The last post, I somewhat dubiously estimated the effect of accelerating ground water from under-shoe during footstrokes. I concluded the kinetic energy was potentially a significant form of energy dissipation. But this is an absolute upper-bound estimate. After all, when considering energy dissipation, one must consider what would happen to the energy if it were not dissipated in water. For example, it might simply be dissipated in the sole of the shoe, in leg muscles and tendons. If this were the case, the total energy dissipation would be the same. Whether the energy would be dissipated in the shoe sole or in the water wouldn't affect speed. You could even argue the water would improve endurance by cushioning the landing, decreasing wear and tear on the body.
The question is what the marginal elastic efficiency of foot strikes on the road. When the foot hits the pavement the sole compresses, absorbing some of the kinetic energy. More kinetic is absorbed by the body. On the next stride, the shoe decompresses, returning some of the stored energy. A total elastic efficiency can be defined as the total returned energy divided by the total absorbed energy. A perfect spring would return all of the absorbed energy, that energy absorbed purely as elastic potential, and so a perfect spring when dropped on a perfect road at absolute zero temperature in a perfect vacuum would bounce essentially perpetually. But since energy absorbed by 2 mm of standing water is a small fraction of the total, what is at issue isn't the total efficiency, but the marginal efficiency: for an extra joule of kinetic energy in foot-strike, what fraction of that joule is returned.
This paper by MR Shorten concludes that shoes can absorb approximately 10 J of elastic energy, to be returned on the next stride. Minetti's study, which I have used extensively here, concluded running has a metabolic cost of approximately 3.4 kJ/kg/km, so if I am running at 4.5 minutes/km, and if I weigh 59 km with clothing, run with a 25% metabolic efficiency, and take 3 steps per second, that's 62 joules per foot strike of mechanical (not metabolic) energy. So the shoe is able to absorb only 16% of that. If some of the kinetic energy is first transferred to water, the returned energy may still be close to 10 J, resulting in a low marginal elastic efficiency. I don't know the details here, but the conclusion might be that the net effect of the water is less than I predicted.
But then there's the issue of pulling the foot off the ground. Does the water suck energy here? You might think that lifting the foot creates a vacuum which must be filled, that the vacuum creates a net downward pressure on the foot which makes it harder to pull off the ground. Or you might conclude the foot simply lifts off the ground and the water, on its own, falls into the hole. I'm not going to speculate on this: Napier-Stokes equation is beyond the scope of this blog.
It was suggested to me a neglected component of the water was that it's heavy. After the run, my clothing and feet were saturated in water. It has been determined (see also PubMed listing) that 100 grams of shoe mass results in an additional 1% of aerobic load during running. 100 grams per foot is of course well under 1% of total mass: in my case, assuming 59 kg total mass including clothing, it would be 0.34%. But as I have tried to analyze before, each stride involves raising and lowering the body center-of-mass, but additionally raising and lowering the foot and portions of the leg as part of the running stride motion. At optimal cadence the energy associated with these two motions is equal: too high a cadence and more energy goes into kicking, and too low a cadence and more energy goes into the ballistic trajectory of the center-of-mass. As part of that analysis I put my foot on a scale to estimate the total energy of my feet moving up and down. I concluded from that analysis that my optimal cadence was 94, which coincided with my typical cadence, which seemed to some extent to validate my model assumptions. From that analysis, the effective mass of each foot (modeling each foot as a point mass, the mass of the leg included) was 7.1 kg (including shoe & sock & compression sleeve on calf). 100 grams per foot is 1.41% of this. Assuming I run so half the energy is center-of-mass, half foot-calf oscillation, I conclude that the effect of 100 grams per foot is the average of 0.34% for the center of mass motion and 1.41% for the foot kicking, or 0.88%. This is in excellent agreement with the reported value of "1%".
So the question is then how much did the weight of my shoes and socks increase relative to when the shoes are dry? (A side question is whether wearing socks was a mistake which slowed me down enough to cost me my Boston qualification time, but I won't go there, as I always run in socks and blisters would have slowed me even more). I should in principle do this experiment, but since drying out soaking wet shoes can take quite awhile and probably isn't great for the shoes, I will resist the temptation.
In addition to shoes and socks, the rest of me also got wet. My shirt will typically be saturated with sweat, but my shorts are not. I would need to conduct the experiment: run on a warm day, weigh my shorts, dunk them in water, re-measure. I'll have to wait for a warm day.
On a side note: my Nike running flats are 172 grams, while New Balance Minimus are 220 grams, a difference of 48 grams, so if there was no difference in trauma the Nikes would obviously be faster. Assuming my calculated 0.88% per 100 grams, I conclude the New Balance should be 0.42% slower. I have not observed any speed difference between the two shoes, but I know I feel slightly better running on pavement in the New Balance shoes, and so suspect reduced trauma offsets the increased energy load, especially at marathon distance where the trauma is time-limiting for me, not aerobic capacity. But then that 0.42% is only 1.1 seconds per km assuming a 3:15 marathon pace, so I wouldn't notice that anyway run-to-run, although over a course of the marathon it would be 52 seconds. So my conclusion is that if the New Balance reduced trauma to save me 52 seconds (perhaps holding off my knee pain by 1 km per knee) then they were worth the extra mass, neglecting differences in water capacity.
In any case, the extra weight of wet shoes and clothes goes at least part of the way to explaining the slower speeds this year. I still need to consider the effect of wind.