Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 11 / b

For a uniform string, let $c=\sqrt{T/\mu }$. A separated mode obeys $Y^{\prime \prime }+k^{2}Y=0$, $\omega =ck$. Fixed ends require $Y(0)=Y(L)=0$, soThe 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 $j=1,\dots ,n-1$.

A free end has zero transverse force, hence $T{Y}^{\prime }(L)=0$, rather than zero displacement. With $Y(0)=0$, write $Y=A\sin kx$. The remaining condition is $\cos kL=0$, givingThe first three shapes have one, three and five quarter-waves. Each has a node of a standing wave at the fixed end and an antinode at the free end; the mode frequencies are in the ratio $1:3:5$.

Free ends give $Y^{\prime }(0)=Y^{\prime }(L)=0$, so the oscillatory normal modes areThere is also the zero mode $k_0=0$, with uniform displacement $y_0(x,t)=a+bt$: a free string can translate without a restoring force. It is not a positive-frequency oscillation. The first three oscillatory shapes are $\cos (\pi x/L)$, $\cos (2\pi x/L)$ and $\cos (3\pi x/L)$, with an antinode at each end. The figure includes the uniform mode too, so that the zero-frequency limit is explicit.