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This is an applied problem, which arises from the problem of reorienting of a sensor axes according to particle displacement directions:

Consider a sensor which is located inside the solid substance. This sensor is capable of detecting the substance oscillations along each of the three axes (usually orthogonal, but generally, any non-degenerated (non-coplanar) basis). This sensor produces a recording of the detected oscillations, called a trace, containing the displacement sampled at high-frequency, capable to capture any oscillation frequency existing in the substance. There are three traces, one for each axis.

Consider that at some period of time the sensor registers an interference of an event, consisting of the compression wave, and two shear waves. Traces now contain a recordings of the event, effectively, a projections of the oscillations on the sensor's axes.

How do I now virtually re-orient the axes, that is, perform the linear transform of the traces, so that they will be oriented each along a corresponding wave displacement vector, and thus will contain only single wave event recording in each of the traces after the transformation?

EDIT1: Actually, we have a 3D-curve $r(t)$ of sensor motion, which is represented by sensor axial readings $r_1(t), r_2(t), r_3(t)$. The task is to find the "primal" directions of $r(t)$ movements and re-orient axes to those directions $q_1(t), q_2(t), q_3(t)$.

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There are two questions here. The first, which Eric answered for orthogonal axes, is "given the axes to transform to, how do I transform?" The second is "given the data, how do I figure out the axes to transform to?. I don't think that is available from a single sensor. If you have some more knowledge you might be able to. If you know the source of the disturbance, one axis might be from the source to the sensor. If you have a model how the longitudinal and transverse disturbances are related, you could try a fit to your data as a function of the axes.

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Yes, the main question is how to figure out the "primal" axis system. We have not the only sensor, but three of them (axial). Since the oscillations are presented in several captured periods, the $r(t)$ curve spans the ellipsoid like a thread in a clew. Figuring out the major axes of the ellipsoid may provide a clue to the "primal" coordinate system. – mbaitoff Jan 19 '11 at 15:14
I was thinking of the sensor as being triaxial, but without information from some other locations it will be difficult/impossible to determine the direction of travel. If you know transverse waves travel at different speeds, you may be able to use the frequency information to help – Ross Millikan Jan 19 '11 at 15:29
Why do you think that direction of travel is necessary? In the presence of anisotropy local substance oscillations generally do not correspond neither to direction of travel (for compression wave) nor the tangential plane (for shear waves). – mbaitoff Jan 19 '11 at 16:57

If you have three unit vectors that describe your current axes $r_1, r_2, r_3$ and you want to know what the linear transform is that will change $a_1 r_1 + a_2 r_2 + a_3 r_3 $ into the coordinates of some new axes $r'_1, r'_2, r'_3$, the matrix of this transform is:

$ \left( \begin{array}{ccc} r'_{11} & r'_{21} & r'_{31} \\ r'_{12} & r'_{22} & r'_{32} \\ r'_{13} & r'_{23} & r'_{33} \end{array} \right) $

where $r'_{j} = r_{j1} r_1 + r_{j2} r_2 + r_{j3} r_3$. So once you've figured out what the axes are for the coordinate system that you want to transform into, you can just plug those numbers into that matrix and apply it to your trace-vector to get the trace in the new coordinate system.

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Thats's right, the formula you provided seems correct, however, the vector is not known. Instead, we have a series of time-sampled values, $r_1(t), r_2(t), r_3(t)$, recorded during the event: $t\in[t_a..t_b]$. Those values are the recorded solid displacement. – mbaitoff Jan 19 '11 at 14:18
However, the matrix you provided is correct only for orthogonal unit base vectors $r_1, r_2, r_3$ – mbaitoff Jan 19 '11 at 14:21

I think you should read the wiki page on the classical stress tensor, especially the section on change of basis. Perhaps what you want to do is use your sensors to measure the stress tensor, and then find the basis which diagonalizes it. Also you might want to see Feynman Lectures on Physics, volume 2 chapter 39, and volume 3 chapter 5, section 5-7.

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'Elastic materials' and 'Spin one'? – mbaitoff Feb 28 '11 at 5:14

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