Wind Speed & Direction (Ultrasonic Anemometer 2D).

Speed of Sound
The speed of a sound wave in air depends upon the properties of the air, namely the temperature and the pressure. At normal atmospheric pressure, the temperature dependence of the speed of a sound wave through air is approximated by the following equation:

v = 331.45 m/s + (0.6 * T)

speed = distance / time

Wind velocity and direction
When the sound wave travels against the wind, the total transit time is increased by a wind speed dependent amount. When the sound wave travels with the wind, the total transit time is reduced.
The different wind speeds and directions are measured over a fixed path.
As the speed of sound is very dependent on the air temperature, the propagation time of the sound is measured on both of the measurement paths in both directions. In this way, the influence of the temperature dependent speed of sound on the measurement result can be eliminated by subtracting the reciprocals of the measured propagation times.
By combining the two measuring paths which are at right angles to each other, one obtains the measurement results of the angle and velocity of the wind.

The time of flight of the first signal (out) is given by:

to = d / (c + v)  (1)

tb = d / (c - v)  (2)

Orthogonal ultrasonic anemometer 2D design

x = (d / 2) * (1 / twe - 1 / tew)

y = (d / 2) * (1 / tsn - 1 / tns)

If wind direction is exactly in line with one measuring path (NS or SN or WE or EW), this will cause turbulence due to transducer structure. This will cause erratic readings and needs to be compensated for in software

Equilateral triangle ultrasonic anemometer 2D design

ab = (d / 2) * (1 / tab - 1 / tba)

bc = (d / 2) * (1 / tbc - 1 / tcb)

ca = (d / 2) * (1 / tca - 1 / tac)

x = ab * Cos (60) + ca

y = ab * Sin (60)

x = ab * Cos (60) + bc * Cos (60)

y = ab * Sin (60) + bc * Sin (60)

x = bc * Cos (60) + ca

y = bc * Sin (60)

x = ab * Cos (60) + bc * Cos (60) + ca

y = ab * Sin (60) + bc * Sin (60)

Magnitude and direction calculated from x and y coordinates

The length of the vector r pointing from the coordinate origin to a point in 2D space is given by Pythagoras' Theorem

r = sqrt(x² + y²)

While the polar or phase angle f is obtained by performing the operation:

First quadrant (+x, +y)

f = 90 - arcsin (y / r)

Fourth quadrant (+x, -y)

f = arcsin (y / r) + 90

Third quadrant (-x, -y)

f = (90 - arcsin (y / r)) + 180

Second quadrant (-x, +y)

f = arcsin (y / r) + 270

Wind speed at above angle:

v = r  m/s

or:

v = r * 3.6  km/hr

or:

v = r * 1.9476  knots

Virtual Temperature
As previously mentioned, the speed of the propagation of sound is highly dependent on the air temperature, but is hardly affected by air pressure and humidity.
As this is a measurement of gas temperature which is made without thermal coupling to a measurement sensor, it is called the "virtual temperature".
The advantages of this measured variable is the avoidance of measurement errors such as those which occur when a solid state temperature sensor is heated up by sun irradiation or cooled by evaporation cooling by rain and wind.

The sonically determined speed of sound can be found from the sum of the inverses of (1) and (2):

c = (d / 2) * (1 / to + 1 / tb)

Ts = ((c * c) / (1.4 * 287.04)) - 273.15

Suppliers of ultrasonic sensors.

Notes.

Multiple feedback bandpass filter
fc = 40kHz
gain = 4
bandwidth = 12kHz
C = 220pF

Q = 40000 / 12000 = 3.32

R1 = 3.32 / (4 * 2 * 3.142 * 40000 * 220 x 10^-12) = 15k ohms

C' = (220 x 10^-12 * 220 x 10^-12) / (220 x 10^-12 + 220 x 10^-12) = 110pF

R3 = 3.32 / (2 * 3.142 * 40000 * 110 x 10 ^-12) = 120k ohms

R' = 1 / ((2 * 3.142 * 40000)² * 120000 * 220 x 10^-12 * 220 x 10 ^-12) = 2725.8 ohms

R2 = (15000 * 2725.8) / (15000 - 2725.8) = 3.3k ohms

What others have built.

Reference.

Alternatives.