13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Figure 2. Incident plane wave The incident angles of plane sound wave are shown in Fig. 2. If neglect the vibration of the plate, the incident and reflected sound pressure can be expressed as Eq. (16-17) [8]: 1 2 2 i t i[sin (cos +sin ) cos ] 1 2 2 2 ( , ,z , ) e 2 h h k x y i i h h p x y t P ω φ θ θ φ + − + + = = (16) 1 2 2 i t i[sin (cos +sin ) cos ] 1 2 2 2 ( , ,z , ) e 2 h h k x y r i h h p x y t P ω φ θ θ φ + − − + = = (17) The dynamic response of an elastic plate excited by the incident wave can modify the resultant sound wave. The plate and acoustic medium have the same normal velocity on the interface. Use Rayleigh’s integral (Eq. 18) to calculate the scattered sound pressure produced by the elastic vibration of the plate [9]. 2 2 0 0 0 i ( ) ( ) 2 i t 0 1 2 0 0 0 2 2 0 0 2 ( , , , ) ( , ) e 2 2π ( ) ( ) k x x y y s S h h e p x y z t w x y dS x x y y ω ρω − − + − + = =− − + − ∫ (18) An additional pressure components, associated with the wave transmitted into the other side of plate, is added to the loading term in the governing equations. The two plate surfaces partake in identical motions, and the acoustic medium on both sides of the plate are of the same. The phase is, however, different. Thus, on the plate surface this transmitted pressure tp takes the value sp− [8]. 2 2 0 0 0 i ( ) ( ) 2 i t 0 1 2 0 0 0 2 2 0 0 2 ( , , , ) ( , ) e 2 2π ( ) ( ) k x x y y t S h h e p x y z t w x y dS x x y y ω ρω − − + − + =− = − + − ∫ (19) Resultant acoustic excitation is of the form: ( , , ) i r s t q x y t p p p p = + + − (20) Considering Eq. (16-19) and Eq. (20), the loading (Eq. 21) can be obtained: 2 2 0 0 0 i ( ) ( ) 2 i i sin ( cos sin ) i 0 0 0 1 2 0 2 2 0 0 ( , ) 2 ( , , ) 2 cos( cos )e e 2 π ( ) ( ) k x x y y t k x y t i S e w x y h h q x y t P k dS x x y y ω φ θ θ ω ρω φ − − + − − + + = − − + − ∫ (21) 2.3.2. Vibration responses
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