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 Wave Reflection inside Enclosures, 1D
   Duct, 1D mode,  with two closed ends
   Duct, 1D mode,  with two open ends
   Duct, 1D mode,  with one end open and one end  closed

Wave Reflection inside Enclosures, 1D

A 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.

image

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.

image

Duct, 1D mode,  with two closed ends

image

The 1D wave equation is

image

Assume harmonic sound wave imply

image

Take the time differential operation, imply

image

And get the Helmoltz equation,

image

Assume the solution of the equation of the form,

image

Assume the two closed ends are rigid wall, the boundary conditions at two ends are with particle velocity equals to zero, imply

image and image

From the equation of momentum conservation

image

At x = 0 or x=L, u=0, therefore the time derivative of u is zero also, imply:

image

From boundary condition at x=0, imply:

image

From boundary condition at x=L, imply:

image

Substitute k, A and B into the pressure function, imply:

imageand image

Therefore the natural frequency and mode shape is:

imageimage and image

Duct, 1D mode,  with two open ends

image

The 1D wave equation is

image

Assume harmonic sound wave imply

image

Take the time differential operation, imply

image

And get the Helmoltz equation,

image

Assume the solution of the equation of the form,

image

With two open ends, the boundary conditions at two ends are with pressure equals to zero, imply

image and image

From boundary condition at x=0, imply:

image

From boundary condition at x=L, imply:

image

Substitute k, A and B into the pressure function, and p<>0, imply:

imageand image

Therefore the natural frequency and mode shape is:

imageimage and image

Duct, 1D mode,  with one end open and one end  closed

image

The 1D wave equaton is

image

Assume harmonic sound wave imply

image

Take the time differential operation, imply

image

And get the Helmoltz equation,

image

Assume the solution of the equation of the form,

image

Assume the closed end is rigid wall, the boundary conditions at closed end is with particle velocity equals to zero, imply

image

From the equation of momentum conservation

image

At x = 0, u=0, therefore the time derivative of u is zero also, imply: o, imply:

image

From boundary condition at x=0, imply:

image

From boundary condition at x=L, imply:

image

Substitute k, A and B into the pressure function, imply:

imageand image

Therefore the natural frequency and mode shape is:

imageimage and image


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ID: 101000021 Last Updated: 10/20/2010 Revision: 0 Ref:

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References

  1. Michael P. Norton; Denis G. Karczub,, 2003, Fundamentals of Noise and Vibration Analysis for Engieer
  2. G. Porges, 1977, Applied Acoustics
  3. Douglas D. Reynolds, 1981, Engineering Principles of Acoustics:; Noise and Vibration Control
  4. Conrad J. Hemond, 1983, Engineering Acoustics &amp; Noise Control
  5. F. Fahy, 2001, Foundations of Engineering Acoustics
  6. D.A. Biew; C.H. Hansen, 1996, Engineering Noise Control: Theory and Practice
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