= Wave equation on a string
{title2=$y_{tt}=(T/\mu)y_{xx},\quad c=\sqrt{T/\mu}$}
An ideal uniform, flexible string under constant <tension> $T>0$ has transverse displacement $y(x,t)$ satisfying
$$
\mu y_{tt}=Ty_{xx},\qquad c=\sqrt{\frac{T}{\mu}},
$$
where $\mu>0$ is <linear mass density>, meaning mass per unit length of the straight reference string. The approximation assumes small slope $|y_x|\ll1$, transverse motion, negligible bending stiffness, negligible <damping>, and no distributed external force. Constant <tension> and uniform <linear mass density> make $c$ constant. Some texts use $\rho$ for this same <linear mass density>, giving $c=\sqrt{T/\rho}$; here $\rho$ must have units $\mathrm{kg\,m^{-1}}$, rather than units of volume <mass density>. <Math 220A: Partial Differential Equations of Applied Mathematics/5 Wave Equation in R>{ref}
The units provide a useful check: $T/\mu$ has units $(\mathrm{kg\,m\,s^{-2}})/(\mathrm{kg\,m^{-1}})=\mathrm{m^2\,s^{-2}}$. For $T=100\,\mathrm N$ and $\mu=0.010\,\mathrm{kg\,m^{-1}}$, the <wave speed> is $100\,\mathrm{m\,s^{-1}}$, so a pulse crosses a one-metre string in $0.010\,\mathrm s$. Quadrupling <tension> doubles <wave speed>; quadrupling <linear mass density> halves it. These square-root laws explain how material and loading set the time scale of the vibration.
For a length-$L$ string with fixed endpoints, the homogeneous <Dirichlet boundary conditions> are $y(0,t)=y(L,t)=0$. Its nonzero <normal modes> have
$$
\phi_n(x)=\sin\frac{n\pi x}{L},\qquad
\omega_n=\frac{cn\pi}{L},\qquad
\nu_n=\frac{\omega_n}{2\pi}=\frac{cn}{2L},\qquad n=1,2,\ldots.
$$
Thus $cn\pi/L$ is an <angular frequency>, in radians per second, whereas $cn/(2L)$ is the <frequency> in hertz. The <wavelength> is $2L/n$, and each <node of a standing wave> lies at $x=jL/n$ for some $j=0,\ldots,n$. For the numerical string above with $L=1\,\mathrm m$, the first three <frequencies> are $50$, $100$ and $150\,\mathrm{Hz}$, with <angular frequencies> $100\pi$, $200\pi$ and $300\pi\,\mathrm{s^{-1}}$. https://farside.ph.utexas.edu/teaching/315/Waveshtml/node28.html[Richard Fitzpatrick, Normal Modes of Uniform String]{ref}
\Image[/string-standing-modes.png]
{title=First three fixed-end string modes, showing opposite extrema and stationary nodes}
Each <normal mode> is a <standing wave>, but it also has a <travelling wave> description:
$$
\sin(kx)\cos(ckt)=\frac12\sin(k(x-ct))+\frac12\sin(k(x+ct)).
$$
At a fixed endpoint an incident displacement pulse reflects with reversed sign, allowing incident and reflected displacements to cancel at the endpoint. <Periodic reflection for a fixed-end string> implements both endpoints through odd, $2L$-periodic extensions of the initial data. This is the bridge between moving pulses and the stationary <Fourier sine basis>. https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/9e6f7f49eea8381d72965b469e1a3fdf_MIT2_062J_S17_Chap2.pdf[MIT, Wave Propagation, Section 2.4]{ref}
A short derivation comes from <Newton's second law>. The transverse force on a small segment is approximately $T[y_x(x+\delta x,t)-y_x(x,t)]$, and its mass is $\mu\delta x$. Dividing the force balance by $\delta x$ and taking the limit gives $\mu y_{tt}=Ty_{xx}$. <Separation of variables> then gives $\phi''+k^2\phi=0$; the fixed-end <boundary conditions> select $k=n\pi/L$, while the temporal equation is a <harmonic oscillator equation> with $\omega=ck$.
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