Friday, March 31, 2023

Solutions To Korteweg deVries Equation Soliton Problems

 1)  Plot a graph of  F   vs.  v  for the case c s   -  v o   <   0 and indicate the position of v max  on the graph.


2)  Integrate  the KdV equation:

a  dV/ dx2   -    (v o     c s ) v  -   v 2/ 2   =  0 

And show how the soliton solution:

v   =   3 (v o     c s ) sech 2  [(v o     c s  / 4 a) ½     x' ]

Is obtained.  Given this is in the fluid frame, show what it would be in the lab frame.


Multiply the KdV eqn. by v'  and then integrate to obtain:

a/ 2  (v' 2)  =  (v o     c s 2/ 2  -    v3/ 6

We choose the constant of integration to be zero because we want v' = 0  when  v = 0.   Then:

dv/ dx'   =   v'  =   ( 2/ a) ½ [ (v o     c s 2/ 2  -    v3/ 6 ] ½ 


dv/  [(v o     c s 2/ 2  -    v3/ 6 ] ½    =   ( 2/ a) ½  dx'

Now integrate both sides to obtain:

ò dv/ Ö (v  2  -  b v  )  =   ò ( 2/ a) ½ dx'

Where   b   = 1/  [3 (v o     c s )]

Now let u  =  Ö (1    -   b v)   so  v  = (1 -  u  ) /  b  

   And du =    (-  b   dv/ 2)/  Ö (1    -   b v)

Then we are in position, with some labor, to work out:

( 2/ a) ½ x'   =  (2/ v o     c s ) ½  ln  (1 - u/  1 + u)  

On introducing another compressed factor with an exponential, viz.

exp (g  x')   =    (1 - u/  1 + u)    

with  1  - u   =    (1 + u) exp (g  x')     

 It is straightforward to get to the final soln.   The key penultimate step is:

v  =   1/ b   =   4 [e  x' /2 g  -  e - x' /2 g ]  

=   1/ b  sech 2 (g  x'/  2) 

Leading to:

(v o     c s ) sech 2  [(v o     c s  / 4 a) ½  x' ] 

In the lab frame (where x =   x'  +   v o t)  we have:

v (x, t) =  
(v o     c s ) sech 2  [(v o     c s  / 4 a) ½  (x'  -   v o t)] 

Trump Indictment - Even With No Prison Time - Ought To Derail Any Presidential Ambitions


                  "N-N-No! Say it ain't so! Where are my MAGA Mutts when I need 'em?"
"Ah well.  Ya win some ya lose some!"

"This is simple: You indict a former president if he deserves it." -  Historian Michael Beschloss on MSNBC

"Bragg has done an important service. By treating Trump as a normal suspect, he has made it easier for other public prosecutors to move ahead with potentially more serious allegations."-  Financial Times Editorial today

"We're glad someone has finally decided to hold this guy accountable for his actions.  And there's hopefully a lot more accountability to come. As for Trump's die hard fans who think his critics suffer from Trump Derangement Syndrome, I gotta ask: Why are you still clinging to this guy? Maybe you suffer from Trump Fetish Syndrome." - Dan Rodricks, Baltimore Sun

As I've written in previous posts, if my Revolutionary War ancestors were still around they'd have hung the orange maggot Trump already for treason.  So let's have no tiny violins playing sob stories or genuine citizens weeping now that this piece of fecal offal has finally been indicted by the Manhattan grand jury.  Besides, it's only a first step. Doesn't mean he's on a one way trip to the slammer ...yet.  But he should be, having already energized enough of his maggots to hurl death threats at the Manhattan D.A. Alvin Bragg and the Traitor himself posting an image with a bat to Mr. Bragg's head on his oxymoronic "Truth Social" site, e.g.

                    Trump posted image attacking DA Alvin Bragg

Let's get clear in our heads no matter which side of the political spectrum we're on that this is not the way a president, or potential president behaves. Alexander Hamilton, one of the key Founders, was very emphatic on this point when he wrote (in The Federalist, #58):  

"The process of election (by state electors) affords a moral certainty that the office of President shall never fall to the lot of any man who is not to an eminent degree endowed with the requisite qualifications...It will not be too strong to say that there will be a constant probability of seeing the station filled by characters pre-eminent for ability and virtue."

And let's also get clear in our heads NONE of the Founders would have tolerated such behavior - as well as Trump's recent vile rhetoric as "a joke", "funny", "Trump being Trump" or any other degraded take which sullies the whole basis of the office as well as the nation that supports such a depraved imp.

We have seen and heard enough of this foul fungus to last a lifetime and there should be no way in hell he gets near the highest office again, to foul like he did in his last term.

