Vroom!

This morning for breakfast, we got some pastries from a favorite bakery. On the way home, another car honked its horn as it passed, and the changing pitch reminded be of something I've often wanted to try: I wondered whether I could simulate the sound of a vehicle moving by an observer, by calculating the Doppler shift at each point.

The Doppler shifted frequency is given by

where v is the velocity of the vehicle, c the speed of sound, and f0 the original frequency. The dot product with r means we only take the part of the velocity that is along the line-of-sight between the vehicle and the observer. That dot product is what leads to the changing pitch: As the vehicle approaches the observer, the angle decreases, lowering the pitch.

Anna, Mom-Eater!

While watching the morning news recently, I noticed their graphic of a wind gauge during the weather segment:

Wikipedia

Initially, I imagined the cups were moving at the same speed as the air, but then I started wondering about how the backs of the cups would slow the rotation – The wind is pushing on both sides, so how much gets cancelled?

The key is that the cup's drag coefficient changes as it rotates. This coefficient is defined as

where F is the force applied to the cup, ρ the density of air, u the speed of the air, and A the area of the cup facing the wind. Wikipedia gives the drag coefficient for the outside as 0.38, and the inside as 1.42. To get the area of the cup we need, we can use the same type of stereographic projection from an earlier post. In theory, the drag coefficient would change continuously as the cup rotates, but we can get an approximation by assigning the inside/outside coefficients to the inside/outside area projections. Putting everything together

This is just for a single cup, so now we turn to the full anemometer design. Initially, they were built with 4 cups at right-angles, but these have mostly been replaced by 3 cups at 120° angles. I decided to try both:

The Wikipedia article I linked to above says, "The three-cup anemometer also had a more constant torque and responded more quickly to gusts than the four-cup anemometer." My model doesn't show this, with the 4-cup design (red) giving less variation in force, and a larger mean force. I imagine this is due to the drag approximation I mentioned, which leads to the sharp knees in the plot above.