# Why does a small air gap under a cylinder damp oscillations of a membrane on the cylinder's opposite end?

This gif shows a table tennis ball bouncing on a membrane stretched over a cylinder.

When some cards are placed under the cylinder to form a thin gap between the cylinder and table, the bouncing is dramatically damped compared to when the cylinder is placed directly on a table.

When the cylinder is lifted more than a few millimeters off the table, the ball bounces well again; only for small air gaps is there a large damping effect.

My guess is that this has to do with impedance matching. The small gap has about the same impedance as the cavity in the cylinder, so there's a large transmission coefficient and lots of energy is lost. The impedance of the table is very high and the impedance of open air is very low, leading to much lower transmission coefficients.

I tried setting the acoustic impedance of the cylinder cavity to $$Z_c = \dfrac{z}{\pi r^2},$$ with $$z$$ the specific impedance of air in the experiment and $$r$$ the radius of the cylinder. Then I set the impedance of the gap at the bottom of the cylinder to $$Z_{\rm gap} = \dfrac{6 \mu w}{\pi d^3 r},$$ with $$\mu$$ the viscosity of air, $$w$$ the width of the rim, $$d$$ the height of the gap. This come from treating the table and bottom rim of the cylinder as parallel plates with flow between them at low Reynolds number. Equating $$Z_c$$ and $$Z_{\rm gap},$$ I find optimal damping at $$d = \left(\frac{6 \mu L R}{z}\right)^{1/3}$$

Estimating numbers from the video, I came up with an optimal gap of around $$0.5 \;\mathrm{mm},$$ a bit smaller than what people on Twitter reported in their experiments, but within a factor of 2 or 3.

Perhaps that error can be explained by entry length effects, uncertainty in measurements from the video, and the gap not being a length $$2\pi r$$ around as I assumed (because people put cards under the cylinder to prop it up, and the cards block a lot of the gap). But perhaps I'm missing some significant part of the physics here.

Is the impedance approach the right track? Are there other significant factors to account for?

Source (longer video with sound): https://twitter.com/DarekDewey/status/1525115833466380290