We  now know campaign finance laws were definitely broken, based on the thousands of records retrieved from Michael Cohen's residence in the April 9, 2018 warrant search conducted by the FBI.  The materials included hard copy financial, legal documents, seized electronic devices, audio recordings made by Cohen, emails,  text messages sent over encrypted applications, records retrieved from the corporation referenced in the charging documents as "Corporation One" and records from the media company also therein referenced.  Cohen had even produced a signed check for $35,000 with Trump's signature. This was to pay Cohen back for using his own $130,000 from a home equity line of credit.  So no matter what cockeyed imbeciles yelp about not trusting Cohen, he did deliver evidence - hard evidence of Trump's crimes.  

Let's also note here that none other than John Edwards - Democratic VP candidate in 2004 -  was similarly charged with the crime of using campaign funds to conceal an affair (with Rielle Hunter). It literally blew up his political career.  A disreputable maggot like Donald Trump deserves no less so don't even cite the trope about "setting a dangerous precedent." 

Best part of all this?  Even if the traitor squirms his way out of jail time, it ought to derail his future political ambitions - at least among the sane voters - of which I still maintain there is a slight majority. Even WSJ op-ed columnist Karl Rove in a recent piece, agreed an indictment will put the scarlet letter of renegade rat  on this POS, i.e.

Trump’s Indictment Won’t Win Him Votes


"All in all, the upsides for Mr. Trump of his indictment are likely temporary. The idea of the married Mr. Trump having an affair with a porn star then illegally funneling money to pay for her silence has the odor of raunchy truth. Nothing about it will convince anyone who voted against Mr. Trump to switch allegiance."

And as Michael Cohen lawyer Lanny Davis put it way back in August, 2019:

"If those payments were a crime for Michael Cohen, then why wouldn’t they be a crime for Donald Trump?" 

Yes, Republican leaders in Congress whined and lamented the moment as “a sad day in the annals of United States history”.  But these are the same aimless, shameless reprobates who haven’t done a damned positive thing since they took over the House in January. Inflation plan? Fuggedaboutit!  Aid to Ukraine for its ongoing war against Russia? Dream on!  Taking on assault weapons after the Nashville slaughter? Gimme a break! Progress on the debt ceiling to avoid default? We prefer to hold the nation hostage!

All these clueless rats know how to do is circle the wagons around their cult leader and top traitor.  Then kiss his orange, putrid ass.  Just look at Lindsey Graham on FOX screeching: "This is gonna destroy America!" No, you pussified poltroon, but maybe it'll destroy Trump.

Yes, there will be weeping and gnashing of teeth among these Trump sycophants and butt kissers, but no serious sane citizen should feel that way or that “it was only a twitchy nothing burger”.  No it was not, and NO one gets to be above the law  - any law, including a swine like Donald J. Trump.  

The takeaway here is elementary: Either we are a nation of laws which apply to all, or we are an aberration among nations in which rich and powerful criminals can make their own 'laws'.

See Also:

by Amanda Marcotte | April 4, 2023 - 7:52am | permalink

— from Salon


Donald Trump has never won the popular vote. His election in 2016 was a fluke, in which he only overcame Hillary Clinton's nearly 3 million vote lead because of a few thousand swing voters in purple states. Since then, he's been an electoral albatross around the GOP's neck, helping them lose in 2018 and 2020 and even, in 2022, nuking Republican chances in many elections they would have otherwise won.

Most Americans hate Donald Trump.

And yet, somehow many people keep imbuing Trump with almost magical powers to spin political straw into gold. Now we get to enjoy the spectacle of some of the dumbest people in politics issuing the same galaxy brain take that Trump will somehow benefit from being smacked with reportedly more than 30 charges by Manhattan District Attorney Alvin Bragg.


What we know about the Donald Trump-Stormy Daniels payment case


by Alex Henderson | April 1, 2023 - 6:49am | permalink

— from Alternet


On June 3, 2011, the Democratic Party suffered a major scandal when a grand jury indicted former Sen. John Edwards of North Carolina on six federal criminal charges. Edwards had been 2004 Democratic presidential nominee John Kerry's running mate, and right-wing media outlets like Fox News were quick to hype the fact that someone who was facing those charges could have become vice president.

Outlets that are more sympathetic to Democrats, including MSNBC, extensively reported on Edwards' legal problems as well — but without Fox News' hysterical tone.

The charges against Edwards were serious; he could have spent 30 years in federal prison if convicted on all counts. But a decade later, a former president — Donald Trump — is potentially facing a lot more legal exposure.


by Elizabeth Preza | April 4, 2023 - 8:00am | permalink

— from Alternet

Former President Donald Trump is hoping his arraignment on Tuesday will be “kind of a Jesus Christ thing” as he plans to turn himself in during daylight hours at the Manhattan courthouse, Rolling Stone reports.

