Technical Brief

Frequencies and Modes of the Helmholtz Equation With Zero Normal Derivatives in Isosceles Triangular and Rhombic Domains

[+] Author and Article Information
C. Y. Wang

Departments of Mathematics and Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: cywang@mth.msu.edu

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 29, 2015; final manuscript received March 1, 2016; published online April 18, 2016. Assoc. Editor: Miao Yu.

J. Vib. Acoust 138(4), 044501 (Apr 18, 2016) (6 pages) Paper No: VIB-15-1457; doi: 10.1115/1.4033062 History: Received October 29, 2015; Revised March 01, 2016

The two-dimensional Helmholtz equation with zero normal derivatives on the boundary is studied using boundary collocation. The frequencies and modes are found for the isosceles triangular and rhombic domains. The solutions are important in predicting the standing waves in containers and also the transverse electric (TE) waves of electromagnetic, optical, and acoustic wave guides.

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Fig. 1

(a) The isosceles triangle. Vertex angle is 2θ0 and (b) the rhombus.

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Fig. 2

Frequency distribution for the isosceles triangle. Solid curves are modes symmetric to y = 0 or the symmetry line (SS), dashed curves are those antisymmetric to the symmetry line (SA). Dotted curve is from the asymptotic formula, Eq. (35).

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Fig. 3

First five mode shapes for the isosceles triangle. The interior nodal curves are shown. From top: θ0=20deg,  40deg,  and  60deg.

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Fig. 4

Instantaneous free surface oscillations for fluid in a triangular basin of vertex angle 40deg. Left column and right column are half a period apart. From top row: k = 3.7502, 5.4428, 6.8512, 9.5016, and 9.7383, corresponding to the first row of Fig. 2.

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Fig. 5

Frequency distributions for the rhombus. — SS mode, — — SA mode, — · — AS mode, and — · · — AA mode. Dotted curves are from Eq. (35) for SA mode and Eq. (37) for AA mode.

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Fig. 6

Mode shapes for the rhombus (first five frequencies). Top row: θ0=15deg, middle rows: θ0=30deg, bottom rows: θ0=45deg. In each group, the modes in the same column have the same frequency.




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