8.4Wavenumber Vectors in Acoustic Waveguides

In the acoustic waveguide shown in the right figure below, the acoustic wave can travel in the z-direction but will reflect on all four sides of the waveguide.

Waveguides are elongated rectangular cavities with no closed walls in the axial direction. We can construct a waveguide by removing the two walls in the z-direction of an elongated rectangular cavity. The figure below shows that the two walls in the z-direction of a resonant cavity are removed to allow acoustic waves to propagate in the z-direction in an acoustic waveguide.

After the two walls in the z-direction are removed, the four wavenumber vectors become two wavenumber vectors because there is no reflection in the z-direction as shown in the figure below:

With the two walls in the z-direction, the wavenumbers in both the y-direction and z-direction are discretized as:

kym ¼ π m kym is DISCRETIZED where m is an INTEGER

Ly

Without the two walls in the z-direction, the wavenumbers in the y-direction are still discretized, but the wavenumbers in the z-direction become continuous numbers as:

In a waveguide, because Lz can be considered as a very large number, and based on the formula above, kzn is continuous:

: Continuous Number

Because the wavenumber kz changes from being discretized numbers to being continuous numbers, a pure tone (single-frequency sound) will propagate along the axial direction with any transverse eigenmode of the waveguide.

Example 8.3: (Wavenumber Vectors in a Waveguide)

Assume a waveguide has a rectangular cross-section of Lx ¼ 13 [m] and Ly ¼ 12 [m].

(a) Calculate all the possible transverse eigenmodes that can carry an 850 [Hz] pure

(b) Calculate the cutoff frequency of the waveguide.

A summary of all the possible transverse eigenmodes found in the table is:

!

k 00 ¼ 5πbezðdirect waveÞ

The detailed calculations are shown below: (l, m, n) ¼ (0, 0, ): . . . this is the direct wave:

h i k2z ¼ k2 0 0 ¼ 25π2 m1

where k is the wavenumber of an 850 [Hz] pure tone as:

ðl, m, nÞ ¼ ð0, 1, Þ : k2z ¼ k2 k2x0 k2y1 ¼ ð5πÞ2 0 ð2πÞ2 ¼ 21π2 ðl, m, nÞ ¼ ð0, 2, Þ : k2z ¼ k2 k2x0 k2y2 ¼ ð5πÞ2 0 ð4πÞ2 ¼ 9π2 ðl, m, nÞ ¼ ð0, 3, Þ : k2z ¼ k2 k2x0 k2y3 ¼ ð5πÞ2 0 ð6πÞ2 ¼ 11π2 ðl, m, nÞ ¼ ð1, 0, Þ : k2z ¼ k2 k2x1 k2y0 ¼ ð5πÞ2 ð3πÞ2 0 ¼ 16π2

ðl, m, nÞ ¼ ð1, 1, Þ : k2 ¼ k2 k2 k2 ¼ ð5πÞ2 ð3πÞ2 ð2πÞ2 ¼ 12π2

z x1 y1

218 8 Acoustic Waveguides

ðl, m, nÞ ¼ ð1, 2, Þ : k2z ¼ k2 k2x1 k2y2 ¼ ð5πÞ2 ð3πÞ2 ð4πÞ2 ¼ 0 ðl, m, nÞ ¼ ð2, 0, Þ : k2z ¼ k2 k2x2 k2y0 ¼ ð5πÞ2 ð6πÞ2 0 ¼ 11π2

Summary of Wavenumber Vectors in Waveguides

For the eigenmodes having a mode number l ¼ 0, there are two eigenmodes (l, m, n) ¼ (0, 1, ) and (0, 2, ) in y-direction. These two eigenmodes have the following properties:

Because kz is continue, eigenmodes can be constructed by the combination of a discretized kym and an arbitrary kz that satisfies:

k ¼ k2x0 þ k2ym þ k2z

Therefore, the possible eigenmodes are intersections (black dots) between a circle with a radius k and horizontal lines with a height kym as shown in the figure below: