Here are some of the pictures from my master thesis. The topic were transmission and reflexion of (sound) waves at the open end of a tube.
All images are made with TikZ and pgfplots. Many thanks to Christian Feuersänger for showing how to do the wave shadings!
All images are shown in one document, which I split in parts for this answer. To compile it on your machine you must put all fragments in one document; and gnuplotis required, too.
% !TeX encoding = utf8
\documentclass[
11pt,cmyk,
multi={tikzpicture},
border=10mm,
]{standalone}
% General packages
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{helvet}
\usepackage[garamond]{mathdesign}
\usepackage{mathtools}
% Colors
\usepackage{xcolor}
\definecolor{spot}{cmyk}{1,0.20,0,0}
\colorlet{gray}{black!60}
\colorlet{wave}{spot}
% Load TikZ and libraries
\usepackage{tikz}
\usetikzlibrary{calc,positioning,decorations.pathmorphing,%
arrows.meta,backgrounds}
% Define TikZ styles etc.
\tikzset{
every plot/.style={
prefix=plots/,
samples=200,
},
every node/.style={
font=\footnotesize,
},
line width=1pt,
shorten/.style={
shorten >=#1,
shorten <=#1,
},
>=Triangle[],
axis/.style={-Stealth[]},
tick/.style={
axis,
shorten <=-0.5\pgflinewidth,
shorten >=-0.5\pgflinewidth,
},
}
% Load and configure pgfplots
\usepackage{pgfplots}
\usepgfplotslibrary{polar}
\pgfplotsset{
compat=1.11,
wave axis/.style={
view={0}{90},
hide axis,
colormap={custom}{color=(white) color=(wave)},
trig format plots=rad,
trig format=rad,
x=1cm,
y=1cm,
z=0cm,
anchor=origin,
},
wave plot/.style={
samples=200,
samples y=2,
surf,shader=interp,
},
polar wave plot/.style={
data cs=polar,
domain=0:2*pi,
samples y=100,
surf,shader=interp,
},
}
% Macro for subfigure captions
\newcommand{\subfig}[1]{\textbf{\textsf{#1)}}~}
\begin{document}
Modes of a wave in a tube
\begin{tikzpicture}[
tube/.style={ultra thick,black},
wave/.style={thick,gray},
]
% define variables
%% tube size
\def\H{.9}
\def\L{5.5}
%% distances
\def\A{0.3}
\def\B{0.5}
%% wave's amplitude
\pgfmathsetmacro\a{0.35}
% both ends open or closed
\foreach \n in {1,2,3,4,5} {
\begin{scope}[shift={(0,-\n*\H-\n*\A)}]
% closed ends
%% wave
\begin{scope}
% air pressure
\begin{axis}[wave axis]
\addplot3[wave plot,domain=0:\L,domain y=-\H/2:\H/2] {abs(sin(pi/\L * \n * x))};
% \addplot[domain=0:\L,samples=100] function {abs(sin(pi/\L * \n * x))};
\end{axis}
% wave form
\draw [wave]
plot [id=moden-gg-1-\n, domain=0:\L] function {\a * sin(pi/\L * \n * x)}
plot [id=moden-gg-2-\n, domain=0:\L] function {-\a * sin(pi/\L * \n * x)};
\end{scope}
%% tube
\draw [tube] (\L,-\H/2) -| (0,\H/2) -- (\L,\H/2) -- cycle;
%% coordinates for later use
\coordinate (GG-\n) at (0,0);
\coordinate (B1) at (0,-\H/2);
% open ends
\begin{scope}[shift={(\L+\B,0)}]
%% wave
\begin{scope}
% air pressure
\begin{axis}[wave axis]
