= Solution
A <standing wave> has a fixed spatial amplitude profile and oscillates in time, with stationary <nodes of a standing wave>. It arises from equal-amplitude waves of the same frequency travelling in opposite directions. For a uniform string with <linear mass density> $\mu$, the <wave speed> is $c=\sqrt{T/\mu}$ and a separated mode has $Y''+k^2Y=0$.
The fixed boundary requires $Y(0)=0$, while the free boundary requires $TY'(L)=0$. Thus $Y=A\sin kx$ and $\cos kL=0$, giving
$$
\boxed{k_n=\frac{(n+1/2)\pi}{L},\qquad
y_n(x,t)=A_n\sin(k_nx)\cos(ck_nt+\phi_n),\quad n=0,1,2,\ldots.}
$$
The fixed end is a <node of a standing wave> and the free end an <antinode>. Arbitrary small motions are superpositions of these <normal modes>; the <tension> alone determines their frequencies only when $\mu$ is also specified.
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