File:Modeab25T.png

Principal mode $f$ of wave guided between the absorbing walls with damping parameter $\alpha=1/4$.

The thin black curve shows $|f(x)|$

The thick green curve shows $\arg(f)$

In the central part, $|x|\!<\!d~$, $~f(x)\!=\!\cos(px)$

Outside, $~f(x)=r \exp\Big( q \big(|x|-d\big)\Big)$

Parameters are expressed through function acosq:

$pd = \text{acosq}(\alpha) \approx 1.30652013112871 - 0.20108562381528\, \mathrm i $

$qd = pd \tan(pd) \approx 2.63604614403057 - 2.93518290714528 \, \mathrm i $

$r = \cos(pd) \approx 0.26650956316919 + 0.19541505906974 \, \mathrm i$

C++ generator of curves

//Files ado.cin and acosc.cin should be loaded to the working directory in order to compile the C++ code below.

#include <math.h> 
#include <stdio.h>
#include <stdlib.h>
#define DB double 
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

#include "ado.cin"

#include "acosc.cin"

z_type acosq(z_type z){ z_type c=z*exp(I*M_PI/4.); c=acosc(c); return c;}

z_type acosqq(z_type z){ z_type c=z*exp(I*M_PI/4.); c=acosc(c); return c*tan(c);}

DB Sazae= 2.798386045783887; // H
DB Tarao= -0.33650841691839534; // J
DB alpha=.25;
z_type pd= acosq(alpha);
z_type qd= pd*tan(pd);
z_type r = cos(pd);

z_type mode(DB x){ x=fabs(x); 
               if( x < 1 ) return cos(pd*x);
               x-=1.;
               return r* exp(- qd * x); }

#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
#define S(x,y) fprintf(o,"S\n",);
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
printf("%19.14lf %19.14lf\n",Re(pd),Im(pd));
printf("%19.14lf %19.14lf\n",Re(qd),Im(qd));
printf("%19.14lf %19.14lf\n",Re(r),Im(r));
FILE *o;o=fopen("modeab25.eps","w");ado(o,412,140);
fprintf(o,"210 14 translate\n 100 100 scale\n");
for(m=-2;m<3;m++){M(m,0)L(m,1)}
for(n=0;n<2;n++){M(-2,n)L(2,n)}
fprintf(o,"2 setlinecap .002 W 0 0 0 RGB S\n");
/*
DO(m,630){x=-2.01+.01*m; z=pd*x; y=Re(cos(z)); if(m==0)M(x,y)else L(x,y) }
fprintf(o,"1 setlinejoin 1 setlinecap .01 W 0 0 .8 RGB S\n");
DO(m,642){x=-2.01+.01*m; z=pd*x; y=Im(cos(z)); if(m==0)M(x,y)else L(x,y) }
fprintf(o,"1 setlinejoin 1 setlinecap .01 W .8 0 0 RGB S\n");
DO(m,190){x=-.13-.01*m; z=qd*(fabs(x)-1.); z_type t;t=r*exp(-z);y=Re(t); if(m==0)M(x,y)else L(x,y)}
DO(m,390){x= .13+.01*m; z=qd*(fabs(x)-1.); z_type t;t=r*exp(-z);y=Re(t); if(m==0)M(x,y)else L(x,y)}
//     y=Re(r*exp(-z)); // for some reasons the C++ dislikes this
fprintf(o,"1 setlinejoin 1 setlinecap .01 W 0 .7 0 RGB S\n");
DO(m,190){x=-.46-.01*m; z=qd*(fabs(x)-1.); z_type t;t=r*exp(-z);y=Im(t); if(m==0)M(x,y)else L(x,y)}
DO(m,390){x= .46+.01*m; z=qd*(fabs(x)-1.); z_type t;t=r*exp(-z);y=Im(t); if(m==0)M(x,y)else L(x,y)}
fprintf(o,"1 setlinejoin 1 setlinecap .01 W .7 0 .7 RGB S\n");
*/
DO(m,402){x=-2.01+.01*m; z=mode(x); y=abs(z); if(m==0)M(x,y) else L(x,y) }
fprintf(o,".005 W 0 0 0 RGB S\n");
DO(m,116){x= .01*m; z=mode(x); DB y=arg(z); if(y<0) y+=2*M_PI;  if(m==0)M(x,y) else L(x,y) }
DO(m,116){x=-.01*m; z=mode(x); DB y=arg(z); if(y<0) y+=2*M_PI;  if(m==0)M(x,y) else L(x,y) }
fprintf(o,".01 W 0 .8 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
     system("epstopdf modeab25.eps");
     system(    "open modeab25.pdf");
     getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% File modeab02.pdf should be generated with the code above in order to compile the Latex document below.

%<br> % Copyleft 2012 by Dmitrii Kouznetsov %<br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> %\paperwidth 1612pt %<br> \paperwidth 808pt %<br> \paperheight 250pt %<br> \topmargin -90pt %<br> \oddsidemargin -106pt %<br> \textwidth 900pt %<br> \textheight 900pt %<br> \pagestyle {empty} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \begin{document} %<br> \parindent 0pt %\sx{2}{ \begin{picture}(840,214) %<br> \sx{2}{ \begin{picture}(420,116) %<br> \put(4,4){\ing{modeab25}} %<br> %\put(215,211){\sx{1.6}{$y$}} %<br> \put(215,112){\sx{1.6}{\bf 1}} %<br> \put(216, 13){\sx{1.6}{\bf 0}} %<br> \put(103,4){\sx{1.6}{\bf -1}} %<br> \put(310,4){\sx{1.6}{\bf 1}} %<br> %\put(410,4){\sx{1.6}{\bf 2}} %<br> \put(390,4){\sx{1.6}{$x\!/\!d$}} %<br> \put(324,100){\sx{1.6}{$\arg\!\big(f(x)\big)$}}%<br> \put(328,44){\sx{1.6}{$|f(x)|$}}%<br> \end{picture} %<br> } %<br> \end{document} %