\addplot3[wave plot,domain=0:\L,domain y=-\H/2:\H/2] {abs(cos(pi/\L * \n * x))};
% \addplot[domain=0:\L,samples=100] function {abs(cos(pi/\L * \n * x))};
\end{axis}
% wave
\draw [wave]
plot [id=moden-oo-1-\n, domain=0:\L] function {\a * cos(pi/\L * \n * x)}
plot [id=moden-oo-2-\n, domain=0:\L] function {-\a * cos(pi/\L * \n * x)};
\end{scope}
%% tube
\draw [tube] (0,-\H/2) -- (\L,-\H/2) (0,\H/2) -- (\L,\H/2);
%% coordinates
\coordinate (B2) at (0,-\H/2);
\end{scope}
\end{scope}
}
\foreach \n in {1,3,5} {
\begin{scope}[shift={(0,-\n*\H-\n*\A)}]
% one end open, one closed
\begin{scope}[shift={(2*\L+2*\B,0)}]
%% wave
\begin{scope}
% air pressure
\begin{axis}[wave axis]
\addplot3[wave plot,domain=0:\L,domain y=-\H/2:\H/2] {abs(sin(pi/\L * (\n-0.5) * x))};
% \addplot[domain=0:\L,samples=100] function {abs(sin(pi/\L * (\n-0.5) * x))};
\end{axis}
% wave
\draw [wave]
plot [id=moden-go-1-\n, domain=0:\L] function {\a * sin(pi/\L * (\n-0.5) * x)}
plot [id=moden-go-2-\n, domain=0:\L] function {-\a * sin(pi/\L * (\n-0.5) * x)};
\end{scope}
%% tube
\draw [tube] (\L,-\H/2) -| (0,\H/2) -- (\L,\H/2);
%% coordinates
\coordinate (B3) at (0,-\H/2);
\end{scope}
\end{scope}
}
% captions/text
\foreach \n in {1,2,3,4,5} {
\node at (GG-\n) [rotate=90,left=4pt,anchor=south,inner sep=0pt] {$n=\n$};
}
\node at (B1) [below right=1.5mm and 0mm,inner sep=0pt] {\subfig{a}{closed/closed}};
\node at (B2) [below right=1.5mm and 0mm,inner sep=0pt] {\subfig{b}{open/open}};
\node at (B3) [below right=1.5mm and 0mm,inner sep=0pt] {\subfig{c}{closed/open}};
\end{tikzpicture}
Wave moves through five points (particles)
\begin{tikzpicture}
% define variables
%% wave
\pgfmathsetmacro\T{9}
\pgfmathsetmacro\A{1.75}
%% oscillations
\pgfmathsetmacro\Ti{1.5}
\pgfmathsetmacro\Ai{0.6}
\pgfmathsetmacro\yMax{1}
\pgfmathsetmacro\xMax{2*\Ti+0.3}
\pgfmathsetmacro\D{2*\Ti+0.2}
\coordinate (S) at (0,-2.7);
% wave
%% axis
\draw [axis] (0,-2) -- (0,2.6) node [left] {$z(x,0)$};
\draw [axis] (0,0) -- (10,0) node [below] {$x$};
\draw [tick, |-] (0,\A) node [left] {$z_\text{m}$} -- (0,0);
%% wave langth
\draw [|-|] (0,2.2) -- ++ (\T,0) node [midway,above] {$\lambda$};
%% wave form
\draw [ultra thick, gray] plot [id=welle, domain=-\yMax-0.1:\T+\yMax]
function {\A*sin(2*pi/\T*x)};
\pgfmathsetmacro\X{0.425*\T}
\pgfmathsetmacro\Y{\A*sin(2*pi/\T*\X r)}
\draw [gray] (\X,\Y) -- ++(35:0.7) node [right,align=left]
{\textbf{snapshot} of wave\\ at time $t=0$};;
%% oscillating points
\coordinate (1) at (0.00*\T,0);
\coordinate (2) at (0.25*\T,\A);
\coordinate (3) at (0.50*\T,0);
\coordinate (4) at (0.75*\T,-\A);
\coordinate (5) at (1.00*\T,0);
\coordinate (6) at (1.25*\T,\A);
\foreach \n in {1,2,3,4,5} {
\node (n\n) at (\n) [
circle,
font=\sffamily\scriptsize,
spot,
draw, ultra thick,
fill=white,
inner sep=0pt,
minimum size=3mm,
