Periodic shading in tikz

Question

Is there an easy way to shade a rectangle in tikz so that the color changes not from one to the other but several times in between. Something like a grayscale value changing with the position x like cos(x).

A linear variation instead of a sinusoidal one would be fine: I'd jsut like it to mimic an interference pattern.

Answer 1

You could try shadings:

\documentclass[tikz,border=5]{standalone}
\begin{document}
\tikz[x=0.125cm,y=0.125cm]
  \foreach \i in {0,1,...,21}
    \path [left color=black, right color=white, shading angle={mod(\i,20)*180+90}]
    (\i*.9,0) rectangle ++(1,10);
\end{document}

Or fadings:

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{fadings}
\begin{tikzfadingfrompicture}[name=interference]
\foreach \i in {0,1,...,21}
    \path [left color=transparent!0, right color=transparent!100, shading angle={mod(\i,2)*180+90}]
    (\i*.9,0) rectangle ++(1,20);
\end{tikzfadingfrompicture}
\begin{document}
\foreach \i in {0,...,24}{
\begin{tikzpicture}
\clip (-5,-5) rectangle ++(10,10);
\path [fill=black, path fading=interference, fit fading=false, fading transform={rotate=\i*7.2}] (-10,-5]) rectangle ++(20,10);
\path [fill=black, path fading=interference, fit fading=false] (-10,-5) rectangle ++(20,10);
\end{tikzpicture}
}
\end{document}

In both cases using \i*.9 is to workaround some annoying white lines that appear between adjacent shadings which may (or may not) be viewer artifacts.

To change the phase of the interference pattern is tricky to do with shading but with fadings it is fairly easy if the fading is specified over a larger area than the required path. Then the fading transform key can be used to shift the fading that is "seen through" the path.

In the following example the red rectangle illustrates the position of the fading relative to the area that is faded. Also, a line is drawn to fill in the white lines between adjacent shadings instead of the workaround described above to get rid of the white lines that appear between adjacent shadings.

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{fadings}
\begin{tikzfadingfrompicture}[name=interference]
\foreach \i in {0,1,...,19}{
  \path [left color=transparent!0, right color=transparent!100, shading angle={mod(\i,2)*180+90}]
  (\i,0) rectangle ++(1,10);
  \ifodd\i\else% Fill in gap
    \path [draw=transparent!0] (\i,0) -- ++(0,10);
  \fi
}
\end{tikzfadingfrompicture}
\begin{document}
\foreach \i in {-10,...,10,9,8,...,-9}{%
\begin{tikzpicture}
\path [draw=red, shift=(0:\i/2)] (-10,-5) rectangle (10,5);
\path [fill=black, path fading=interference, fit fading=false, fading transform={shift=(0:\i/2)}] (-5,-5) rectangle (5,5);
\useasboundingbox (-15,-5) rectangle (15,5);
\end{tikzpicture}%
}
\end{document}

Using fadings means you can also do cool stuff like this:

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{fadings}
\begin{tikzfadingfrompicture}[name=interference]
\foreach \i in {1,...,15}
    \foreach \j in {1,...,25}
       \path [line width=\j, draw=transparent!0,opacity=1/30]
    (0:\i) arc (0:180:\i);
\end{tikzfadingfrompicture}
\begin{document}
\begin{tikzpicture}
\clip (-5,-5) rectangle ++(10,10);
\path [fill=black, path fading=interference, fit fading=false] (-10,-5) rectangle ++(20,10);
\path [fill=black, path fading=interference, fit fading=false, fading transform={rotate=45}] (-10,-5) rectangle ++(20,10);
\end{tikzpicture}

\end{document}

Answer 2

You can draw yourself the shade pattern, by filling lots of thin rectangles, each one with a solid gray computed via a sin() function:

\usetikzlibrary{calc}

\begin{tikzpicture}
\def\detail{300}
\pgfmathsetmacro{\width}{8/\detail}
\foreach \i in {0,...,\detail} {
  \pgfmathsetmacro{\shade}{45+45*sin(\i*10)}
  \fill[fill=black!\shade!white] (\i*\width,0) rectangle +(\width, 2);
}

\end{tikzpicture}  

Update

Addressing Steve's question in a comment, here is a new version of the code, more customizable and with better variable names:

