\begin{problem*}{13.10}
A transverse wave on a string is described by the wave function
\begin{equation}
  y = (0.120\U{m}) \sin\p({\frac{\pi}{8}x + 4\pi t}) \;.
\end{equation}
\Part{a} Determine the transverse speed and acceleration of an
element of the string at $t = 0.200\U{s}$ for the point on the string
located at $x = 1.60\U{m}$.  \Part{b} What are the wavelength, period,
and speed of propagation of this wave?
\end{problem*}

\begin{solution}
\Part{a}
Notice that the phase $\phi = \pi x/8 + 4\pi t = 0.2\pi + 0.8\pi=\pi$.
Since the transverse displacement is a $\sin$ curve, that means that
the transverse velocity will be most negative and the acceleration
will be zero.
\begin{align}
 \pderiv{t}{y}(x=1.60\U{m}, t=0.200\U{s}) &= A\omega = \ans{-1.51\U{m/s}} \\
 \npderiv{2}{t}{y}(x=1.60\U{m}, t=0.200\U{s}) &= \ans{0\U{m/s$^2$}} \;.
\end{align}

\Part{b}
To increase the phase by $2\pi$, we need a $\Delta x_{2\pi} =
\ans{\lambda = 16.0\U{m}}$.  Similarly, we would need a $\Delta
t_{2\pi} = \ans{T = 0.500\U{s}}$.  The wave speed then is $v = \Delta
x / \Delta t = \lambda / T = \ans{32.0\U{m/s}}$.
\end{solution}
