= Fixed-end standing wave frequencies
{title2=$k_n=n\pi/L,\quad f_n=nc/(2L)$}
For the ideal uniform <wave equation on a string>, the <wave speed> is $c=\sqrt{\tau/\mu}$, where $\tau$ is constant <tension> and $\mu$ is <linear mass density>. Assume small transverse slopes, negligible bending stiffness and no <damping>. Fix both ends at zero <displacement>, giving the homogeneous <Dirichlet boundary conditions> $y(0,t)=y(L,t)=0$. Our <standing wave> already satisfies the first. For a nonzero <amplitude>, the second requires $\sin(kL)=0$, hence
$$
k_n=\frac{n\pi}{L},\qquad \lambda_n=\frac{2L}{n},\qquad \omega_n=ck_n,\qquad f_n=\frac{\omega_n}{2\pi}=\frac{nc}{2L},\qquad n=1,2,3,\ldots.
$$
The value $n=0$ gives zero <displacement> in this sine family, not a vibrating <normal mode>. Each positive $n$ fits $n$ half-<wavelengths> into the string: it has $n$ <antinodes>, $n-1$ interior <nodes of a standing wave>, and two endpoint <nodes of a standing wave>. For $L=1\,\mathrm m$ and $c=20\,\mathrm{m\,s^{-1}}$, $f_n=10n\,\mathrm{Hz}$; the third <normal mode> has $f_3=30\,\mathrm{Hz}$ and <wavelength> $2/3\,\mathrm m$.
\Image[/standing-wave-nodes-mode-spectrum.png]
{title=Third-mode nodes and antinodes, with the discrete frequency spectrum of a one-metre ideal string at wave speed twenty metres per second}
The equally spaced <eigenfrequencies> follow from this string's uniform <wave speed> and fixed-end <boundary conditions>. They are not a universal spectrum for every vibrating object. Changing an endpoint constraint changes the allowed <mode shapes>; changing <tension> changes <wave speed>. At fixed $L$ and $\mu$, quadrupling $\tau$ doubles every $f_n$. The more general construction of arbitrary initial motion belongs to <normal mode> and <wave equation on a string>.
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