= Solution
Fixed endpoints require $k_n=n\pi/L$, with $n=1,2,\ldots$. Writing $c_s=\sqrt{T/\rho}$,
$$
\boxed{\omega_1=\frac{\pi c_s}{L},\qquad\omega_2=\frac{2\pi c_s}{L}.}
$$
For equal <amplitudes> $A$, the two requested profiles are
$$
y(x,0)=A[\sin(\pi x/L)+\sin(2\pi x/L)],\qquad
y(x,t_1)=-A\sin(2\pi x/L).
$$
The first is asymmetric, with an interior zero at $x=2L/3$; the second has its interior <node> at $L/2$. For <two-mode string recurrence>, <orthogonality> means the whole initial profile recurs only when both <cosine> coefficients equal one. Since $\omega_2=2\omega_1$,
$$
\boxed{t=\frac{2\pi n}{\omega_1},\qquad n=0,1,2,\ldots.}
$$
\Image[/past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1-ia2012-string-modes.png]
{title=Two-mode string profiles and the small change produced by weak positive dispersion at the original quarter-period}
The <weak-stiffness string frequency shift> follows by expanding the supplied <dispersion relation>:
$$
\omega(k)=c_sk\left(1+\frac{\alpha k^2}{2c_s^2}\right)+O(\alpha^2),\qquad [\alpha]=\mathrm{m^4\,s^{-2}}.
$$
Higher <normal modes> acquire larger fractional <frequency> shifts. Let $\eta=\alpha k_1^2/c_s^2\ll1$ and retain $\omega_1^{(0)}=c_sk_1$ for the original ideal <frequency>. At the originally defined time $t_1=\pi/(2\omega_1^{(0)})$,
$$
\cos(\omega_1t_1)=-\frac{\pi\eta}{4}+O(\eta^2),\qquad
\cos(\omega_2t_1)=-1+O(\eta^2).
$$
Thus the $t=0$ sketch is unchanged, whereas the original quarter-period sketch gains a small negative fundamental component. If $t_1$ is redefined using the new fundamental <frequency>, its fundamental component is exactly zero and the second-mode change there is only second order. Both conventions agree that generic later shapes change because of relative <phase> drift.
The new <frequency> ratio is
$$
r=\frac{\omega_2}{\omega_1}=2\sqrt{\frac{1+4\eta}{1+\eta}}=2+3\eta+O(\eta^2).
$$
An exact recurrence now requires $\omega_1t=2\pi q$ and $\omega_2t=2\pi p$ for integers $p,q$. \b[If $r=p/q$ in lowest terms, the first recurrence is $2\pi q/\omega_1$; for irrational $r$ there is no exact finite recurrence.] Near-recurrences remain possible. Positive stiffness alone does not prove that the ratio is irrational.
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