After some runs with my analytical underwater pile noise model CAMPRADOP using different cylindrical-shell motion models a few years ago, I came to believe that for axisymmetric shell vibration (n = 0), the membrane (M) model is just as accurate as the other models, such as Donnell-Mushtari (D). On completing some further runs recently, I have found that the difference is negligible in a heavy liquid such as water or seabed, but in a light gas such as air the difference can be large.

A pile can be considered to have three segments determined by its adjacent external and internal media: aerial, immersed and embedded. If the pile head is in an aerial medium (usually the case), M predicts the vibration spectrum for any point on the immersed or embedded pile segments to have a null (deep minimum) between the pile's bar and plate ring frequencies, due to high attenuation along the aerial segment in that frequency band. See for example FIG. 2 in my 2015 paper (http://dx.doi.org/10.1121/1.4927034, henceforth H2015). As a consequence, the underwater radiated sound pressure spectrum also has a null there (FIG. 3 in H2015). The D model predicts no null in that band but does predict the general level of the spectrum over a wide band of frequencies above the plate ring frequency to be several dB lower than M does.

To obtain the complex phase velocity V, M yields a linear equation in V^2 with a straightforward solution; op. cit., Eqs. (17) and (18). The equation in V^2 yielded by D is obtained by starting with Eq. (2.21) in Leissa 1993, deleting all azimuthal terms, inserting a radiation loading term, as per Eq. (12) in H2015, and solving for axial wavenumber "lambda" as a function of omega, rather than vice-versa. This yields a cubic equation in 1/V^2 that has complex coefficients. Depending on how one's cubic solver works, obtaining the appropriate results may be a little tricky. The large differences mentioned above occur even though the extra terms in the equation to be solved (cubic rather than linear) are proportional to a small quantity h^2 /12 a^2 (Junger-Feit's "beta-squared" or Leissa's "k") in which “a” is the shell radius, and h is the wall thickness.

I won't go into detail here and now, but if a reader should present results that seem to be relevant, I would be happy to discuss them further.

Footnote: On the issue of attenuation in the embedded segment of the pile, my current view is that loss of vibration energy in the sub-bottom is largely due to downward radiation from the annulus at the pile toe. In lieu of setting the toe reflectivity to 0.8 (as in H2015, section II B) I now use an analytical model for loss due to downward radiation from the annular toe. For realistic seabed acoustic properties, it gives a reflectivity that varies with frequency with an average around 0.85.