One of the advantages of the digest form is that you get a well
defined bunch of messages to consider at a time.
Rather than try to sort out who is saying what about who said what, I
shall restate with maximum clarity the point where I seem to differ
markedly from some other views:
The elements: Two elastic spheres, i.e. two balls of wobbly jelly.
The contrasting conditions: (i) the two spheres are close together,
each impaled (not necessarily through a diameter) by a stake which
keeps them a constant distance apart... the stakes are parallel to
each other and perpendicular to the line joining the centres of the
spheres. (ii) the spheres are in space falling freely towards each
other.
I am not considering oceans, rotations, revolutions... the effects of
these may be added later.
Claim: In case (i) there will be one bulge only, the spheres
distorting so that on their near sides the jelly moves away from the
stake, towards the other sphere, while on their far sides the jelly
moves towards the stake, ALSO TOWARDS THE OTHER SPHERE. In
case (ii) there will be two bulges. On the near sides the jelly moves
away from the stake, towards the other sphere, while on the far sides
the jelly moves away from the stake, AWAY FROM THE OTHER SPHERE.
(But bear in mind that in this case the spheres are accelerating
towards each other, so that while the distortion is away from the
other sphere the motion is not).
Justification: case (i) is an equilibrium case. A jelly element on
the near side is attracted away from the stake, and the distortion of
the jelly - a stretch - supplies the elastic force to maintain
equilibrium. A jelly element of the far side is attracted in the same
direction, which is now towards the stake, and a compression provides
the elastic force for equilibrium. Each sphere is compressed on one
side and stretched on the other side. I call this one bulge, but that
is not important. The important thing is that the situation is
different from...
Case (ii): Each sphere is accelerating. An element on the near side
has a relatively greater gravitational pull on it, and this must be
compensated by the elastic force of the stretched jelly pulling the
element back. An element on the far side has a relatively weaker
gravitational force, which must also be compensated by an elastic
force, pulling the element forwards: it is clear that now the jelly
must stretch in both directions. I call this two bulges.
Leigh: are you convinced that your analysis using potential functions
is valid for the case of the staked spheres. Here we have elastic
spheres restrained by a rigid stake passing through some part of the
spheres. The situation is messy, and I think irrelevant to the
problem of the tides. This is why I rather objected to Donald's
"nailing down" approach. Changing to a rotating reference frame is
NOT like nailing things down, or rather it is only if you nail down
every point. But isn't this a relevantly different case?
The ultimate point that I'm after is whether Donald is correct in
saying that one can dispense altogether with acceleration in
discussing the tides, and explain everything in terms of the gradient
in the gravitational force only. It seems to me that both are
required, or else equivalently one must use an accelerated reference
frame.