By Walter E Thirring

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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**Example text**

The dimension of this last space is increased by the presence of dependent variables in Lie’s analysis (in the case under consideration, U, I/ and a are dependent variables). 22 Group Properties of the Acoustic Difirential Equation Another part of the Lie infinitesimal operators only acts in the space of dependent variables and has no analogy among symmetry operators (in the case under consideration this part is represented by the operators 2,). It should be emphasized that for the separation of variables, as for generation of exact solutions, only the first part of the operator set is essential: symmetry operators and its analogous among the Lie infinitesimal operators.

Y&j. 2) with minimal difficulties the operators { Yf)} need to be determined in the simplest form. 2) and, then, to solve more complicated equations. This recommendation only has a practical character. In the process of finding the eigenfunctions of operators {Y} we are determining the dependences of a common eigenfunction from different variables. Typically, some operators’ subset, {Y}, determines all dependences apart from one. 2), we obtain ordinary differential equations with respect to the remaining independent variable.

So, all automodel equations may be exactly intePO Separation of Variables. 7. Non-homogeneous media with cylindrical symmetry For non-homogeneous elastic media characterized by full cylindrical symmetry, the acoustic equation admits the next three-operator subset of the commuting (separating) operators of symmetry F, Jz, P, This subset is enough for a complete separation of variables in cylindrical coordinates. 2) where A,(z), p&), p,,(7) are real and positive arbitrary functions of the cylindrical radius 7 = (x2 + y2)l12, and k is a real parameter.