outer sep=1mm,
] {\n};
}
%% movment of points
\draw [spot,thick,->] (n1) -- ++(0,-0.5*\A);
\draw [spot,thick,->] (n2) -- ++(0,-0.5*\A);
\draw [spot,thick,->] (n3) -- ++(0,0.5*\A);
\draw [spot,thick,->] (n4) -- ++(0,0.5*\A);
\draw [spot,thick,->] (n5) -- ++(0,-0.5*\A);
% oscillations
\begin{scope}[shift={($(1)+(S)$)}, rotate=-90]
%% axis
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_1(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
%% sine form
\draw [thick, spot] plot [id=welle-schwingung-1, domain=0:\D]
function {-\Ai*sin(2*pi/\Ti*x))};
%% ponts
\node (n) at (0,0) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {1};
%% movment
\draw [spot,thick,-{Triangle[scale=0.7]}] (n) -- ++(0,-\Ai);
%% origin coordinate for later use
\coordinate (U1) at (0,0);
%% root coordinate for later use
\coordinate (N1) at (3*\Ti/4,0);
\end{scope}
\begin{scope}[shift={($(2)+(S)+(0,-\A)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_2(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=welle-schwingung-2, domain=0:\D]
function {\Ai*sin(2*pi/\Ti*x+pi/2)};
\node (n) at (0,\Ai) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {2};
\draw [spot,thick,-{Triangle[scale=0.7]}] (n) -- ++(0,-\Ai);
\coordinate (N2) at (4*\Ti/4,0);
\end{scope}
\begin{scope}[shift={($(3)+(S)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_3(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=welle-schwingung-3, domain=0:\D]
function {-\Ai*sin(2*pi/\Ti*x+pi)};
\node (n) at (0,0) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {3};
\draw [spot,thick,-{Triangle[scale=0.7]}] (n) -- ++(0,\Ai);
\coordinate (N3) at (5*\Ti/4,0);
\end{scope}
\begin{scope}[shift={($(4)+(S)+(0,\A)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_4(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=welle-schwingung-4, domain=0:\D]
function {\Ai*sin(2*pi/\Ti*x+3*pi/2)};
\node (n) at (0,-\Ai) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {4};
\draw [spot,thick,-{Triangle[scale=0.7]}] (n) -- ++(0,\Ai);
\coordinate (N4) at (6*\Ti/4,0);
\end{scope}
\begin{scope}[shift={($(5)+(S)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_5(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=welle-schwingung-5, domain=0:\D]
function {-\Ai*sin(2*pi/\Ti*x+2*pi)};
\node (n) at (0,0) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {5};
\draw [spot,thick,-{Triangle[scale=0.7]}] (n) -- ++(0,-\Ai);
\coordinate (U5) at (0,0);
\coordinate (N5) at (7*\Ti/4,0);
\end{scope}
% help lines
\foreach \x in {0.25,0.5,...,2} {
\begin{scope}[on background layer]
\draw [dotted] ($(U1)+(-\yMax,-\x*\Ti)$) -- ($(U5)+(\yMax,-\x*\Ti)$);
\end{scope}
}
% pahse shift
\foreach \n [remember=\n as \lastn (initially 1)] in {2,3,4,5} {
\draw [gray,thick] (N\lastn) -| ($(N\lastn)!0.5!(N\n)$) |- (N\n);
}
