Last week I went swimming and met a couple who were very helpful and friendly. They were both professors at the Art Center College of Design here in Pasadena. Anyway, one of them pointed out that the pool does have a lap timer clock which I hadn't bothered to notice before, so I timed myself swimming 50 and 100 yard sets (two and four laps of the pool). My time for 50 was about 40 seconds (1.14m/s), and 100 was 1:40 (0.914 m/s) these were fairly consistent after the first set or two. I consider this not bad since I'm a father of two small boys who hasn't been exercising enough for the last 4 years or so. But another thing I discovered was that I know I swim 25 yards in about 13 or 14 strokes, sometimes 15. This, together with my lap times gives me a range of stroke rates which is around 36 or so strokes per minute. Now typical stroke rates are advocated to be around 60 to 70 per minute, which suggests that somehow I should increase my stroke rate. Intuitively, it seemed to me that this wouldn't work for me, but I thought I'd analyze the math of it. Thinking about how this would affect power output led me to the following analysis:

Let's add in the energy cost of stroking to the previous analysis. To get an energy cost, consider that your muscle "wastes" the kinetic energy of your arm at each end of the stroke. So the power you're putting out overcomes drag, and also accelerates and decelerates your arm every stroke.

Where $a$ is the length of the arm stroke, which is different from $sL$, so $m(2ar)^2r/2$ has the dimensions of power (kinetic energy per stroke time). Also, note that $s= v/(Lr)$ which I substitute for. We're going to try to control $v$ and $r$ so having things in terms of those variables is more helpful.

Another analysis I'm going to do here is to change from using $h$ and $d$ to $h$ and $t$, where $h$ is still the height of the wake above the water, but $t$ is the full cross sectional profile of the body, so $t = h+d$. I do this because for people who have a fairly flat profile in the water, $t$ is something fixed by their body shape, whereas $h$ can be controlled by things like head position. The previous analysis normalized by drag using $d$ but normalizing by full submersion drag using $t$ makes a little more sense because it's a single fixed point rather than something that changes when $h$ changes. Substituting $d=t-h$, and normalizing by $K\rho t wv^3$ gives the following dimensionless power equation (note, as in the previous analysis, I've dropped the lateral viscous drag term as we've already seen it's multiplied by a trivially small constant).

This is a slightly different form than before, we see in this analysis that we have three dimensionless groups $h/t, gh^2/(v^2t),ma^2r^3/(\rho t v^3 w)$.

From this analysis, if we take a very low in the water body position, then $h/t=0$, and we see that we can decrease our drag by riding up a little bit so that $h>0$. Since the wave drag grows like $h^2$ when $h$ is small the decrease in piston drag $(-h/t)$ exceeds the increase in wave drag. This probably accounts for the fact that elite swimmers often have a relatively forward looking head position. This lets them reduce their piston drag significantly while suffering only a small wave drag penalty. It also probably helps to produce the trough that allows for effective breathing.

Plugging in typical values for the mass of the arm and the density of the fluid and soforth, the order of magnitude estimates for the dimensionless groups are about 0.2 for the wave group, and $0.02r^3$ for the arm group. So, although the equation predicts that if I double my stroke rate, I will waste 8 times more arm power, this still brings it into the same range as the wave and piston drag power. But why would I want to do that? The only reason I can see is that it would give me significantly more opportunities to breathe. One thing that I find with a 30 or so strokes per minute stroke rate is that I pretty much have to breathe one side only because otherwise I build up carbon dioxide. But doing so gives me an asymmetric stroke and might increase the wave and piston drag coefficients, and it makes it hard to empty my lungs fast enough. A faster stroke rate will have to be combined with a shorter stroke length, but will allow me to have more opportunities to breathe and may allow me to have a better sustained power output, actually making me faster even though my power output increases.

At least, that's how I interpret the math. I plan to get a little metronome that beeps regularly to help me time my stroke and see how it works for me.