# On axially symmetric elastic wave propagation in a fluid-filled cylindrical shell

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## Abstract

The early stages of propagation of a water hammer disturbance are investigated, water hammer constituting a special case of axially symmetric elastic wave propagation in a fluid-filled cylindrical shell. Many of the objectionable features of the elementary (Joukowsky) water hammer theory are removed, and particular emphasis is placed upon consideration of the effects of radial inertia of the fluid and of the shell. The formulation is appropriate for consideration of any axially symmetric acoustic disturbance which originates in the fluid and any of the usual engineering boundary conditions which describe constraints on motion of the end, or ends, of the shell.

Motion of the shell is described by a thin-shell theory, and motion of the fluid is described by the axially symmetric wave equation, nonhomogeneous boundary conditions providing coupling of the fluid and shell motions. Application of a finite Hankel transform to the axially symmetric wave equation yields an infinite system of one-dimensional wave equations representing motion of the fluid. Integration of a finite set of these wave equations in conjunction with equations governing motion of the shell is accomplished numerically after a straight-forward application of the method of characteristics.

An analysis which includes bending, rotary inertia, and shear deformation in the shell is conducted for the case of sudden termination of uniform flow in a semi-infinite shell with a"built-in" end. For a relatively thick steel shell filled with water it is found that bending stresses and transverse shearing stresses at the end of the shell are significant, but that nowhere are there significant longitudinal membrane stresses. Maximum stresses and displacements are found to occur within the time required for an acoustic disturbance in unbounded fluid to traverse one diameter of the shell. The maximum radial displacement of the middle surface of the shell is found to exceed the value predicted by the elementary theory by about fifty percent.

A solution based on the classical membrane theory of shells, neglecting longitudinal stresses, also is obtained by numerical integration of the ordinary differential equations arising from application of the method of characteristics to the governing partial differential equations. Considerable simplification of the numerical calculations results from the fact that only one pair of families of characteristic lines are involved in the membrane analysis as compared to three pairs of such families in the bending analysis. The membrane analysis is employed principally to show the adequacy of using the first five of the infinite set of one-dimensional wave equations governing motion of the fluid. The membrane formulation takes account of two important omissions of the elementary theory, namely radial inertia of the fluid and radial inertia of the shell.

A representation of the fluid motion by a single one-dimensional wave equation is investigated. Radial inertia of the fluid is taken into account by attributing additional mass to the shell. This formulation is shown to produce the same results as the best available long-time asymptotic solution to a water hammer problem, but it is found, on the basis of an analysis employing the membrane theory of shells, to be inadequate for describing the early stages of a water hammer disturbance.