stationary modes infinite cylindric tube

STATIONARY MODES (AZIMUTHAL) IN AN INFINITE CYLINDRIC GUIDE

Animations

You can use images and animations included in this page for teaching using, but please acknowledge where you obtained the animation!

here: Catherine Potel and Michel Bruneau (Université du Maine - France)

See the slides associated to chapter 5 of the fluid acoustic course (in french)

Modes in an infinite cylindric tube

Modes in an infinite cylindrique tube with a meridian half wall

INFINITE CYLINDRIC TUBE
The acoustic field created by a monochromatic source in an infinite cylindric tube (diameter 2a) can be expressed as the superposition of the fields associated to each mode (,m) : ,
where , the numbers corresponding to the (m+1) zeroes of the derivative of the Bessel function of the 1st kind J, i.e. to the (m+1) extremas of this function. Thus, the acoustic field is propagative following the axis z of the cylinder, is stationary following the radius r, and is either propagative or stationary following the azimuth , according to the nature of the source.
Case of the stationary mode in . As the volume of the fluid at the interior of the cylinder permits to bypass the Oz-axis when the azimuth varies, the quantic number is an integer and can be denoted n. The pressure associated to each mode (,m) is thus given by: .
The animations below represent the variation of the acoustic pressure of a mode (n,m) in a section of the cylindric tube, for a given z, with color level (red = maximum, blue = minimum).

=0 ; m=0	=0 ; m=1	=0 ; m=2	=0 ; m=3
=1 ; m=0	=1 ; m=1	=1 ; m=2	=1 ; m=3
=2 ; m=0	=2 ; m=1	=2 ; m=2	=2 ; m=3
=3 ; m=0	=3 ; m=1	=3 ; m=2	=3 ; m=3

INFINITE CYLINDRIC TUBE WITH A MERIDIAN HALF WALL

The acoustic field created by a monochromatic source in an infinite cylindric tube (diameter 2a) can be expressed as the superposition of the fields associated to each mode (

,m), with here

=N/2. Indeed, as the volume of the fluid at the interior of the cylinder does not permit to bypass the Oz-axis when the azimuth psi

varies (due to the presence of the meridian half wall), the quantic number

is no more necessary an integer. The acoustic pressure associated to each mode (

,m) is thus given by:

For N being an even number, the stationary modes are those of the ordinary infinite tube.
For N being an odd number, the stationary modes are anti-symmetric with respect to the meridian half wall.

The animations below represent the variation of the acoustic pressure of a mode (

,m) in a section of the cylindric tube, for a given z, with color level (red = maximum, blue = minimum).

=0.5 ; m=0	=0.5 ; m=1	=0.5 ; m=2	=0.5 ; m=3
=1.5 ; m=0	=1.5 ; m=1	=1.5 ; m=2	=1.5 ; m=3