= Solution
For a uniform string, let $c=\sqrt{T/\mu}$. A separated mode obeys $Y''+k^2Y=0$, $\omega=ck$. Fixed ends require $Y(0)=Y(L)=0$, so
$$
\boxed{k_n=\frac{n\pi}L,\qquad
y_n(x,t)=A_n\sin\frac{n\pi x}L\cos(ck_nt+\phi_n),\qquad n=1,2,\ldots.}
$$
The first three <normal modes> have one, two and three half-waves respectively. Their interior <nodes of a standing wave> are at $x=jL/n$ for $j=1,\ldots,n-1$.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit-string-boundary-modes.png]
{title=The first oscillatory string modes for fixed and free boundaries}
{description=Original analytic mode shapes for three boundary conditions. Vertical offsets separate the shapes; the free-free column also includes the uniform zero mode.}
Back to article page
