= Solution
Using
$$
\sinh(a+ib)=\sinh a\cos b+i\cosh a\sin b
$$
in the quotient from part (a), multiplying numerator and denominator by the complex conjugate of the denominator, and taking the real part after multiplication by $e^{i\omega t}$ gives
$$
\boxed{
\begin{aligned}
\frac{u(y,t)}{U_0}
={}&\frac{\cos\omega t\,
[\cosh\Delta_+\cos\Delta_-
-\cosh\Delta_-\cos\Delta_+]}
{\cosh(2\Delta)-\cos(2\Delta)}\\
&+\frac{\sin\omega t\,
[\sinh\Delta_+\sin\Delta_-
-\sinh\Delta_-\sin\Delta_+]}
{\cosh(2\Delta)-\cos(2\Delta)},
\end{aligned}}
$$
where
$$
\Delta_\pm=\Delta(1\pm\widehat y).
$$
The denominator follows from
$$
2|\sinh((1+i)\Delta)|^2
=\cosh(2\Delta)-\cos(2\Delta).
$$
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