Wave equation from linear Euler equations

The linear Euler equations for pressure $p = p_0 + p_a$, density $\rho = \rho_0 + \rho_a$, and velocity field $\mathbf{v}$ are \begin{gather} \frac{\partial\rho_a}{\partial t} + \rho_0\nabla\cdot\mathbf{v} = 0 \\ \rho_0\frac{\partial\mathbf{v}}{\partial t} + \nabla p = \mathbf{0}. \end{gather}

By taking the gradient of the first equation, and substituting the expression for $\nabla p$ from the second equation (while also using $p_a = c^2\rho_a$), I find that \begin{align} \frac{\partial\mathbf{v}}{\partial t} = c^2\nabla^2\mathbf{v}. \end{align}

However, this text (pages 17-18) says that my wave equation form only holds when $\nabla\times\mathbf{v}=\mathbf{0}$. In general, it states, only \begin{align} \frac{\partial\left(\nabla\cdot\mathbf{v}\right)}{\partial t} = c^2\nabla^2\left(\nabla\cdot\mathbf{v}\right) \end{align} holds. What then did I miss?

Note that $\nabla(\nabla \cdot \mathbf{v}) \neq \nabla^2 \mathbf{v}$.