>>16757412
h = 7
t = theta = pi/9

The blue curve is v_2 = v_3.
The yellow curve is Re[v_4].

The left side of the graph is at r = r_12 = h/(csc(t) + 1).

If r_12 < r < r_23, then only the blue curve is valid and v = v_2.
The maximum of the blue curve is in this interval.
Thus the sphere is more than half submerged.

If r = r_23 = h/csc(t), then the blue and yellow curves intersect.

et cetera