Click here for reference. Then as far as tidal potential energy goes in terms of displacements...
$$\frac{\Delta {{r}_{sun}}}{\Delta {{r}_{moon}}}\simeq \frac{{{m}_{sun}}}{{{m}_{moon}}}\frac{{{R}_{moon}}^{3}}{{{R}_{sun}}^{3}}\simeq \frac{1.99\times {{10}^{30}}}{7.35\times {{10}^{22}}}\frac{{{\left( 3.84\times {{10}^{8}} \right)}^{3}}}{{{\left( 1.5\times {{10}^{11}} \right)}^{3}}}\simeq 4.29\times {{10}^{30-22+24-33}}\\\simeq 4.29\times {{10}^{-1}}\simeq 43\%$$
UPDATE: (based on comments)
Consider two masses ${{m}_{1}}$ and ${{m}_{2}}$ whose co-ordinates are measured relative to the centre of a third mass, M, via distances ${{r}_{1}}$ and ${{r}_{2}}$. Point E lies on the surface of mass M (assumed to be spherical) and for now is assumed to also be in the plane of all three masses for convenience. Let $\angle EO{{R}_{1}}=\theta $and $\angle {{R}_{2}}O{{R}_{1}}=\phi $. We have using the same approximations as before
$${{V}_{T}}={{V}_{m1}}+{{V}_{m2}}=-\frac{G{{m}_{1}}{{r}^{2}}}{2R_{1}^{3}}\left( 3\cos \left( \theta \right)-1 \right)-\frac{G{{m}_{2}}{{r}^{2}}}{2R_{2}^{3}}\left( 3\cos \left( \theta -\phi \right)-1 \right),0\le \phi ,\theta \le \frac{\pi }{2}$$
Tidal displacement due to both moon, ${{m}_{1}}$and sun, ${{m}_{2}}$ as a first approximation can be given by
$$\Delta r=\frac{{{r}^{4}}}{2M}\left\{ \frac{{{m}_{1}}}{R_{1}^{3}}\left( 3\cos \left( \theta \right)-1 \right)+\frac{{{m}_{2}}}{R_{2}^{3}}\left( 3\cos \left( \theta -\phi \right)-1 \right) \right\}$$
From which we have then
$$\Delta {{r}_{spring}}=\frac{{{r}^{4}}}{2M}\left\{ \frac{{{m}_{1}}}{R_{1}^{3}}+\frac{{{m}_{2}}}{R_{2}^{3}} \right\}\left( 3\cos \left( \theta \right)-1 \right)$$
and
$$\Delta {{r}_{neap}}=\frac{{{r}^{4}}}{2M}\left\{ \frac{{{m}_{1}}}{R_{1}^{3}}\left( 3\cos \left( \theta \right)-1 \right)+\frac{{{m}_{2}}}{R_{2}^{3}}\left( 3\sin \left( \theta \right)-1 \right) \right\}$$
Q: Are there positions on the earth where the tidal displacement for both a spring and neap tide are the same? Equating the two displacements we find $\cos \left( \theta \right)=\sin \left( \theta \right)\Rightarrow \theta =\frac{\pi }{4}$ independent of mass. More generally we might ask when the system is not in a neap-tide, then where will the displacements become equal to that when it is in a neap-tide. Then we are considering
$$\sin \left( \theta \right)=\cos \left( \theta -\phi \right)=\sin \left( \tfrac{1}{2}\pi -\theta +\phi \right)\Rightarrow \theta =\tfrac{1}{4}\pi +\tfrac{1}{2}\phi $$
Which defines a function $\theta \left( \phi \right)$ describing a point on the earth which has a displacement equivalent to that at neap tide. Note: $\theta \left( 0 \right)=\arctan \left( 1 \right)=\frac{\pi }{4}$ which was our previous result. So the point moves linearly with respect to angular separation, starting at $\pi /4$ in a spring tide and converging upon itself at a neap tide (which we logically expect). Note that this analysis delivers a single point where rather I think it would be a locus of points created at the intersection of two spheroids. We have the prolate spheroidal bulge due to the moon, which we hold fixed, and then add to that a tilted spheroid from the sun which as it moves towards its position for a neap-tide, becomes an oblate-spheroid. As it tilts it creates a locus of intersections on the surface where tidal displacements are equivalent to that at a neap-tide. Also there are other solutions here i haven't considered (due to the periodicity of the functions) in which a more careful analysis will bring to the fore. However i'd suggest moving directly to a more general coordinate set and try and capture the full locus.
Is there any legitimacy to the claim that tidal energy that is provided during "spring tides" (which the Sun has additive effects with the moon for higher tide) is cancelled out with "neap tides" in which the Sun's forces are in opposition to the Moon's?
– user699279 Apr 27 '21 at 18:49