\usetikzlibrary{calc}

\begin{tikzpicture}

% ---- Customizable section
\def\numsamples{200}   % The greater, the thinner each sample
\def\numcycles{8}      % Number of "white bars" in the result
\def\totalwidth{10}    % Width (in cm) of the resulting box
% ---- End of customizable section

\pgfmathsetmacro{\samplewidth}{\totalwidth/\numsamples}
\pgfmathsetmacro{\frequency}{360/\numsamples*\numcycles}
\foreach \i in {0,...,\numsamples} {
  \pgfmathsetmacro{\shade}{50+50*sin(\i*\frequency)}
  \fill[draw=none, fill=black!\shade!white] (\i*\samplewidth,0) rectangle +(\samplewidth*1.1, 2);
}
\end{tikzpicture} 

The above code produces the following "low res" result:

Increasing \numsamples to 500, you get a better result:

Answer 3

Actually, using pgfplots and surface plots from gnuplot turned out to be more flexible.

Here is an exemple of the interference pattern of 7 point sources with the following parameters * \Lambda is the wavelength * \DistanceSources is the constant distance between the sources * the amplitude can be chosen with the function "amorti" but will be constant within a radius of \Cutoff from the source * \RetardIIvsI is the constant phase shift between two adjacent sources * \Date is the constant phase added to all sources

\documentclass{standalone}
\usepackage{tikz,pgfplots}
\begin{document}
\begin{tikzpicture}[]
   \pgfmathsetmacro{\xmin}{-}6
   \pgfmathsetmacro{\ymin}{-6}
   \pgfmathsetmacro{\xmax}{6}
   \pgfmathsetmacro{\ymax}{6}
   \pgfmathsetmacro{\Lambda}{.5}
   \pgfmathsetmacro{\DistanceSources}{\Lambda/2}
   \pgfmathsetmacro{\CentreI}{-4*\DistanceSources}
   \pgfmathsetmacro{\CentreII}{\CentreI+\DistanceSources}
   \pgfmathsetmacro{\CentreIII}{\CentreII+\DistanceSources}
   \pgfmathsetmacro{\CentreIV}{\CentreIII+\DistanceSources}
   \pgfmathsetmacro{\CentreV}{\CentreIV+\DistanceSources}
   \pgfmathsetmacro{\CentreVI}{\CentreV+\DistanceSources}
   \pgfmathsetmacro{\CentreVII}{\CentreVI+\DistanceSources}
   \pgfmathsetmacro{\Cutoff}{\Lambda/10}
   \pgfmathsetmacro{\RetardIIvsI}{pi/2}
   \pgfmathsetmacro{\Date}{0}

   \begin{axis}[colormap/blackwhite,
      view ={0}{90},
      xlabel = $x$,
      ylabel = $y$,
      extra x ticks = ,
      extra x tick labels = ,
      extra y ticks = ,
      extra y tick labels = ,
      xmin = \xmin,
      ymin = \ymin,
      xmax = \xmax,
      ymax = \ymax,
      domain = \xmin:\xmax,
      samples = 50
      ]
      \addplot3[surf,shader=interp,raw gnuplot] gnuplot {
          set isosamples 50,50;
          amorti (centre,xy,y,cutoff)= 1;
          interf(centre,date,x,y,cutoff,lambda,retard) =%
          amorti(centre,x,y,cutoff)%
          * cos(date-2*pi*sqrt((x-centre)**2 + y**2)/lambda-retard);
          splot [x=\xmin:\xmax] [y=\ymin:\ymax] 1./7.*(%
             interf(\CentreI,\Date,x,y,\Cutoff,\Lambda,0)%
           + interf(\CentreII,\Date,x,y,\Cutoff,\Lambda,\RetardIIvsI)%
           + interf(\CentreIII,\Date,x,y,\Cutoff,\Lambda,2*\RetardIIvsI)%
           + interf(\CentreIV,\Date,x,y,\Cutoff,\Lambda,3*\RetardIIvsI)%
           + interf(\CentreV,\Date,x,y,\Cutoff,\Lambda,4*\RetardIIvsI)%
           + interf(\CentreVI,\Date,x,y,\Cutoff,\Lambda,5*\RetardIIvsI)%
           + interf(\CentreVII,\Date,x,y,\Cutoff,\Lambda,6*\RetardIIvsI)%
           ) 
          };
      \end{axis}
  \end{tikzpicture}
 \end{document}