Drawing wave propagation using Tikz

Question

I would like to demonstrate a homogenous spreading of a wave over the surface of a sphere using Tikz. I have had a look for similar projects on the Tikzample site but found nothing.

I would like to show a shaded 3D sphere, which only a sector of it demonstrating the inverse square law (similar to that shown here).

The arrows should demonstrate the homogeneous (even) spreading of the wave across the surface of the sphere.

Please let me know how this could be done. I am new to using Tikz. Thank you.

Answer 1

Here's one possibility:

The code (includes comments):

\documentclass[border=5pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing,arrows,positioning}

\definecolor{arrowred}{RGB}{255,16,16}
\definecolor{gridyellow}{RGB}{255,255,220}

% The 3D code is based on The drawing is based on Tomas M. Trzeciak's 
% `Stereographic and cylindrical map projections example`: 
% http://www.texample.net/tikz/examples/map-projections/
\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,thin,black] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,thin,dashed] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}%this is fake: for drawing the grid
\newcommand\DrawLongitudeCirclered[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (150:1) arc (150:180:1);
  %\draw[current plane,dashed] (-50:1) arc (-50:-35:1);
}%for drawing the grid
\newcommand\DLongredd[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,black,dashed, ultra thick] (150:1) arc (150:180:1);
}
\newcommand\DLatred[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane,dashed,black,ultra thick] (-50:1) arc (-50:-35:1);

}
\newcommand\fillred[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,red,thin] (\angVis:1) arc (\angVis:\angVis+180:1);

}

\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane,thin,black] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,thin,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}%Defining functions to draw limited latitude circles (for the red mesh)

\newcommand\DrawLatitudeCirclered[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  %\draw[current plane,red,thick] (-\angVis-50:1) arc (-\angVis-50:-\angVis-20:1);
\draw[current plane] (-50:1) arc (-50:-35:1);
}


\newcommand\DrawLatitudeCircleredCoord[4][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  %\draw[current plane,red,thick] (-\angVis-50:1) arc (-\angVis-50:-\angVis-20:1);
\draw[current plane] (-50:1) coordinate (#3) arc (-50:-35:1) coordinate (#4);
}

\newcommand\DrawLongitudeCircleredCoord[4][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (150:1) coordinate (#3) arc (150:180:1) coordinate (#4);
  %\draw[current plane,dashed] (-50:1) arc (-50:-35:1);
}%for drawing the grid

\tikzset{%
  >=latex,
  inner sep=0pt,%
  outer sep=2pt,%
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
    fill=black,circle}%
}

\begin{document}

\begin{tikzpicture}[scale=1,node/.style={minimum size=1cm}]
\def\R{4} % sphere radius
\def\angEl{15} % elevation angle
\def\angAz{-100} % azimuth angle
\def\angPhiOne{-50} % longitude of point P
\def\angPhiTwo{-35} % longitude of point Q
\def\angBeta{30} % latitude of point P and Q

%% working planes
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole

%drawing the sphere
\fill[ball color=white] (0,0) circle (\R);
\coordinate (O) at (0,0);
\coordinate[mark coordinate] (N) at (0,\H);
\coordinate[mark coordinate] (S) at (0,-\H);

\DrawLongitudeCircle[\R]{\angPhiOne} % pzplane
\DrawLongitudeCircle[\R]{\angPhiTwo} % qzplane
\DrawLatitudeCircle[\R]{\angBeta}
\DrawLatitudeCircle[\R]{0} % equator

%drawing the outermost grid and
%locating some special points in its border
\DrawLongitudeCircleredCoord[\R+6]{130}{11}{41}
\DrawLongitudeCircleredCoord[\R+6]{135}{12}{42}
\DrawLongitudeCircleredCoord[\R+6]{140}{13}{43}
\DrawLongitudeCircleredCoord[\R+6]{145}{14}{44}

\DrawLatitudeCircleredCoord[\R+6]{0}{41}{44}
\DrawLatitudeCircleredCoord[\R+6]{10}{31}{34}
\DrawLatitudeCircleredCoord[\R+6]{20}{21}{24}
\DrawLatitudeCircleredCoord[\R+6]{30}{11}{14}

%drawing the middle grid and
%locating some special points in its border
\DrawLongitudeCircleredCoord[\R+3]{130}{m11}{m31}
\DrawLongitudeCircleredCoord[\R+3]{137.5}{m12}{m32}
\DrawLongitudeCircleredCoord[\R+3]{145}{m13}{m33}

