For varnish or tinfoil the resistance varied as v1.83. For sand the resistance varied as v2.00.

3. The average resistance per square foot of surface was much greater for short than for long boards; or, what is the same thing, the resistance per square foot at the forward part of the board was greater than the friction per square foot of portions more sternward. Thus,

Mean Resistance in
℔ per sq. ft.
Varnished surface 2 ft. long 0.41 
50 ” 0.25 
Fine sand surface 2 ” 0.81 
50 ” 0.405

This remarkable result is explained thus by Froude: “The portion of surface that goes first in the line of motion, in experiencing resistance from the water, must in turn communicate motion to the water, in the direction in which it is itself travelling. Consequently the portion of surface which succeeds the first will be rubbing, not against stationary water, but against water partially moving in its own direction, and cannot therefore experience so much resistance from it.”

§ 69. The following table gives a general statement of Froude’s results. In all the experiments in this table, the boards had a fine cutwater and a fine stern end or run, so that the resistance was entirely due to the surface. The table gives the resistances per square foot in pounds, at the standard speed of 600 feet per minute, and the power of the speed to which the friction is proportional, so that the resistance at other speeds is easily calculated.

Length of Surface, or Distance from Cutwater, in feet.
2 ft. 8 ft. 20 ft. 50 ft.
A B C A B C A B C A B C
Varnish 2.00 .41 .390 1.85 .325 .264 1.85 .278 .240 1.83 .250 .226
Paraffin .. .38 .370 1.94 .314 .260 1.93 .271 .237 .. .. ..
Tinfoil 2.16 .30 .295 1.99 .278 .263 1.90 .262 .244 1.83 .246 .232
Calico 1.93 .87 .725 1.92 .626 .504 1.89 .531 .447 1.87 .474 .423
Fine sand 2.00 .81 .690 2.00 .583 .450 2.00 .480 .384 2.06 .405 .337
Medium sand 2.00 .90 .730 2.00 .625 .488 2.00 .534 .465 2.00 .488 .456
Coarse sand 2.00 1.10 .880 2.00 .714 .520 2.00 .588 .490 .. .. ..

Columns A give the power of the speed to which the resistance is approximately proportional.

Columns B give the mean resistance per square foot of the whole surface of a board of the lengths stated in the table.

Columns C give the resistance in pounds of a square foot of surface at the distance sternward from the cutwater stated in the heading.

Although these experiments do not directly deal with surfaces of greater length than 50 ft., they indicate what would be the resistances of longer surfaces. For at 50 ft. the decrease of resistance for an increase of length is so small that it will make no very great difference in the estimate of the friction whether we suppose it to continue to diminish at the same rate or not to diminish at all. For a varnished surface the friction at 10 ft. per second diminishes from 0.41 to 0.32 ℔ per square foot when the length is increased from 2 to 8 ft., but it only diminishes from 0.278 to 0.250 ℔ per square foot for an increase from 20 ft. to 50 ft.