= Solution
A spatially uniform external <gravitational acceleration> moves a body's <centre of mass> without stretching it. <Tidal forces> arise from the variation of that <acceleration> across the body. Let a distant <mass> $M$ lie at distance $D$ along <unit vector> $\mathbf n$, and let $|\mathbf r|\ll D$ denote position relative to the body's centre. Subtract the centre's <acceleration>. To first order,
$$
\boxed{\mathbf a_{\mathrm{tidal}}=\frac{GM}{D^3}
\left[3(\mathbf n\cdot\mathbf r)\mathbf n-\mathbf r\right].}
$$
The <tidal tensor> is $T_{ij}=GM(3n_in_j-\delta_{ij})/D^3$. It stretches along the line to the source with <eigenvalue> $2GM/D^3$ and compresses in the two perpendicular directions with <eigenvalues> $-GM/D^3$. Its <trace> vanishes outside the source, consistently with <Laplace's equation> for the <gravitational potential>.
After removal of the constant and uniform-acceleration terms, the tidal potential is
$$
\Phi_{\mathrm{tidal}}=-\frac{GM}{2D^3}
\left[3(\mathbf n\cdot\mathbf r)^2-r^2\right].
$$
An ideal equilibrium ocean on a <sphere> of radius $R$ follows an equipotential of the body's gravity plus this perturbation. For surface gravity $g$, the leading <displacement> is
$$
\delta h(\psi)=-\frac{\Phi_{\mathrm{tidal}}(R,\psi)}g
=\frac{GMR^2}{2gD^3}(3\cos^2\psi-1),
$$
up to a constant fixing total <water> <volume (thermodynamics)>. The quadrupolar form produces bulges on both the near and far sides; the far-side bulge follows from differential <acceleration>, without postulating a repulsive gravitational <force>.
The Moon and Sun both raise terrestrial <tides>, with relative strength set mainly by $M/D^3$, rather than by the total gravitational pull $M/D^2$. Their contributions reinforce near new and full Moon to produce <spring tides>, and partly cancel near quadrature to produce <neap tides>. Earth's <rotation> relative to the tidal pattern gives a strong semidiurnal component in the simplest geometry. Real <tide> heights and timing depend on ocean depth, coastlines, basin resonances, <Coriolis acceleration>, friction and the declination of the forcing bodies. <Tidal dissipation> produces phase lags and <torques>, exchanging rotational and <orbital angular momentum> and gradually altering the <spin> and orbit. The equilibrium bulge is therefore a useful starting model, not an exact local <tide> forecast.
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