\draw [gray,thick] (N1) -- ++(-\yMax,0);
\draw [gray,thick] (N5) -- ++(\yMax,0);
\end{tikzpicture}
Standing wave
\begin{tikzpicture}
% define variables
%% wave
\pgfmathsetmacro\T{9}
\pgfmathsetmacro\A{1.75}
%% oscillations
\pgfmathsetmacro\Ti{0.9}
\pgfmathsetmacro\Ai{0.3}
\pgfmathsetmacro\yMax{0.5}
\pgfmathsetmacro\xMax{2*\Ti+0.3}
\pgfmathsetmacro\D{2*\Ti+0.1}
\coordinate (S) at (0,-2.7);
% wave
%% axis
\draw [axis] (0,-2) -- (0,2.6) node [left] {$z(x,t_0)$};
\draw [axis] (0,0) -- (10,0) node [below] {$x$};
\draw [tick, |-] (0,\A) node [left] {$z_\text{m}$} -- (0,0);
%% wave length
\draw [|-|] (0,2.2) -- ++ (\T,0) node [midway,above] {$\lambda$};
%% wave form
\draw [ultra thick, gray] plot [id=stehende-welle, domain=-\yMax-0.1:\T+\yMax]
function {\A*sin(2*pi/\T*x)};
\pgfmathsetmacro\X{0.425*\T}
\pgfmathsetmacro\Y{\A*sin(2*pi/\T*\X r)}
\draw [gray] (\X,\Y) -- ++(35:0.7) node [right,align=left]
{\textbf{snapshot} of wave\\ at time $t=t_0$};
%% osizllationg points (II)
\coordinate (1) at (0.00*\T,0);
\coordinate (2) at (0.25*\T,\A);
\coordinate (3) at (0.50*\T,0);
\coordinate (4) at (0.75*\T,-\A);
\coordinate (5) at (1.00*\T,0);
\pgfmathsetmacro\X{0.125*\T}
\pgfmathsetmacro\Yvi{\A*sin(2*pi/\T*\X r)}
\coordinate (6) at (\X,\Yvi);
\pgfmathsetmacro\X{0.375*\T}
\pgfmathsetmacro\Yvii{\A*sin(2*pi/\T*\X r)}
\coordinate (7) at (\X,\Yvii);
\pgfmathsetmacro\X{0.625*\T}
\pgfmathsetmacro\Yviii{\A*sin(2*pi/\T*\X r)}
\coordinate (8) at (\X,\Yviii);
\pgfmathsetmacro\X{0.875*\T}
\pgfmathsetmacro\Yix{\A*sin(2*pi/\T*\X r)}
\coordinate (9) at (\X,\Yix);
\foreach \n in {1,2,3,4,5,6,7,8,9} {
\node (n\n) at (\n) [
circle,
font=\sffamily\scriptsize,
spot,
draw, ultra thick,
fill=white,
inner sep=0pt,
minimum size=3mm,
outer sep=1mm,
] {\n};
}
%% movment of points
\draw [spot,thick,->] (n2) -- ++(0,-0.5*\A);
\draw [spot,thick,->] (n4) -- ++(0,0.5*\A);
\draw [spot,thick,->] (n6) -- ++(0,-0.5*\Yvi);
\draw [spot,thick,->] (n7) -- ++(0,-0.5*\Yvii);
\draw [spot,thick,->] (n8) -- ++(0,-0.5*\Yviii);
\draw [spot,thick,->] (n9) -- ++(0,-0.5*\Yix);
% oscillations
\begin{scope}[shift={($(1)+(S)$)}, rotate=-90]
%% axis
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_1(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
%% sine form
\draw [thick, spot] plot [id=stehende-welle-schwingung-1, domain=0:\D]
function {0};
%% oscillating point (particle)
\node (n) at (0,0) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {1};
%% origin coordinate for later use
\coordinate (U1) at (0,0);
%% root coordinate for later use
\coordinate (N1) at (5*\Ti/4,0);
\end{scope}
\begin{scope}[shift={($(2)+(S)+(0,-\A)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_2(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=stehende-welle-schwingung-2, domain=0:\D]
function {\Ai*sin(2*pi/\Ti*x+pi/2)};
