= Solution
For a uniform taut sheet in the small-displacement approximation, the two-dimensional <wave equation> is
$$
\partial_t^2\psi=v^2(\partial_x^2\psi+\partial_y^2\psi).
$$
Seek a harmonic solution $Y(y)e^{i\omega t-ik_xx}$. The clamped edges impose <Dirichlet boundary conditions> $Y(0)=Y(b)=0$. Hence $Y''+k_y^2Y=0$ has nontrivial solutions only for
$$
Y=A\sin(n\pi y/b),\qquad k_y=n\pi/b,\quad n=1,2,\ldots.
$$
Taking the real part gives the stated sinusoidal transverse profile. There is no nonzero $n=0$ mode satisfying both clamped boundaries.
Substitution gives the <dispersion relation>
$$
\boxed{\omega_n^2=v^2\left[k_x^2+(n\pi/b)^2\right].}
$$
The three branches have cutoffs $\omega_{c,n}=n\pi v/b$ at $k_x=0$ and approach $\omega=v|k_x|$ at large $|k_x|$. Only frequencies above a branch's cutoff give traveling <waves> along an infinitely long guide; below it, the axial dependence is evanescent. For a forward propagating mode,
$$
\boxed{v_{p,n}=\frac{\omega}{k_x}
=\frac{v}{\sqrt{1-(nv/(2bf))^2}},\qquad
v_{g,n}=\frac{d\omega}{dk_x}=\frac{v^2}{v_{p,n}}.}
$$
The <phase velocity> exceeds $v$, while the <group velocity> is smaller: the carrier phase speed is not the signal speed.
Here the first two cutoff frequencies are $50$ and $100\ {\rm Hz}$, so both modes propagate at $200\ {\rm Hz}$. With $\omega=400\pi\ {\rm s^{-1}}$,
$$
\boxed{
\begin{array}{c|cc}
n&k_x\ ({\rm m^{-1}})&v_p\ ({\rm m\,s^{-1}})\\
1&10\pi\sqrt{15}=121.673&40/\sqrt{15}=10.328\\
2&20\pi\sqrt3=108.828&20/\sqrt3=11.547
\end{array}}
$$
Each transverse mode independently acquires its axial phase. The second mode first changes sign relative to the first when $(k_1-k_2)x_1=\pi$, giving
$$
\boxed{x_1=\frac{\pi}{10\pi\sqrt{15}-20\pi\sqrt3}
=\frac{1}{10(\sqrt{15}-2\sqrt3)}
\simeq0.2446\ {\rm m}.}
$$
At that distance a common temporal phase is $\phi=-k_1x_1$ modulo $2\pi$; the second term is negative relative to the first for every $y$ and every time, not just at one selected instant.
In an optical fibre, different guided modes have different <group velocities>, so an input pulse becomes a sum of differently delayed pulses. This <intermodal dispersion> broadens pulses and can make adjacent digital symbols overlap. A <single-mode optical fiber> removes this particular mode-delay mechanism; a <graded-index optical fiber> can reduce it in a multimode design. Single-mode propagation still permits wavelength-dependent group delay, so it does not remove every kind of <optical dispersion>.
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