1.953

The other approach, usually easier, is to bring each side of an exponential equation to the same base. For example, say: 5^{x -1} = 0.04

The trick here is to get both sides in base 5 and go from there. It helps if we can see the decimal 0.04 can be expressed as a fraction, 0.04 = 4/ 100.

But 4/100 is just 1/ 25 and we immediately see  5² =  25, and we have our base 5.

Then,  working through we get:  5^{x -1} = 1/ 4^x = 15
 =    5^-2

So we can now equate the exponents:  x - 1 =  -2 or x = -2 + 1  = -1

Practice Problems: 

Solve each equation for the unknown and check:

 (1) 2a^-3 - 1 = 15

(2) 9^x =  55

3) 5x^1/2 + 7   = 22

4) 3a³   = 81

5) 4^x = 8 ^x+1

6) b^-5 = 1/32

 

7)  3 +   7 ^x-1  =  10

For each of the following, write as a power with positive exponents in simplest form:

8)

9)

10)