\node (n) at (0,\Ai) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {2};
\end{scope}
\begin{scope}[shift={($(3)+(S)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_3(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=stehende-welle-schwingung-3, domain=0:\D]
function {0};
\node (n) at (0,0) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {3};
\end{scope}
\begin{scope}[shift={($(4)+(S)+(0,\A)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_4(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=stehende-welle-schwingung-4, domain=0:\D]
function {-\Ai*sin(2*pi/\Ti*x+pi/2)};
\node (n) at (0,-\Ai) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {4};
\end{scope}
\begin{scope}[shift={($(5)+(S)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_5(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=stehende-welle-schwingung-5, domain=0:\D]
function {0};
\node (n) at (0,0) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {5};
\coordinate (U5) at (0,0);
\coordinate (N5) at (5*\Ti/4,0);
\end{scope}
\begin{scope}[shift={($(6)+(S)-(0,\Yvi)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_6(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=stehende-welle-schwingung-6, domain=0:\D]
function {\Yvi/\A*\Ai*sin(2*pi/\Ti*x+pi/2)};
\node (n) at (0,\Yvi/\A*\Ai) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {6};
\coordinate (U6) at (0,0);
\end{scope}
\begin{scope}[shift={($(7)+(S)-(0,\Yvii)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_7(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=stehende-welle-schwingung-7, domain=0:\D]
function {\Yvii/\A*\Ai*sin(2*pi/\Ti*x+pi/2)};
\node (n) at (0,\Yvii/\A*\Ai) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {7};
\coordinate (U7) at (0,0);
\end{scope}
\begin{scope}[shift={($(8)+(S)-(0,\Yviii)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_8(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=stehende-welle-schwingung-8, domain=0:\D]
function {\Yviii/\A*\Ai*sin(2*pi/\Ti*x+pi/2)};
\node (n) at (0,\Yviii/\A*\Ai) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {8};
\coordinate (U8) at (0,0);
\end{scope}
\begin{scope}[shift={($(9)+(S)-(0,\Yix)$)}, rotate=-90]
\draw [axis] (0,-\yMax) -- (0,\yMax) node [above,midway] {$z_9(t)$};
\draw [axis] (0,0) -- (\xMax,0) node [right] {$t$};
\draw [thick, spot] plot [id=stehende-welle-schwingung-9, domain=0:\D]
function {\Yix/\A*\Ai*sin(2*pi/\Ti*x+pi/2)};
\node (n) at (0,\Yix/\A*\Ai) [
circle,
font=\sffamily\tiny,
spot,
draw, thick,
fill=white,
inner sep=0pt,
minimum size=2mm,
outer sep=0.5mm,
] {9};
\coordinate (U9) at (0,0);
\end{scope}
% help lines
\foreach \x in {0.25,0.5,...,2} {
\begin{scope}[on background layer]
\draw [dotted] ($(U1)+(-\yMax,-\x*\Ti)$) -- ($(U5)+(\yMax,-\x*\Ti)$);
\end{scope}
}
% (no) phase shift
