Draft for Information Only Content Wave Reflection inside Enclosures, 1D Wave Reflection inside Enclosures, 1DA sound source radiates outwardly in any direction in a free source will always encounter reflecting surfaces unless the sound source is place in an anechoic chamber where all sound energy is almost absorbed with any reflection. And in such case then only wave equation should be satisified without any boundary conditions. Multiple reflection in an enclosure due to the wall creates standing waves with acoustic modes charactered by frequencies and mode shapes. These modes should satisfied the wave equation and boundary conditions simultaneously. The boundary conditions are usually defined in terms of continuity between the wall motion and the particle velocity. The wall characteristics are usually expressed in term of impedance. A rigid wall is used to describe a wall with higher impedance while a soft wall is used to describe a wall with lower impedance. Duct, 1D mode, with two closed endsThe 1D wave equation is Assume harmonic sound wave imply Take the time differential operation, imply And get the Helmoltz equation, Assume the solution of the equation of the form, Assume the two closed ends are rigid wall, the boundary conditions at two ends are with particle velocity equals to zero, imply and From the equation of momentum conservation At x = 0 or x=L, u=0, therefore the time derivative of u is zero also, imply: From boundary condition at x=0, imply: From boundary condition at x=L, imply: Substitute k, A and B into the pressure function, imply: and Therefore the natural frequency and mode shape is: , and Duct, 1D mode, with two open endsThe 1D wave equation is Assume harmonic sound wave imply Take the time differential operation, imply And get the Helmoltz equation, Assume the solution of the equation of the form, With two open ends, the boundary conditions at two ends are with pressure equals to zero, imply and From boundary condition at x=0, imply: From boundary condition at x=L, imply: Substitute k, A and B into the pressure function, and p<>0, imply: and Therefore the natural frequency and mode shape is: , and Duct, 1D mode, with one end open and one end closedThe 1D wave equaton is Assume harmonic sound wave imply Take the time differential operation, imply And get the Helmoltz equation, Assume the solution of the equation of the form, Assume the closed end is rigid wall, the boundary conditions at closed end is with particle velocity equals to zero, imply From the equation of momentum conservation At x = 0, u=0, therefore the time derivative of u is zero also, imply: o, imply: From boundary condition at x=0, imply: From boundary condition at x=L, imply: Substitute k, A and B into the pressure function, imply: and Therefore the natural frequency and mode shape is: , and ©sideway References
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