= Solution
This is the \b[<wave equation>], again taking $y$ as time. Zero is \b[not unique]: for $k=\pi/2$,
$$
\boxed{u(x,y)=\sin(kx)\sin(ky)}
$$
satisfies $u_{xx}=u_{yy}$ and all the stated <boundary conditions>. The <initial displacement> is zero but the <initial velocity> is not prescribed.
A sufficient extra condition is \b[$u_y(x,0)=0$]. For the resulting homogeneous data use wave energy
$$
E(y)=\frac12\int_0^1(u_y^2+u_x^2)\,dx.
$$
The equation and <integration by parts> give $E'(y)=[u_xu_y]_0^1=0$. At $x=1$ the factor $u_x$ vanishes, and at $x=0$ the factor $u_y$ vanishes because $u(0,y)=0$. The <initial displacement> implies $u_x(x,0)=0$, so the extra <initial velocity> condition gives $E(0)=0$. Consequently $u_x=u_y=0$ and $u=0$. With the stated boundary regularity, integrate the local <energy conservation> law over inset rectangles and let their edges approach the boundary; the energy and flux terms converge by <continuity> of the first derivatives.
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