Tension in a waving rope

Derivation of the wave equation, by focussing on a small segment of a waving rope/string.

Edit and compile if you like:

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% Author: Izaak Neutelings (December 2020)
\documentclass[border=3pt,tikz]{standalone}
\usepackage{physics}
\usepackage{tikz}
\usetikzlibrary{calc} % for pic
\usetikzlibrary{angles,quotes} % for pic (angle labels)
%\usetikzlibrary{arrows.meta} % for arrow size
\tikzset{>=latex} % for LaTeX arrow head
\usepackage[outline]{contour} % glow around text
\contourlength{1.2pt}
\colorlet{xcol}{blue!70!black}
\colorlet{vcol}{green!60!black}
\colorlet{myred}{red!65!black}
\colorlet{metalcol}{blue!25!black!30!white}
\tikzstyle{vvec}=[->,vcol,very thick,line cap=round]
\tikzstyle{force}=[->,myred,thick,line cap=round]
\tikzstyle{Fproj}=[->,myred!50,thick,line cap=round]
\tikzstyle{rope}=[brown!30!black,double=brown!90!black,double distance=1.2,line width=0.4,line cap=round]
\tikzstyle{dark rope}=[brown!10!black,double=brown!60!black,double distance=1.2,line width=0.4]
\tikzstyle{metal}=[draw=metalcol!10!black,rounded corners=0.1,
  top color=metalcol,bottom color=metalcol!80!black,shading angle=10]
\tikzstyle{wood}=[draw=brown!80!black,rounded corners=0.1,
  top color=brown!80,bottom color=brown!80!black!80,shading angle=10]
\tikzstyle{width}=[<->] %{Latex[length=4,width=3]}-{Latex[length=4,width=3]}]
\newcommand{\vbF}{\vb{F}}
\def\wave#1#2{
  ({(#1-0.36)*\xmax},0) to[out=0,in=180,looseness=1] (#1*\xmax,#2*\A)
  to[out=0,in=180,looseness=1]++ (0.36*\xmax,#2*-\A)
}
\begin{document}
% VELOCITY - FORCES
\def\xmax{3.0}
\def\A{0.8}
\def\t{0.12}
\def\v{0.25*\xmax}
\def\R{0.60*\A}
\def\FT{0.7*\A} % tension
\def\Ft{0.8*\A} % tangential tension
 
 
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