# Analysis of Shockwaves on Liquid Surfaces

I have completed several analyses of recordings collected from liquid surfaces using my laser microphone array. The first of these tests attempts to tune the parameters of the analysis method I am developing, and verify the reliability of the results.

The initial test involved two datasets, derived from the same configuration of measurement points along the same small dish of water. The excitation signal was a droplet produced by a pipette, from approximately the same position relative to the dish. The excitation responses were taken a few seconds apart from one another. The analysis method does not require precision with regard to the measurement positions, and in fact was designed to be largely robust despite incidental variances in these measurement positions.

Both datasets were recorded in 4 channels of 96 kHz, 24 bit. This first test only involves one dataset, named "drop 0", which can be found here, in .txt format.

trial one:
"drop 0"
channels: 4
order: 20
winsize: 152200

The 80 resulting eigenvalues:

Since the data was real-valued, the eigenvalues appear in complex conjugate pairs. They appear to be spaced evenly across the available bandwidth, in groups of four. In the right-hand half of the complex plane, the groups' variance appears to increase rapidly as the groups approach 0.

A typical artifact of 1 dimensional modal analysis techniques is to line the unit circle with eigenvalues. This effect appears when the order of analysis is higher than the number of modes present in the dataset. Perhaps, in a n-dimensional dataset, the effect manifests as groups of n tightly grouped eigenvalues. If this is true, reducing the order of the analysis should ameliorate this effect.

trial two:
"drop 0"
channels: 4
order: 10
winsize: 152200

The 40 resulting eigenvalues:

Once again, the eigenvalue conjugate pairs appear evenly spaced across the bandwidth, in groups of four. The algorithm may be too highly influenced by the transient, rather than the homogeneous response. If this is true, perhaps taking the later half of the dataset will ameliorate this effect.

trial three:
"drop 0"
channels: 4
order: 10
winsize: 76100

The 40 resulting eigenvalues:

The eigenvalue conjugate pairs continue to appear evenly spaced across the bandwidth, in groups of four. To test whether the grouping is the result of nearly matched modes being measured from the four different points, two of the four channels are removed, and the process repeated.

trial four:
"drop 0"
channels: 2
order: 10
winsize: 76100

The 20 resulting eigenvalues:

The eigenvalue pairs are once again evenly spaced, now in groups of two. This is consistent with the hypothesis that the grouping is the result of nearly matched modes measured from different points on the surface. The variations in frequency are likely the result of slight deviations in the shape of the dish, and perhaps also the depth of the liquid, as a function of angle. The variations in damping are likely the result of the mode shapes as they are expressed across the surface.

Future experiments on this dataset will consider the eigenvectors as well, and compare the analysis data across the two different samples.