Claim
The angle of a trolled fishing line at a given point, measured at the surface, does not change as more line is deployed and the rig is lowered into the water.
Forces
The forces acting on a trolled line are characterized as weight, hydrodynamic drag and hydrodynamic lift of both the terminal tackle and the line. The diving force of a lure due to its shape is a form of negative lift. These forces control the resultant line angle at every point of the line.
The forces at the terminal tackle (sinker weight and lure) create a drag force and usually a downward force. The vector sum of these forces result in a force angle and magnitude that is balanced by an equal and opposite force provided by the fishing line. At each point of the line, additional forces are generated by the hydrodynamic drag and lift of the line at that point. These forces are incrementally added to those of the line below, until we reach the surface, where we have the total drag and downward force of the whole line and terminal tackle resulting in the final line angle at the surface.
So, how do the forces at a particular point of the the line and lure change as the line is deployed deeper or shallower? If the forces do not change, neither does the angle.
Equations
The drag equation states that drag D is equal to the drag coefficient Cd times the density of water R times half of the square of the velocity V times the reference area A.
D = Cd * R * .5 * V^2 * A
The lift equation is exactly analogous: Lift L is equal to the lift coefficient Cl times the density of water R times half of the square of the velocity V times the area A.
L = Cl * R * .5 * V^2 * A
For small angles of attack,
Cl = 2 * pi * angle_of_attack
Variables
How do each of the variables of the equations change with respect to depth?
- Density R of water does not change significantly with depth; water is less than 1% more dense at 100 meters than at the surface.
- Density of water does change with temperature, and deeper water is generally cooler, but the maximum difference in water density between 80F and 40F degrees is less than a half percent, so temperature is not significant in this application.
- Salty seawater is somewhat more dense than fresh water, but salinity does not change significantly with depth.
- Insignificant changes in water density are also balanced out to a large degree, as density effects all of drag, lift, and weight/buoyancy.
- Velocity V does not change with depth.
- The frontal or top/bottom area A of the terminal tackle or the line do not change with depth.
- The coefficients Cd and Cl do change with shape, but the shape of the line or of the terminal tackle does not change with depth.
- The coefficients Cd and Cl do change significantly from one part of the line to another due to the angle of attack of the line in a way that is not trivial to predict except for small angles, so that lift and drag at one segment of the line is quite different than that at another segment. But the drag or lift at a particular point of the line does not change with depth.
- The coefficients Cd and Cl can also change somewhat due to the surface condition (roughness) of the line at a particular point, but again the condition of a particular point of the line does not change with depth.
Conclusion
None of the variables in the drag and lift equations change as depth changes.
The forces of weight, drag, and lift on each point do not change significantly with depth, so the line below any point keeps the same shape as the point is lowered or raised.
If we were able to supply accurate values for all the variables in the equations shown above plus the line length, and use some advanced mathematics and computation, we could determine the xy coordinates of any point on the line. However, the patented Trolling Angles method does not do this.
We simply measure a series of deployed line lengths and the respective observed line angle at the surface, we assume that the angle does not change as the line is submerged, and we find a curve approximation which fits that observed data.