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A few illustrative “mode” slides

A few illustrative “mode” slides. Courtesy of Peter Dahl APL-UW. SOURCE. RECEIVER. R. p = (1/ R ) e ikR k= w /c. Spherical Wave in Free Space. PH Dahl University of Washington Mechanical Engineering. r. z. Cylindrical Spreading in an Underwater Waveguide. Source.

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A few illustrative “mode” slides

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  1. A few illustrative “mode” slides Courtesy of Peter Dahl APL-UW

  2. SOURCE RECEIVER R p = (1/R) eikR k=w/c Spherical Wave in Free Space PH Dahl University of Washington Mechanical Engineering

  3. r z Cylindrical Spreading in an Underwater Waveguide Source Between the ro and r closest to source, spreading follows a spherical law: Between the ro and r farther away from source, spreading follows a cylindrical law: PH Dahl University of Washington Mechanical Engineering

  4. spherical spreading cylindrical spreading and trapped modes Ro sea surface r z H qc bottom qc The critical angle from analysis of the plane wave reflection coefficient R(q) PH Dahl University of Washington Mechanical Engineering

  5. sea surface r z r=1026 kg/m3 c =1450 m/s H = 10 m sea water bottom r = 1800 kg/m3 c =1750 m/s bottom continues to z = infinity PH Dahl University of Washington Mechanical Engineering

  6. This doesn’t matter: field calculation ends at H and BC satisfied p = 0 Sea Surface z = H Mode 1 Mode 2 Mode 3 Sea Bottom z = 0 = 0 Waveguide analysis Depth H = 10 m Frequency = 240 Hz Sound speed = 1500 m/s PH Dahl University of Washington Mechanical Engineering

  7. Depth Eigenfunctions for Modes 1-3 surface 3 kH=10 Freq=240 Mode 2 has zero-crossing (null) at 6.68 m Mode 3 has nulls at 2 and 4 m All modes have nulls at the sea surface, or 0 m 2 1 bottom References for today’s lecture March 5, 2010 Ref 1: George Frisk Ocean and Seabed Acoustics: A theory of wave propagation Ref 2: L.E. Kinsler, A.R. Frey, A.B. Coppens, and J.V. Sanders Fundementals of Acoustics 3rd Edition PH Dahl University of Washington Mechanical Engineering

  8. Sea Surface Sea Surface Sea Surface Z (m) Z (m) X (m) X (m) 15 Modes 1 + 2 10 5 0 0 10 20 30 40 50 60 70 80 90 100 Absolute value of pressure field freq = 240 Hz waveguide depth H = 10 m Field 0 here 15 Modes 1 + 2 + 3 10 5 0 0 10 20 30 40 50 60 70 80 90 100 What would similar image of only Mode 1 look like? PH Dahl University of Washington Mechanical Engineering

  9. Mode 1 Spherical spreading Plot 20 log10 |f(r,z)| This shows the decrease in the level of the field, in dB, increasing range from source. For example, at range 500 m, the acoustic field has been reduced by about 30 dB—although the precise reduction is a function range. Notice that for spherical spreading (dashed line), the decrease is much greater, ~53 dB. kH=10 Freq=240 Source Depth 5 m, Receiver Depth 5 m All 3 modes (total field) Modes 1 & 2 PH Dahl University of Washington Mechanical Engineering

  10. Mode 1 Spherical Spreading Some very interesting effects happen if either the source depth or receiver depth are co-located with a the null of a mode kH=10 Freq=240 Source Depth 6.68 m, Receiver Depth 5 m All 3 modes (total field)—but mode 2 is missing because source depth is at its null at 6.68 m. Only Modes 1 & 3 contribute PH Dahl University of Washington Mechanical Engineering

  11. Spherical Spreading kH=10 Freq=240 Source Depth 6.68 m, Receiver Depth 8 m All 3 modes (total field)—but modes 2 are 3 missing because source depth at 6.68 m, and receiver depth is at a null of mode 3. Now only mode1 contributes! PH Dahl University of Washington Mechanical Engineering

  12. sea surface r z r=1026 kg/m3 c =1450 m/s H = 10 m sea water bottom r = 1800 kg/m3 c =1750 m/s bottom continues to z = infinity PH Dahl University of Washington Mechanical Engineering

