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3D ORIGAMI by Ikuko Mitsuoka

By Ikuko Mitsuoka

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OM ^ and Μ i s the mass of the body causing the g r a v i t a t i o n a l a t t r a c t i o n , t h e E u l e r i a n equation becomes: ^ + ( » V ) r + igrad;, = - « ^ . (5: 25) Mathematical and Thermodynamic Methods o f Gas Dynamics In the Lagrangian form ajid in r e c t a n g u l a r equation i s w r i t t e n as du . \ dp _ 'W~^ Ρ dx ~~ ^ GMx , r3 • _L I- dv dt . I dp _ ρ dy ~ — — dt ~^ ρ dz " coordinates, GM r» h3 this (5; 26) Γ3 • v/here r = ΐΛ:^ + y ^ + , and x / r , y / r , and z / r a r e d i r e c t i o n c o s i n e s of the r a d i u s v e c t o r R.

During an e l e m e n t a r y time i n t e r v a l d t , t h e mass o f t h e medium c o n t a i n e d i n t h e volume element at the i n s t a n t t changes by an amount (5;54) l-^(fydx)dt = ^{f9)dxdt. 55), ät = - we dx dt. (5; 55) get |r(/p)+¿(/P«)-0- (5; 66) For a s p e c i f i c equation o f s t a t e , t h e system o f e q u a t i o n s ( 5 . 5 3 ) and ( 5 . 5 6 ) d e s c r i b e s t h e motion o f t h e medium i n t h e tube c o m p l e t e l y . In the Lagrangian f o r m , t h e c o r r e s p o n d i n g c o n t i n u i t y e q u a t i o n w i l l be Mathematical and Thermodynamic Methods o f Gas Dynamics ^9 (5: 57) where a i s the Lagrangian c o o r d i n a t e .

We s h a l l l a t e r determine the v a l u e o f t h e momentum i n s p e c i f i c cases o f propagation o f plane, c y l i n d r i c a l , or spherical waves. I t i s easy t o imderstand why t h e t o t a l momentum d i f f e r s from z e r o f o r a weak sound v;ave. The reason i s t h a t a mass m+m* moves i n t h e compression r e g i o n i n t h e d i r e c t i o n o f p r o p a g a ­ t i o n o f t h e wave w i t h an a v e r a g e speed v , and a mass m-m» moves i n t h e o p p o s i t e d i r e c t i o n i n the r a r e f a c t i o n r e g i o n v/ith a p p r o x i m a t e l y the same a v e r a g e speed; t h u s , t h e r e i s a t r a n s f e r o f mass 2m' - p» w i t h a speed ν - p' i n the d i r e c t i o n o f wave p r o p a g a t i o n .

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