By Prof. Dr. Fridolin P. Mechel (auth.), Prof. Dr. Fridolin P. Mechel (eds.)
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E. : fTm (r)· Tn (r)· g(r) dr = Om,n . N m (I) A g(r) is weight function which is induced by some co-ordinate systems; it is independent of the mode order m; Om,n is the Kronecker symbol; N m is the norm of the mode. e. kO, Zo complex). 738 ff) if the defining boundaries normal to r are either hard or soft or locally reacting, and if in this case the derivative opfor does not appear in the separated wave equation of the co-ordinate r. Modes may be one, two, or three dimensional to the number of pairs of walls that define the boundary conditions.
JK TO ---:-:--:-_~) Po ideal gas TO characteristic mean free molecular path length for heat conduction effects; f mean free path length; (79) (80) 36 Formulas of Acoustics B_ General Linear Fluid Acoustics characteristic mean free molecular path length for (81) shear visCQsity effects; characteristic mean free molecular path length for (82) shear and bulk viscosity; Linearised fundamental equations for a density wave (time factor e +joo t): - K-l( (0) -f> temperature variation: T=--I+j-Ph UK Co density variation: P = 2" Co entropy variation: S=J----Ph-p To UK Co longitudinal particle velocity: P~) ve = ( -j- - - -gradp Aa a sk + {k} . a is J au·J s JS JS aaf L{k} ak . a·s - { JS} JS 1 k V·a· -+ J 1 =a Uj s 1 •. ;.. aks ·k a·IS s Uj Jl J Vj(aib k ) = Vj(ai)bk +aiVj(b k ) It holds that: ik k V j (8 ik ) = Vjg ik = Vjg j = Vjg = 0 (29) Orthonormal basis vectors: Orthonormal basis vectors named ei ; i=I,2,3. The basis vector components are : R·1 =Re· 1 1 with: (30) 1 gJi" =--8 .. RH. I,J 1 ~ 1; J (31) g = det(gij) = HIH2 H 3 H~ Vector components: a= L a~ ei (32) 30 Formulas of Acoustics B. General Linear Fluid Acoustics Scalar product: (33) Vector (cross) product: axb= 1 Hlel H 2e 2 H 3e3 al bl a2 b2 a3 b3 HIH2 H 3 el al* b; e2 e3 a2* a3* b; b; (34) Vector triple product: abc = 1 HIH2 H 3 al bl a2 b2 a3 b3 ci c2 c3 al* a2* a2* = b~ b; b; cl* c2* (35) c3* Differential operators : The gradient of a scalar function is a vector : (36) Nabla operator (a vector): The divergence of a vector is a scalar : (37) The rotation of a vector is a vector : (38) The Laplacian of a scalar function : (39) The Laplacian of a vector is a vector: Identities : a 11 = grad( div it) - rot( rot it) (40) 31 Formulas of Acoustics B.