  13. spherical spreading cylindrical spreading and trapped modes Ro sea surface r z H qc bottom qc The critical angle from analysis of the plane wave reflection coefficient R(q) PH Dahl University of Washington Mechanical Engineering

  14. c2=1450 c2=1800 r1=1025 r2=1800 Any trapped mode will be found here |R| Grazing angle (q) PH Dahl University of Washington Mechanical Engineering

  15. c2=1450 c2=1800 r1=1025 r2=1800 Don’t look here Grazing angle (q) |R| |R e(2ik1sin(qn)H) +1| H=10 m Freq=100 Hz 200 Hz 2 modes at for 200 Hz, 1 mode for 100 Hz PH Dahl University of Washington Mechanical Engineering

  16. SD Water Depth = 50 m, c1=1480 m/s r1=1025 kg/m3 c2=1500 m/s r2=1100 kg/m3 Freq 250 Hz 3 trapped modes SD=20 m, which is at a null of mode 3) Field magnitude plotted in dB with 50 dB range shown Depth (m) Range (m) aspect ratio = 1:1 plot Depth Range PH Dahl University of Washington Mechanical Engineering

  17. Water Depth = 50 m, c1=1480 m/s r1=1025 kg/m3 c2=1500 m/s r2=1100 kg/m3 Freq 250 Hz 3 trapped modes SD=20 m, which is at a null of mode 3) Field magnitude plotted in dB with 50 dB range shown SD Depth (m) Range (m) Modal interference pattern produces weak spots (and hot spots) in the field strength PH Dahl University of Washington Mechanical Engineering

  18. 0 10 SD 20 30 40 50 0 50 100 150 200 250 300 350 400 450 500 Water Depth = 50 m, c1=1480 m/s r1=1025 kg/m3 c2=1500 m/s r2=1100 kg/m3 Freq 2500 Hz 28 trapped modes—more complicated interference SD=20 m Depth (m) Range (m) PH Dahl University of Washington Mechanical Engineering

  19. 0 10 SD 20 30 40 50 0 50 100 150 200 250 300 350 400 450 500 Water Depth = 50 m, c1=1480 m/s r1=1025 kg/m3 c2=1500 m/s r2=1100 kg/m3 Freq 2500 Hz 28 trapped modes—more complicated interference SD=20 m Depth (m) Range (m) Examine the field along this receiver depth track, RD = 20 m. Plot -20log(|f(r,zr)|), now a positive quantity called transmission loss (TL) PH Dahl University of Washington Mechanical Engineering

  20. Simple and handy approximation: TL=20log(r) r < Ro TL=20log(Ro)+10log(r/Ro) r > Ro 0 10 20 TL (dB) 30 40 50 60 70 80 0 50 100 150 200 250 300 350 400 450 500 Range (m) -20log(|f(r,zr)|), PH Dahl University of Washington Mechanical Engineering

  21. 0 10 20 Depth (m) 30 40 50 0 50 100 150 200 250 300 350 00 450 500 Range (m) Water Depth = 50 m, c1=1480 m/s r1=1025 kg/m3 c2=1500 m/s r2=1100 kg/m3 Freq 25000 Hz 100s of trapped modes (275) SD=20 m PH Dahl University of Washington Mechanical Engineering

  22. 0 10 20 Depth (m) 30 40 50 0 50 100 150 200 250 300 350 00 450 500 Range (m) Water Depth = 50 m, c1=1480 m/s r1=1025 kg/m3 c2=1500 m/s r2=1100 kg/m3 Freq 25000 Hz 100s of trapped modes (275): Ray Theory is a better approach SD=20 m qc PH Dahl University of Washington Mechanical Engineering

  23. TL=20log(r) r < Ro TL=20log(Ro)+10log(r/Ro) r > Ro TL (dB) Range (m) -20log(|f(r,zr)|), PH Dahl University of Washington Mechanical Engineering

  24. PH Dahl University of Washington Mechanical Engineering

  25. PH Dahl University of Washington Mechanical Engineering

  26. MORE EXACT CALCULATIONS: THE PARABOLIC WAVE EQUATION DEPTH (m) PH Dahl University of Washington Mechanical Engineering RANGE (m)

  27. Caustic at depth 10 m, range ~190-200 m Ray theory shows the effect --but does poorly in quantify it PH Dahl University of Washington Mechanical Engineering

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