= Solution
A free end has zero transverse force, hence $TY'(L)=0$, rather than zero displacement. With $Y(0)=0$, write $Y=A\sin kx$. The remaining condition is $\cos kL=0$, giving
$$
\boxed{k_n=\frac{(n+1/2)\pi}L,\qquad
y_n=A_n\sin(k_nx)\cos(ck_nt+\phi_n),\qquad n=0,1,2,\ldots.}
$$
The 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 $1:3:5$.
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