By F. W. J. Olver
A vintage reference, meant for graduate scholars, mathematicians, physicists, and engineers, this ebook can be utilized either because the foundation for educational classes and as a reference device.
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Extra resources for Asymptotics and special functions
If the water is deep compared to the wave length, then dk % 1, while if the wave height is small compared to wavelength, hk % 0. When all of these approximations hold, then, we expect Ψ to be roughly constant, so ω2 is proportional to gk. In fact, we find ω2 ¼ gk in the limit, but this requires analysis that is more detailed. For very short waves (ripples) in deep water, it seems reasonable to assume that only surface tension is responsible for the wave motion, so that g does not enter the problem.
There is, therefore, a one-to-one corresponding between the null space N(A) and the set of dimensionless combination of the variables. It may not come as a big surprise that dimensionless combinations have a value independent of the system of units. We simply use Eq. λf ! m n n Y m Y Y Y aij λf a λ ¼ ρj x i ¼ ρj xi if f j¼1 i¼1 j¼1 j¼1 i¼1 ! n m Y n n Y Y Y λ a λ λ ¼ ρj f xi if f ¼ ρj f n Y j¼1 since Aλ ¼ 0 implies n Y a λf xi if i¼1 j¼1 j¼1 ¼ 1. j¼1 Moreover, if you pick a basis for N(A) and take the corresponding dimensionless combinations, π 1 , Á Á Á, π nÀr (here r is the rank of A), then any nÀr where dimensionless combination can be written as a product π c11 Á Á Áπ cnÀr the exponents are uniquely given (they are the coefficients of a member of N (A) in the chosen basis).
In analyzing the dynamics of a liquid flow, for example, one might argue that the liquid temperature is a significant variable. It is important, however, only in its influence on other properties such as viscosity and should therefore not be included along with them. 4. The Buckingham Pi theorem, if applied to the actual number of dimensions being used, tells only that there must be at least a certain number of dimensionless numbers involved. Unless one resorts to one of the tedious techniques that have been devised for discovering the minimum number of dimensions needed, the theorem gives little assurance that all the dimensionless numbers have been found—an assurance that can very quickly be secured from the step-by-step approach, if assurance is needed.
Asymptotics and special functions by F. W. J. Olver