= Solution
{c}
Direct differentiation gives the convective acceleration
$$
(\mathbf u\cdot\nabla)\mathbf u
=(\sin x\cos x,-\sin y\cos y).
$$
The steady <Euler equations> therefore require $p_x=-\rho\sin x\cos x$ and $p_y=\rho\sin y\cos y$. Integrating both derivatives consistently gives
$$
\boxed{p=C+\frac\rho4(\cos2x-\cos2y).}
$$
The additive constant is arbitrary. With a corresponding redefinition of this constant, the Bernoulli function is $p/\rho+|\mathbf u|^2/2=C_0-\psi^2$, making its constancy along streamlines explicit.
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