Laplacian of the cosine of a dot product
In 2D,
$$\dfrac{\partial^2}{\partial x^2}\cos(a_xx+a_yy)=-a_x\dfrac{\partial}{\partial x}\sin(a_xx+a_yy)=-a_x^2\cos(a_xx+a_yy).$$
We can immediately deduce
$$\nabla^2\cos(\mathbb a\cdot\mathbb r)=-\mathbb a^2\cos(\mathbb a\cdot\mathbb r).$$