Surface and acoustic waves natural frequencies estimation by boundary element method

Grzegorz Waśniewski

doi:10.2478/sgem-2026-0004

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Surface and acoustic waves natural frequencies estimation by boundary element method

Studia Geotechnica et Mechanica

Volume 48 (2026): Issue 1 (March 2026)

By: Grzegorz Waśniewski

Open Access

|Apr 2026

Figures & Tables

Determinant sign – Equation (30) for 84 BEM dof.

Determinant sign – Equation (22) for N = 9. ● – Proper frequencies.

Proper values identification – Equation (26). Selected node – l V {l}_{{\rm{V}}} .

Proper values identification – Equation (26). Selected node – l H {l}_{{\rm{H}}} .

Tank and acoustic waves (●) frequencies: t = 0.05 m, B ≠ 0.

Tank and surface waves (●) frequencies: t = 0.05 m, B ≠ 0.

Tank and acoustic waves (●) frequencies: t = 0.002 m, B ≠ 0.

Tank and surface waves (●) frequencies: t = 0.002 m, B ≠ 0.

The lowest natural frequencies for surface waves (Hz)_

No of dof	f 1 {f}_{1}	f 2 {f}_{2}	f 3 {f}_{3}	f 4 {f}_{4}	f 5 {f}_{5}	f 6 {f}_{6}	f 7 {f}_{7}
36	0.8496	1.226	1.532	1.782	2.009	2.202	2.352
60	0.8438	1.210	1.493	1.740	1.962	2.167	2.357
84	0.8417	1.203	1.481	1.721	1.935	2.133	2.319
Exact	0.8375	1.191	1.459	1.685	1.884	2.064	2.229

The lowest natural frequencies for acoustic waves (Hz)_

N N	f 1 {f}_{1}	f 2 {f}_{2}	f 3 {f}_{3}	f 4 {f}_{4}	f 5 {f}_{5}	f 6 {f}_{6}
1		300.2
3	393.7	838.1	589.7		665.4
5	401.0	502.3	785.2	1,030	970.3	1,461
7	401.0	502.0	772.7	1,115	1,226	1,577
9	401.0	502.0	772.6	1,137	1,209	1,311
14	401.0	502.0	772.6	1,138	1,206	1,326
15	401.0	502.0	772.6	1,138	1,206	1,326
Exact	402.8		772.4		1,208

Acoustic waves natural frequencies vs sound speed (Hz)_

a ( m / s ) {a}(\text{m}/\text{s})	Case	f 1 {f}_{1}	f 2 {f}_{2}	f 3 {f}_{3}	f 4 {f}_{4}	f 5 {f}_{5}	f 6 {f}_{6}
5	Equation (20)	1.553	1.749	2.693	3.921	4.238	4.515
5	Equation (22)	1.383	1.731	2.664	3.922	4.170	4.520
10	Equation (20)	2.859	3.464	5.342	7.844	8.374	9.038
10	Equation (22)	2.766	3.462	5.328	7.844	8.340	9.040
30	Equation (20)	8.330	10.39	15.99	23.53	25.03	27.12
30	Equation (22)	8.297	10.39	15.98	23.53	25.02	27.12
100	Equation (20)	27.67	34.62	53.28	78.44	83.40	90.40
100	Equation (22)	27.66	34.62	53.28	78.44	83.40	90.40
500	Equation (20)	138.3	173.1	266.4	392.2	417.0	452.0
500	Equation (22)	138.3	173.1	266.4	392.2	417.0	452.0
1,450	Equation (20)	401.0	502.0	772.6	1,137	1,209	1,311
1,450	Equation (22)	401.0	502.0	772.6	1,137	1,209	1,311

Surface waves natural frequencies vs sound speed (Hz)_

a ( m / s ) {a}(\text{m}/\text{s})	f 1 {f}_{1}	f 2 {f}_{2}	f 3 {f}_{3}	f 4 {f}_{4}	f 5 {f}_{5}	f 6 {f}_{6}
5	0.8113	1.183	1.464	1.708	1.921	2.119
10	0.8340	1.198	1.477	1.717	1.932	2.130
30	0.8408	1.203	1.481	1.720	1.935	2.133
100	0.8416	1.203	1.481	1.721	1.935	2.133
500	0.8417	1.203	1.481	1.721	1.935	2.133
1,450	0.8417	1.203	1.481	1.721	1.935	2.133
Example 1	0.8417	1.203	1.481	1.721	1.935	2.133

