By K. Huseyin
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Additional info for Nonlinear theory of elastic stability
The b o u n d a r y x = 0 is perfectly rigid. 46) 0 at x = 0, Vt /5(x, t ) = / 5 ' ( x , t) at x = L, Vt 17"(x, t ) = V'(x, t) at x = L, Vt outgoing waves conditions at x--~ c~, Vt C H A P T E R 2. 59 A C O U S T I C S OF E N C L O S U R E S p,c iOt ~Ct 0 L Fig. 2. Scheme of the one-dimensional enclosure. where /5(x, t) (resp. /5'(x, t)) stands for the acoustic pressure in the first (resp. (x, t) and V'(x, t) are the corresponding particle velocities; Y(t) is the Heaviside step function ( = 0 for t < 0, = 1 for t > 0).
Remark. y- b/2) COS correspond to the same wavenumber rTr/a which is said to have a multiplicity order equal to 2. But for this wavenumber, the homogeneous N e u m a n n problem has two linearly independent solutions. It is also possible to have wavenumbers with a multiplicity order of 3, to which three linearly independent eigenmodes are associated. 2. 26), if it exists, describes the free oscillations of the fluid which fills the domain f~. Such an oscillation is called a resonance mode; the corresponding wavenumber (resp.
With V • f f l _ 0 and V . fi'~- 0, where 4~ and ~ are the scalar and vector potentials. Similarly for the applied forces: ffl _ VG 1 + V x / q l . Then the linearized motion equation is decomposed into p0 ~O2ffl _ (A0 + 2/z0)V(V . ff~) = VG 1 Ot 2 pO 02 ffl 0V +u Ot 2 xVx~s~ = v x # ~ or, since V2ff - V . ( V f f ) = V ( V . i f ) - V x V x #, V x # ~ - 0 and V . 94) are wave equations. e. pressure waves zT~, derived from a scalar potential 4~ by ff~ - V4~, characterized by V • ff~ - 0 and V .
Nonlinear theory of elastic stability by K. Huseyin