\draw [gray,thick] ($(N1)-(\yMax,0)$) -- ($(N5)+(\yMax,0)$);
\end{tikzpicture}
Organ pipe
\begin{tikzpicture}
% define varaibles
%% pipe foot
\def\F{2.25}
\def\r{0.1}
%% wave length
\def\w{20}
%% amplitude
\def\z{0.5}
%% wave body
\def\R{0.6}
\pgfmathsetmacro\l{0.6*\R}
\pgfmathsetmacro\L{\w/2-\l}
%% cut
\def\A{0.8}
\def\hA{0.4}
%% distance for captions
\pgfmathsetmacro\B{\R+0.3}
% wave
\draw [ultra thick, spot,fill=spot!20] plot [id=pfeifen-welle-1,domain=0:\w/2] function
{\z*sin(2*pi/\w*x)};
\draw [ultra thick, spot,fill=spot!20] plot [id=pfeifen-welle-2,domain=0:\w/2] function
{-\z*sin(2*pi/\w*x)};
% axis
\draw [axis] (-\F-\hA-\A/2,0) -- (\w/2+0.35,0) node [below left] {$x$};
% pipe
\draw [ultra thick] (-\F-\hA-\A/2,-\r) -- (-\hA-\A/2,-\R) -- (\L,-\R)
(-\F-\hA-\A/2,\r) -- (-\hA-\A/2,\R) -- ++(\hA,0) ++(\A,0) -- (\L,\R);
;
\draw [line width=3pt] (0,-\R) -- (0,\R-\A/8);
\draw [dashed] (\L,-\R) -- (\L,\R);
% captions
\node at (\w/4,0) [spot,fill=spot!20] {$\Delta p(x,0)$};
\draw [tick, |-|] (-\F-\hA-\A/2,-\B) -- (0,-\B) node [midway,below] {foot length};
\draw [tick, |-|] (0,-\B) -- (\L,-\B) node [midway,below] {reduced length $L_\text{r}$};
\draw [tick, |-|] (\L,-\B) -- (\L+\l,-\B) node [midway,below] {$\ell$};
\draw [tick, |-|] (0,\B) -- (\w/2,\B) node [midway,above] {theoretical length $L$};
\end{tikzpicture}
Helmholtz’ model for the open end
\begin{tikzpicture}
% define variables
\def\H{0.275}
\def\S{1.8}
\def\B{8}
\def\T{4.5}
\def\l{2.6}
\def\s{0.3}
\pgfmathsetmacro\w{atan((\S+\H/2)/\T)}
\pgfmathsetmacro\R{sqrt(\T^2+(\S+\H/2)^2)}
%% wave parameters
\pgfmathsetmacro\wL{6.5*\H}
\pgfmathsetmacro\wA{\H/2}
\pgfmathsetmacro\D{10*\wL}
% plane wave
\begin{axis}[wave axis]
\addplot3[wave plot,domain=-\B:0.05,domain y=-\wA:\wA] {abs(sin(pi/\wL * x))};
% \addplot[domain=-\D:0,samples=200] function {abs(cos(pi/\wL * x))};
\end{axis}
% radial wave
\begin{scope}
% \clip (0,-\H/2-\S) -- (-\w:\R) arc [start angle=-\w, end angle=\w, radius=\R]
% -- (0,\H/2+\S) -- cycle;
\clip (0,-\H/2-\S) rectangle (\T,\H/2+\S);
\begin{axis}[wave axis]
\addplot3[polar wave plot,domain y=0:2*\T] function {abs(sin(pi/\wL * y))*exp(-0.2*y)};
% \addplot[domain=0:2*\T,samples=200] function {abs(cos(pi/\wL * x))*exp(-0.2*x)};
\end{axis}
\end{scope}
% tube
\draw [ultra thick] (-\B,\H/2) -| (0,\H/2+\S);
\draw [ultra thick] (-\B,-\H/2) -| (0,-\H/2-\S);
% axis
\draw [axis] (-\B,0) -- ($(\T,0)-(0.25,0)$) node [below left] {$x$};
\draw [tick,|-] (0,0) node [below right=2pt and 2.5pt,inner sep=0pt] {$0$} -- (1,0);
\draw [axis] (0,0) -- (25:2.5) node [below] {$\vec{r}$};
% captions
\draw (-2.5*\wL,0.25*\H) -- ++(65:.6) node [above] {$\psi_\text{i}$};
\draw (60:\wL/1.9) -- ++(180:1.1) node [left] {$\psi_\text{a}$};
\node at (0,-\H/2) [below left, align=right] {cross sectional\\area $A$};
\end{tikzpicture}
\end{document}