According to a law enforcement official involved in security planning for the high-profile booking, Trump declined an option for a Zoom arraignment even as his Secret Service detail “argued in favor of holding the proceedings outside of court business hours, at night with minimal cameras and less risk.”


Commentary: The GOP made a devil’s bargain with Trump; the bill is coming due


by Robert Reich | March 31, 2023 - 7:07am | permalink

— from Robert Reich's Substack


Donald Trump has been indicted.

You’re going to hear three basic criticisms of this indictment. Let me rebut each in turn.

  1. It sets a dangerous precedent.

Rubbish. In order for the justice system to work, there must be trust that the system will not play favorites or ignore the wrongdoing of the powerful.

Donald Trump has done everything possible over the last seven years to destroy that trust for his own political gain.

It is true that no former president has ever been indicted, but no former president has done what Donald Trump has done — repeatedly defied laws and disregarded the U.S. Constitution. America never quite recovered from Gerald Ford’s decision to pardon Richard Nixon for all crimes he might have committed.


WSJ Columnist Peggy Noonan Is Off The Mark Asserting Trump Indictment Is "Below Us" 


by David Badash | March 31, 2023 - 6:39am | permalink

— from The New Civil Rights Movement


by Jaime O’Neill | March 30, 2023 - 6:18am | permalink

"Aren't people absurd! They never use the freedoms they do have but demand those they don't have; they have freedom of thought, they demand freedom of speech."
— Soren Kierkegaard, 1843

"Democracy never lasts long. It soon wastes, exhausts, and murders itself. There was never a democracy yet that did not commit suicide."
— John Adams, 1814

Wednesday, March 29, 2023

Revisiting Basic Electrodynamics And E&M Theory (1)

 Electrodynamics plays a crucial role in both space physics and solar physics.  From the aurora, to the magnetosphere's response to the solar wind and the triggering phase of certain solar flares, electrodynamics factors in. So being exposed to the basic concepts of electrodynamics, as well as electricity and magnetism (E & M theory) is crucial to grasping the more advanced aspects of both space and solar physics as well as the fundamentals of electricity & magnetism.

For the most rudimentary intro we can begin by looking at some basic field derivations using the diagram shown above, based on a rectangular wire loop placed in a crossed (EB) field in a rectangular coordinate system (x,y, z). The field intensities EB are shown in the directions indicated.

We begin by the general definition for the magnetic flux, φ:

φ = B (A)

where B is the magnetic induction and A the area of the region through which the flux passes.

Now, from the diagram, since the area of the wire loop is: 

A =  dx

Then the magnetic flux through it is:

φ= B dx

and the rate of change of magnetic flux (from Faraday's law) is:

dφ/dt = d/dt {B ℓ dx} = dB/dt (ℓ dx)

Further, taking the integral around the loops we find:

 E d s  = -dφ/dt


E /  x) dx  = - dx (B /  t)

Thence: E /  x    =   B /  t 


 B d s = mo  ε   dx (d F/ d t )

And the electric flux through the rectangle is expressed:

¶ FE t  dx  ( E /  t)

where mo  is the magnetic permeability = 4 π x 10-7  H/m

and εo  is the electric permittivity of free space, viz.

εo = 8.85 x 10-12  F/m

Then write:

(¶B /  x)    dx  =  mo  ε   dx  ( E /  t)

or finally,

(¶B /  x)   =  - mo  ε  ( E /  t)

and if one takes the derivative of

E /  x    =   B /  t 

and combines with the preceding equation, one obtains the Maxwell wave equations:

1)  2E /  x2  =   moε E /  t2

2) ¶ 2B   x2  =   moε B /  t2

Note that in all these cases we are looking at plane wave solutions of the forms:

EE(max) cos (kx - wt)

B = B(max) cos (kx - wt)

where the 'max' indices refer to maximum amplitude values of the field components, E, B, and w = 2 πf, where f is the frequency. We also may use the magnetic intensity H in the wave equations, which is related to B by:
B =  mo H

Using the above, interested readers ought to be able to show that the respective equations for E, B satisfy the Maxwell wave equations given above,

Suggested Problems:

1) The generic wave equation for propagation of transverse waves, say on a string, is:

 2y/  x2  =   1/ v2   2y/  t2  

Where v is the wave velocity. For the Maxwell wave equations, however, we have v = c. Use this information to obtain the exact value of c - the speed of light - from one or other of the wave equations.

2) For a particular electromagnetic wave traveling in free space the power is given as P = 1300 Wm-2 .  Find the magnitudes of the wave vector components,  y  and z   if:

P = E y  z / c

Hint: The impedance of free space is given by:

Ömo  Ö εo    E y  z 

b)Find the magnitude of the magnetic flux density:  z

Write out all the component wave equations for E.

See Also: