rotations and tides
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Short derivation dated 1.19.17 and signed PhL, in a folder of support documents on rotating frames. It models the tidal bulge as an ellipsoid with r(θ)=R²+a cos2θ, relates two spherical coordinate systems by rotations Rz(φ1)Ry(θ1), and writes cosθ(t) for a fixed latitude with φ'=ωt. It gives the tide height h(t), with an equator example where h=a cos(2ωt) and a polar limit where h=-a. The text has dropped symbols and some equations are missing.
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Rotation and Tides PhL 1.19.17
Since the red ellipse of Fig ** and its equation (8.8.31) is based on the angle θ of Fig (8.8.17), the red ellipse is really an ellipsoid in a three-dimensional version of (8.8.32). If it happened that the axis of the Earth's rotation pointed toward the viewer in (8.8.18), one could calculate the intersection of the red ellipsoid with a line of latitude,
Plan B. The tide height is given by r(θ) = R2 + acos2θ . Think of θ as an angle of spherical coordinates where the z axis points to the right in Fig ***.
Imagine a different set of spherical angles θ', φ' based on a different axis ' . How are these angles related to θ,φ? I think I can answer that question.
Let's assume that ' = R where R then defines your new ' in Fig **. I think we then know that
r' = R-1r in Cartesian coordinates.
Start over. We have two frames
Frame S r = x,y,z
Frame S' r' = x',y',z'
For an active rotation R we would say r' = Rr and the axes stay fixed.
This is equivalent to a passive rotation of the axes "backward" by R-1.
Start over. Start with north pole vector . I want to rotate that vector in a simple manner so it lines up with some new vector ' . Here is an easy way to do that
' = Rz(φ1) Ry(θ1) as in (E.2.2)
Then
' =
Now I want to define a set of coordinates with respect to this new 'axis. But I know from (E.2.2) that
' = ' =
So I now know the spherical coordinates unit vectors in this new Frame S'. As we do this rotation, ANY point in Frame S gets rotated in this manner, so here is what happens to a general vector r,
r' = Rz(φ1) Ry(θ1) r
which says from App E
= with 1 subscripts
Now I have it hand entered so I can then write
r' = = = R1 r
So this says have the Cartesian coordinates move, but how to the Spherical coordinates move??? If I cancel the radius R2 I can write the above as
=
I am interested in a latitude line with θ' = constant and how this affects cosθ which in turn determines the tides. I would like to know the inverse of this matrix.
I guess I will just create a whole new Maple file to compute all this stuff. It is done. I can say then that
= with 1's added, so
= with 1's added, so
I then know that
cosθ = sinθ1cosφ1sinθ'cosφ' + sinθ1sinφ1sinθ'sinφ' + cosθ1cosθ'
Now suppose we are at a fixed latitude and φ' = ωt for earth rotation. Then
cosθ(t) = sinθ1cosφ1sinθ'cosωt + sinθ1sinφ1sinθ'sinωt + cosθ1cosθ'
and this tells how θ(t) varies during the day for latitude line θ' on the true Earth
Meanwhile
cos2θ = 2cos2θ - 1
So
r(θ) = R2 + a cos2θ
= R2 + a ( 2cos2θ - 1 )
= R2 + a ( 2[sinθ1cosφ1sinθ'cosωt + sinθ1sinφ1sinθ'sinωt + cosθ1cosθ']2 - 1 )
More interesting to just do the height of the tide effect
h(t) = a cos2θ = a ( 2cos2θ - 1 )
= a ( 2[sinθ1cosφ1sinθ'cosωt + sinθ1sinφ1sinθ'sinωt + cosθ1cosθ']2 - 1 )
Example: Suppose ' points out of the plane of paper in Fig *** so
' = Rz(φ1) Ry(θ1) = Rz(0) Ry(π/2)
Then
cosθ(t) = sinθ1cosφ1sinθ'cosωt + sinθ1sinφ1sinθ'sinωt + cosθ1cosθ'
= sinθ'cosωt
Then we get
h(t) = a ( 2cos2θ - 1 ) = a ( 2 sin2θ'cos2ωt - 1 )
If θ' = π/2 so we are at the equator , then get cosθ = cosωt so simply θ = ωt and
h(t) = a cos2θ = acos(2ωt)
and we get maximum tides.
As we move to the pole, θ'→ 0, we get
h(t) = -a
and we just sit at low tide all the time.