In this post, I mention of my immediate plan for the week and also describe the problem I am looking at in brief.
In the calculation of TR noise for the beam passing through a cylindrical substrate, Heinert has considered thermal fluctuations
by means of an sinusoidally oscillating heat source, scaled according to the intensity profile of the beam, present along the length of the cylinder.
The heat equation is then solved to evaluate the temperature field. The gradient of temperature is used to calculate the work dissipated.
The work dissipated then is used to find the spectral density according to the Fluctuation Dissipation Theorem.
On the other had, the case of TE noise is considered in the work by Liu and Thorne where the beam reflecting from one of the faces is
considered. In their work, they have used an oscillating pressure, scaled according to the beam, at the face. The equation of stress balance
is solved to get the strain field, the strain results in heating and as a result, a temperature gradient. The work dissipated is calculated using
the gradient of temperature and the spectral density is calculated using the FDT.
Now, in case of Heinert's work, it is to be noted that the heating term is the one that contains the TR coefficient, beta = dn/dT. This is the where
the TR noise that we are looking into comes in the picture. Liu and Thorne's work on the other had, never has an explicit heat source. In their case,
it is the strain that generates the heat implicitly. They have related the expansion with the temperature perturbation in their paper from an expression
in Landau Lifshitz's, Theory of Elasticity. The important point to note is that the physical parameters that characterize the TE noise like coefficient of
linear expansion, alpha, or Poisson ratio, sigma, come into the picture through this relation. Another point to note is that the expression
for the work dissipated ( W_diss ) uses the gradient of temperature( It is the same formula that Heinert has used). This expression is derivable from the heat
equation. Thus, one could have also done the exercise by injecting the right heat source and solving the heat equation instead, since its ultimately
fluctuations in temperature that cause these noises TR or TE.
Our problem is to evaluate TE noise for a beam that passes through the cylindrical substrate instead of reflection off the face. It is suspected that using an
oscillating pressure on the surface will not be the correct approach since the beam is going through the material and not just reflecting from the surface. We
want to solve it by means of a heat injection, as done in Heinert, calculating the gradient of temperature, the work dissipated and then the spectral density. It is realized that the
heat source should be oscillating but the correct coefficients is what is undetermined i.e. we realize the heat source is q_dot = [- - -] * cos(omega * t) . In case
of Heinert it is at this point that the 'beta' comes in. However, in the TE case this is not yet determined. The literature doesn't deal with the case of beam goin through
a material. The equations in Heinert must be looked at more deeply to realize how the 'beta' comes in and then drawing an analogy, we may be able to figure out the
right heat source for the TE noise case.
Any comments on references, the approach that should be taken, or any thoughts on the problem is most welcome.