We will be using a 555 timer and a hex inverter in place of an Arduino to generate the required 40 kHz square waveforms. The original and inverted signals from the 555 will be used as inputs to the motor driver to excite the ultrasonic transmitters.

Supplies

One 555 timer (eg LM555CN)
Two 10 nF capacitors
One 0.1 uF capacitor
One 220 Ohm Resistor
One 1.5k + 36 Ohm Resistors
(In place of resistors, you can use potentiometers)
Protoboard + Soldering Iron or Bread board
One Hex Inverter (eg MM74HC04N)
One L289N Motor Driver
Two 10 mm or 16 mm 40 kHz ultrasonic transmitters
Some kind or spacer to position transmitters (eg Q-tip or wooden dowel)
12V Power Source (I used a variable voltage DC adapter bought off Amazon)
5V Power source (Not needed if you are using L298N to supply 5V)
22 AWG Wires
One ultrasonic receiver (optional, but useful for final debugging step)

Step 1: Power the L298N With 12V

Use the L298N as your source for 5V from its onboard voltage regulator, (by feeding it 12 V). Make sure the rest of the circuit is also connected to the same ground.

Step 2: Build the Square Wave Generator With the 555 Timer

Using the attached diagram from my TinkerCAD, first build a test circuit with the 555 timer as a astable multivibrator to verify that the choice of resistor and capacitor values produces a square waveform with a ~49-50% duty cycle and 40 kHz frequency. While there are astable 555 calculators that give theoretical values for the capacitors and resistors to achieve the intended waveform, I have found these give different results than theoretically predicted. I chose R1 as 220 Ohms, R2 as 1.536 kOhms, and C at 10 nF. I also used a 0.1 uF decoupling capacitor. It may help to use potentiometers to tweak you resistor values to achieve the desired waveform.

Step 3: Generate an Inverted Copy of the Square Wave and Use As Inputs to the Motor Driver

Next feed the output from the 555 to an input pin on the inverter. Use both the output of the 555 and the output of the inverter as logic inputs to the L298N motor diver. Depending on which pair if inputs you chose, this will produce amplified square waves on the corresponding side of the driver board. Verify the waveforms on each on the output pins are in fact inverted and have 5 Vpp compared to ground.

Step 4: Verify the Transducers Are Emitting 40 KHz Sinusoidal Outputs

You can then connect the outputs to pins on the transducers. If the circuit is working, then the ultrasonic receiver (attached to an oscilloscope) should detect sinusoidal waveforms at 40 kHz.

Step 5: Set Up the Transducers and Levitate Some Styrofoam Beads!

Position the transducers such that they are aligned and facing each other, spaced about 2 cm apart. Use a spacer or guide to help keep them in place. A tooth pick or Q-tip are simple enough. Use a coffee filter mesh or something similar to insert a Styrofoam bead at the center of the setup! If perfectly aligned and spaced apart, you should be able to levitate several particles at once!.