Boskone this past weekend was held at the Westin Waterfront in Boston, which has these funky double showerheads that they charmingly call the “Heavenly(R) Shower” (hype aside, they are very nice showers). The picture at right is courtesy of lannalee on Twitter, as I didn’t bring a camera.

Why am I telling you this? Because there was a sign glued to the wall in the shower that read:

Refresh yourself, restore our world

One of your Heavenly(R) Shower heads has been turned off in an effort to minimize water usage and protect one of our most precious natural resources.

The smarmy enviroweenieness of this was undercut somewhat by the next paragraph, which explained that you could turn it back on by pushing a little button on the showerhead (you can see one side of it on the lower head in the picture). And also by the fact that it’s a completely stupid statement.

Turning off one of the two showerheads does essentially nothing to reduce the water usage. The flow rate of water coming into the shower is determined by the pressure and cross-sectional area of the pipes. If you turn off one of the two showerheads, it just makes the water come out of the other one faster– at twice the speed, in the ideal case, which means you use just as much water per second in the shower with one head as with two. This is why putting your thumb over the end of the garden hose makes the water spray out so much farther– the same amount of water needs to pass through a much smaller opening, so it has to move much faster on the way out. The only way turning one showerhead off can reduce the water usage by making showering slightly less pleasant, and thus getting people to take shorter showers.

But that’s the ideal case– does it hold up in reality? And, more importantly, can we test this?

Of *course* we can test this– we’re physicists. Well, I am. Also, I’m enough of a dork to want to check this out experimentally.

So, here’s the configuration for the simple test. I tilted the upper showerhead up as close to horizontal as it would go, and let the water hit the far wall of the shower, like so:

The showerhead was roughly 1.9 m above the floor (measured by counting the number of tiles on the wall, which were 20cm high according to the ruler on the inside back cover of Matter and Interactions (which I had with me so I could field student questions by email)). The far wall was roughly 1.2m away, measured by counting 40cm floor tiles, and I could measure the height on the wall where the water hit.

With water flowing through both showerheads, the maximum height of the spray from the top showerhead hit the wall 0.9m above the floor. With one of them closed, the water hit higher– somewhere between 1.1 and 1.2 m above the floor. This clearly shows that the water comes through the single showerhead faster than when both are open. But is it really twice as fast, as it would need to be to completely undo the supposed conservation benefits?

Explaining that requires a bit of math, but nothing too horrible. The height at which water hits the far wall is determined from simple kinematics, using the following equations:

Here, x_{f} is the 1.2m distance to the far wall, y_{i} is the 1.9m height of the showerhead, and y_{f} is the height of the water on the far wall. The angle θ is the downward angle of the showerhead (which wouldn’t go all the way to the horizontal position– I guesstimated it at around 30 degrees, as I’m not nearly dorky enough to carry a protractor with me), and g is the acceleration due to gravity, 9.8m/s^{2}.

Δt is the time required for water to travel from the showerhead to the wall. We don’t know this, of course, but we can solve the top equation to find Δt in terms of the distance and the speed, giving us the following expression for the distance the water falls in terms of the angle and the distance to the wall:

That looks scary, I know, but the important fact here is that we know everything in this equation except for the speed v. We can do a bit of algebra and find the following expression for the speed of the water, v:

This looks faintly horrible, but it’s pretty easy to crank these numbers into a calculator or a spreadsheet, and get a value for the speed of the water.

Using the numbers given above, the two-showerhead speed comes out at 5.5 m/s, which is a reasonable enough speed. Putting in 1.1m for the height with one showerhead closed off gives the speed as 9.4 m/s, not quite twice the speed; 1.2 m/s would require a speed of 36 m/s. The real height was somewhere between those two– splitting the difference (1.15m) gives a speed of 12.8m/s. If instead of faffing about with approximate heights, we assume that the speed is doubled, and predict the height, we get 1.13m, which is entirely consistent with my eyeball measurements.

So, at best the “shut off one showerhead” strategy might be saving 15% of the water usage (using the lower bound on the speed); the real savings is probably considerably less, and consistent with zero, as a simple analysis of the plumbing would suggest.

So, if the folks at the Westin really believe this saves water, they’re being tripped up by their own lack of physics intuition. Of course, the really important question (from the Westin’s point of view) is not so much whether turning off one showerhead really does save water, as whether turning off one showerhead makes the people who stay at the Westin think they’re saving water, and gives them a warm fuzzy feeling about how “green” they are. The answer to that is probably “yes,” which suggests that the sign is either canny marketing or a depressing indicator of the innumeracy of the general population.

For the rest of us, a little bit of physics confirms that there’s no reason to feel guilty about turning the second showerhead back on. As long as you don’t linger longer in the double shower, you’re not wasting any water by making full use of the facilities. And given what they charge for the rooms, you might as well get your money’s worth.