Download e-book for iPad: Analysis, Manifolds and Physics [Part II] (rev.) [math] by Y. Choquet-Bruhat, et. al.,

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Example text

For example, these approximations all consist of two surfaces, namely, a conical surface and a spherical surface. This may complicate theoretical study of these particles because the two surfaces constitute a mixed boundary problem, which is usually more difficult to solve than a simple boundary. The shapes of falling raindrops have been investigated theoretically by Pruppacher and Pitter (1971), who used a cosine series to represent the shape of a deformed drop. Their method is useful when dealing with the detailed drop shape under the influence of certain forces.

It has been pointed out by some investigators (Charles Knight, private communication; Albert Waldvogel, private communication) that many hailstones have elliptical horizontal cross sections. Hence it is desirable to have a formula describing such conical particles. This can be easily done by first writing down the 3-D form of Eq. 25) and then generalizing it to include the case of elliptical cross sections. The 3-D form of Eq. 56) is shown in Fig. 23. 5 FIG. 23. Two z-axisymmetric conical particles generated by Eq.

The large value of b and b^ are chosen in order to make the branches very thin. FIG. 18. A radiating dendrite generated by Eq. 001. 30 PAO K. WANG Fitting observed 3-D crystals by the above expressions is analogous to the procedure for 2-D crystals, except the measurements may be difficult to perform as mentioned previously. The easiest way to do this is probably by measuring the 2-D projection of the crystals and then performing the fitting process for the 2-D crystals as described in Wang and Denzer (1983) and Wang (1987).

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Analysis, Manifolds and Physics [Part II] (rev.) [math] by Y. Choquet-Bruhat, et. al.,

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