8.2Wavenumber Vectors in Resonant Cavities

In Chap. 7, the natural frequency flmn of the eigenmode (l, m, n) in resonant cavities was calculated without referring to the wavenumber vector based on the formulas below:

where:

k2lmn ¼ k2xl þ k2ym þ k2zn

and:

π π π

kxl ¼ Lx l; kym ¼ Ly m; kzn ¼ Lz n

In Chap. 8, we will relate the natural frequencies to the wavenumber vectors in resonant cavities.

In this section, the formulas of wavenumber vectors will be introduced. In the next section, the physics of wavenumber vectors as traveling waves in resonant

cavities will be explained.

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The discretized wavenumber vectors k lmn of an eigenmode (l, m, n) can be calculated using the formulas developed in the previous chapter as:

!

k lmn ¼ kxlbex þ kymbey þ kznbez

where:

π π π

kxl ¼ Lx l; kym ¼ Ly m; kzn ¼ Lz n

and:

For example, the wavenumber vector and natural frequency of the eigenmode (l, m, n) ¼ (0, 4, 7) in terms of the cavity dimensions (Lx, Ly, Lz) are:

!

The discretized wavenumber vectors k lmn can be visualized with a wavenumber grid.

In an example of a special case with the eigenmode ¼0 , uniform pressure in the

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x-direction, the discretized wavenumber vectors k 047 of the eigenmode (l, m, n) ¼ (0, 4, 7) can be visualized by sketching a wavenumber grid as:

208 8 Acoustic Waveguides

Wavenumber Vector of the eigen-mode ,, = 0, 4, 7

In another example of a special case with the eigenmode ¼0 , uniform pressure in

An example of a general case with an eigenmode with nonzero indexes, l ¼6 0, m ¼6 0, and n ¼6 0, is (l, m, n) ¼ (2, 3, 6). The wavenumber vector can be drawn in 3D coordinates as:

=

=

=

The discretized wavenumber vector has a tail located at the origin and a head (the black dot) in the discretized grid as shown in the figure above. The formula of the discretized wavenumber vector is:

!

k lmn ¼ kxlbex þ kymbey þ kznbez

where:

π π π

kxl ¼ Lx l; kym ¼ Ly m; kzn ¼ Lz n

The natural frequency of resonant cavities is related to the wavenumber vector as:

!

where klmn is the length of the discretized wavenumber vector k lmn as:

or:

k2lmn ¼ k2xl þ k2ym þ k2zn