\DrawLatitudeCircleredCoord[\R+3]{0}{m31}{m34}
\DrawLatitudeCircleredCoord[\R+3]{15}{m21}{m24}
\DrawLatitudeCircleredCoord[\R+3]{30}{m11}{m14}

%drawing the grid on the sphere and
%locating some special points in its border
\DrawLongitudeCircleredCoord[\R]{130}{i11}{i21}
\DrawLongitudeCircleredCoord[\R]{145}{i12}{i22}

\DrawLatitudeCircleredCoord[\R]{0}{i21}{i22}
\DrawLatitudeCircleredCoord[\R]{30}{i11}{i12}


%locating coordinates for the arrows from the origin
\coordinate (P1) at  ( $ (11)!0.2!(43) $ ); 
\coordinate (P2) at  ( $ (13)!0.3!(24) $ );
\coordinate (P3) at  ( $ (12)!0.23!(34) $ );

\coordinate (P4) at  ( $ (11)!0.4!(43) $ );
\coordinate (P5) at  ( $ (21)!0.54!(34) $ );
\coordinate (P6) at  ( $ (12)!0.66!(34) $ );

\coordinate (P7) at  ( $ (31)!0.8!(42) $ ); 
\coordinate (P8) at  ( $ (31)!0.8!(43) $ ); 
\coordinate (P9) at  ( $ (34)!0.7!(43) $ ); 


%locating the coordinates for the dots on the grids
\foreach \Valor in {1,2,3,4,5,6,7,8,9} 
{
  \path[] (O) -- (P\Valor)
    coordinate[pos=1] (punto-out-\Valor) {}
    coordinate[pos=0.71] (punto-middle-\Valor) {}
    coordinate[pos=0.406] (punto-on-\Valor) {};
}

% draw arrows from the origin to the grid on the sphere
\foreach \Valor in {1,...,9}
{
  \draw[arrowred] (O) -- (punto-on-\Valor);
}

%drawing the frames from the origin to the grid on the sphere
\foreach \Valor in {11,12,21,22}
{
  \draw[thick] (O) -- (i\Valor);
}

% filling the grid on the sphere
% and redrawing it
\fill[gridyellow]
  (i11) -- 
  (i12) to[out=-57,in=90,looseness=0.6] 
  (i22) to[out=-150,in=20,looseness=0.6] 
  (i21) to[out=90,in=-68,looseness=0.6] 
  (i11);

%placing the dots on the grid on the sphere
\foreach \Valor in {1,2,3,4,5,6,7,8,9} 
{
  \node[fill,circle,fill=arrowred,inner sep=1.2pt] at (punto-on-\Valor) {};
}

% draw arrows from the grid on the sphere to the middle grid
\foreach \Valor in {1,...,9}
{
  \draw[arrowred] (punto-on-\Valor) -- (punto-middle-\Valor);
}

%drawing the frames from the grid on the sphere
% to the middle grid
\foreach \iValor/\fValor in {11/11,12/13,21/31,22/33}
{
  \draw[thick] (i\iValor) -- (m\fValor);
}

% filling the middle grid
% and redrawing it
\fill[gridyellow]
  (m11) -- 
  (m13) to[out=-57,in=90,looseness=0.6] 
  (m33) to[out=-150,in=20,looseness=0.6] 
  (m31) to[out=90,in=-68,looseness=0.6] 
  (m11);
\DrawLongitudeCircleredCoord[\R+3]{130}{m11}{m31}
\DrawLongitudeCircleredCoord[\R+3]{137.5}{m12}{m32}
\DrawLongitudeCircleredCoord[\R+3]{145}{m13}{m33}

\DrawLatitudeCircleredCoord[\R+3]{0}{m31}{m34}
\DrawLatitudeCircleredCoord[\R+3]{15}{m21}{m24}
\DrawLatitudeCircleredCoord[\R+3]{30}{m11}{m14}

%placing the dots on the middle grid
\foreach \Valor in {1,2,3,4,5,6,7,8,9} 
{
  \node[fill,circle,fill=arrowred,inner sep=1.2pt] at (punto-middle-\Valor) {};
}

% draw arrows from the middle grid to the outermost grid
\foreach \Valor in {1,...,9}
{
  \draw[arrowred] (punto-middle-\Valor) -- (punto-out-\Valor);
}

%drawing the frames from the middle grid 
% to the outermost grid
\foreach \iValor/\fValor in {11/11,13/14,31/41,33/44}
{
  \draw[thick] (m\iValor) -- (\fValor);
}

% filling the outermost grid
% and redrawing it
\fill[gridyellow]
  (11) -- 
  (14) to[out=-57,in=90,looseness=0.6] 
  (44) to[out=-150,in=20,looseness=0.6] 
  (41) to[out=90,in=-68,looseness=0.6] 
  (11);
\DrawLongitudeCircleredCoord[\R+6]{130}{11}{41}
\DrawLongitudeCircleredCoord[\R+6]{135}{12}{42}
\DrawLongitudeCircleredCoord[\R+6]{140}{13}{43}
\DrawLongitudeCircleredCoord[\R+6]{145}{14}{44}

\DrawLatitudeCircleredCoord[\R+6]{0}{41}{44}
\DrawLatitudeCircleredCoord[\R+6]{10}{31}{34}
\DrawLatitudeCircleredCoord[\R+6]{20}{21}{24}
\DrawLatitudeCircleredCoord[\R+6]{30}{11}{14}

%placing the dots on the outermost grid
\foreach \Valor in {1,2,3,4,5,6,7,8,9} 
{
  \node[fill,circle,fill=arrowred,inner sep=1.2pt] at (punto-out-\Valor) {};
}

%drawing arrows from the outermost grid
\foreach \Valor in {1,2,3,4,5,6,7,8,9} 
{
  \coordinate (end-arrow-\Valor) at ( $ (O)!1.2!(P\Valor) $ );
}  

%drawing arrows from the outermost grid
\foreach \Valor in {1,2,3,4,5,6,7,8,9} 
{
  \draw[arrowred,-latex] (punto-out-\Valor) -- (end-arrow-\Valor);
}  


%drawing grid on the spere
\foreach \t in {130,145} { \DrawLongitudeCirclered[\R]{\t} }
\foreach \t in {0,30} { \DrawLatitudeCirclered[\R]{\t} }

%drawing the middle grid
\foreach \t in {130,137.5,145} { \DrawLongitudeCirclered[\R+3]{\t} }
\foreach \t in {0,15,30} { \DrawLatitudeCirclered[\R+3]{\t} }

  %labels
\node[left] at (O) {$O$};
\node[below left=4pt and 1pt] at (i21) {$r$};
\node[below left=4pt and 1pt] at (m31) {$2r$};
\node[below left=4pt and 1pt] at (41) {$3r$};
\end{tikzpicture}

\end{document} 

The code is based on Marco Miani's Spherical and cartesian grids

Answer 2

Fairly minimal, very slow...

\documentclass[tikz,border=5]{standalone}
\tikzset{cs/3d/.code args={#1:#2:#3}{%
  \pgfpointxyz{(#1)*sin(#2)*cos(#3)}{(#1)*sin(#2)*sin(#3)}{(#1)*cos(#2)}}}
\tikzdeclarecoordinatesystem{3d}{\tikzset{cs/3d={#1}}}%     
\tikzset{spherical patch/.style args={#1:#2:#3:#4}{insert path={
  [smooth, samples=20, line join=round]
  plot [domain=0:#3] (3d cs:{#4}:{#1+\x}:{#2}) --
  plot [domain=0:#3] (3d cs:{#4}:{#1+#3}:{#2+\x}) --
  plot [domain=#3:0] (3d cs:{#4}:{#1+\x}:{#2+#3}) --
  plot [domain=#3:0] (3d cs:{#4}:{#1}:{#2+\x}) -- cycle}}}
\begin{document}
\tikz[x=(225:0.707cm),y=(0:1cm),z=(90:1cm), >=stealth]{
\foreach \i [evaluate={\z=30/(\i+1); \p=\i*2; \q=(\i+1)*2;}] in {0, 1, 2, 3}{
  \foreach \j/\k in {80/65, 80/85, 90/65, 100/75}
    \draw [red, -{\ifnum\i=3>\fi}] (3d cs:\p:\j:\k) -- (3d cs:\q:\j:\k);
  \ifnum\i<3
    \foreach \j/\k in {75/60, 75/90, 105/60, 105/90}
      \draw (3d cs:\p:\j:\k) -- (3d cs:\q:\j:\k);
    \foreach \t in {0,...,\i}\foreach \a in {0,...,\i}
      \draw [gray, fill=gray!10, fill opacity=0.75, 
        spherical patch={75+\t*\z:60+\a*\z:\z:\q}];
  \fi}}
\end{document}