Acoustic waves natural frequencies vs wall thickness (Hz)_

t ( m ) {t}\left(\text{m})	Variant	f 1 {f}_{1}	f 2 {f}_{2}	f 3 {f}_{3}	f 4 {f}_{4}	f 5 {f}_{5}	f 6 {f}_{6}
0.000	Exact		772.4			1,376	1,378
0.000	equation (22)	497.4	774.7	1,135	1,333	1,355	1,368
0.001	B = 0 {\boldsymbol{B}}={\bf{0}}	497.1	767.4	1,132	1,330	1,358	1,371
0.001	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	498.3	763.0	1,135	1,351	1,381	1,382
0.002	B = 0 {\boldsymbol{B}}={\bf{0}}	498.9	774.7	1,136	1,328	1,362	1,374
0.002	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	497.2	757.8	1,135	1,338	1,494	1,534
0.005	B = 0 {\boldsymbol{B}}={\bf{0}}	499.0	770.1	1,137	1,332	1,357	1,370
0.005	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	498.3	710.4	1,135	1,288	1,488	1,527
0.010	B = 0 {\boldsymbol{B}}={\bf{0}}	499.0	785.4	1,138	1,336	1,349	1,365
0.010	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	497.9	710.2	1,134	1,179	1,460	1,498
0.020	B = 0 {\boldsymbol{B}}={\bf{0}}	498.1	743.4	1,137	1,352	1,482	1,497
0.020	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	491.4	653.0	1,107	1,142	1,465	1,503
0.050	B = 0 {\bf{B}}={\bf{0}}	499.5	796.3	1,107	1,135	1,218	1,311
0.050	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	491.7	518.8	970.5	1,138	1,256	1,301
0.100	B = 0 {\boldsymbol{B}}={\bf{0}}	496.9	542.2	836.6	1,137	1,237	1,310
0.100	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	379.5	500.3	873.6	1,137	1,210	1,307
0.200	B = 0 {\boldsymbol{B}}={\bf{0}}	345.8	500.7	751.0	1,137	1,491	1,497
0.200	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	410.4	502.1	762.1	1,137	1,231	1,306
0.500	B = 0 {\boldsymbol{B}}={\bf{0}}	397.6	501.9	758.6	1,137	1,181	1,313
0.500	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	391.7	503.4	779.2	1,137	1,213	1,309
	Rigid	401.0	502.0	772.6	1,137	1,209	1,311

Surface waves natural frequencies vs wall thickness (Hz)_

t ( m ) {t}\left(\text{m})	Variant		f 1 {f}_{1}	f 2 {f}_{2}	f 3 {f}_{3}	f 4 {f}_{4}	f 5 {f}_{5}	f 6 {f}_{6}
0.0000	Exact		0.8375	1.191	1.459	1.685	1.884	2.064
0.0000	Equation (45)		0.8417	1.203	1.481	1.720	1.934	2.131
0.0005	B = 0 {\boldsymbol{B}}={\bf{0}}	0.0505	0.8361	1.195	1.484	1.773	1.962	2.141
0.0005	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	0.0502	0.8322	1.189	1.470	1.758	1.946	2.123
0.0010	B = 0 {\boldsymbol{B}}={\bf{0}}	0.1418	0.8416	1.292	1.525	1.757	1.921	2.095
0.0010	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	0.1403	0.8317	1.277	1.508	1.735	1.897	2.070
0.0020	B = 0 {\boldsymbol{B}}={\bf{0}}	0.3793	0.9165	1.220	1.391	1.589	1.744	1.955
0.0020	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	0.3732	0.9006	1.201	1.376	1.575	1.741	1.952
0.0050	B = 0 {\boldsymbol{B}}={\bf{0}}	0.7803	1.164	1.452	1.694	1.913	2.101	2.293
0.0050	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	0.7787	1.161	1.449	1.688	1.906	2.078	2.268
0.0100	B = 0 {\boldsymbol{B}}={\bf{0}}	0.8341	1.199	1.478	1.718	1.933	2.132	2.317
0.0100	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	0.8340	1.199	1.478	1.718	1.933	2.132	2.317
0.0500	B = 0 {\boldsymbol{B}}={\bf{0}}	0.8416	1.203	1.481	1.721	1.935	2.133	2.319
0.0500	B ≠ 0 {\boldsymbol{B}}\ne {\bf{0}}	0.8416	1.203	1.481	1.721	1.935	2.133	2.319
	Rigid – Equation (30)	0.8417	1.203	1.481	1.721	1.935	2.133	2.319
		f 1 {f}_{1}	f 2 {f}_{2}	f 3 {f}_{3}	f 4 {f}_{4}	f 5 {f}_{5}	f 6 {f}_{6}	f 7 {f}_{7}

References

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DOI: https://doi.org/10.2478/sgem-2026-0004 | Journal eISSN: 2083-831X | Journal ISSN: 0137-6365

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Language: English

Page range: 31 - 44

Submitted on: Jun 26, 2025

Accepted on: Apr 9, 2026

Published on: Apr 27, 2026

Published by: Wroclaw University of Science and Technology

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

Fluids,

Waves,

Natural frequencies,

Boundary element method,

Multiple reciprocity method

Related subjects:

Physics,

Mechanics and fluid dynamics

© 2026 Grzegorz Waśniewski, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 48 (2026): Issue 1 (March 2026)

Surface and acoustic waves natural frequencies estimation by boundary element method

Figures & Tables

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

The lowest natural frequencies for surface waves (Hz)_

The lowest natural frequencies for acoustic waves (Hz)_

Acoustic waves natural frequencies vs sound speed (Hz)_

Surface waves natural frequencies vs sound speed (Hz)_

Acoustic waves natural frequencies vs wall thickness (Hz)_

Surface waves natural frequencies vs wall thickness (Hz)_

Paradigm

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