Effect of sound waves striking an object

Question

I am trying to understand the effect an acoustic wave has on an object. I have two speakers, one on top of another, and both emitting a Sine wave of 64Hz generated in MATLAB with a sampling rate of 44100. I place a sheet of paper at a distance 'd' in front of the paper. I then observe the vibrations on the paper using a Dynamic vision sensor(DVS) based event-based camera which can capture changes in intensity at every pixel with very high resolution(in the order of microseconds). So this means when the paper vibrates, I will have the paper's pixel locations and their timestamps(when the paper moves). The entire setup is as shown in the figure below(this is what the camera sees -the side views of the paper and the speaker)image of the left is the camera's the theoretical view and image on the right is an actual frame that the camera sees:

I am trying to understand the time of arrival of the acoustic wave on the surface of the paper.

1) From the event camera's data, I observe vibrations at different time stamps at different positions on the paper. How do I theoretically predict and understand the different time instances at which the sound wave would strike different parts of the paper?

2) Would it be possible to calculate the approximate distance between the speaker and paper using d=c*t where c is the speed of sound in air and t=tp-ts where tp = time at which wave hits the paper and ts = time at which wave was emitted from the speaker?

3) Would I be able to achieve 2) better if I placed one speaker facing the center of the paper?

I would appreciate any tips, pointers or resources to understand acoustic wave propagation. Thank you!

Answer 1

Interesting topic!

If you first imagine that sound waves emanating from a point source (which you are going to have to approximate as a speaker) will emit spherically. Where the sphere intersects first with the paper, you will notice a dot on the paper appearing in your image. That is the point at which the wavefront hits the paper and is the earliest possible time where this can happen. As the wavefront pushes forward, the dot should turn into a circle (assuming uniform wave propagation).

Your equations look right, so I would follow up by stating that you should have the following in place:

• Rock solid event lock between replay device and camera. (Camera must start recording as soon as the speaker starts emitting sound. Also take into account any latency through the digital replay system.)
• extremely high frame-rate camera in order to measure time between wavefront launch and paper impact (which I think you do).

You would get a much better result with a single speaker as there will be considerable interference from the second speaker to deal with.

You should also do this in a controlled environment where both the temperature and atmospheric pressure is known so that the speed of sound can be accurately calculated.

I'm not entirely sure of this next part, but I fear that 64Hz may be too low a frequency for this to be effective. 64Hz gives you approximately a 5.36m wavelength which is way bigger than the distance between the speakers and the paper. This may cause an issue with your measurements. I would consider using a higher frequency so that you at least have some complete wavelengths between the speaker and the paper.