ORE 654 Applications of Ocean Acoustics. Homework Problem Set #2. Assigned 27 October 2011 Due 10 November 2011

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1 ORE 654 Applications of Ocean Acoustics Homework Problem Set #2 Assigned 27 October 2011 Due 10 November 2011 Please use standard 8.5x11 paper. Write clearly in dark pencil/ink, or you can use this document as a template. Use annotated sketches as appropriate. Give text description of steps, making your reasoning clear. Turn in MatLab source and output listings. Start early. Ask questions. Discuss with each other but make sure the work is your own. Note if work is shared. 1. True False Discuss briefly. a) If the reflection coefficient R 12 = 1, then the transmission coefficient T 12 = 0, which conserves energy transfer across the boundary. b) In general, the fastest rays are ones that lie on the sound channel axis. c) An acceptable solution form for the radiated pressure field generated by an omnidirectional (spherical) harmonic sound source at position R is P = (P 0 R 0 / R) exp[i ( t + kr)]. d) If the acoustic wavelength is approximately equal to the ocean depth, ray tracing should not be used to predict the acoustic pressure field. e) A large piston transducer has a more directional beam pattern (narrower main lobe) than a smaller one provided both transducers are operated at the same frequency. f) Sound cannot penetrate into shadow zones. g) The ocean ph affects low frequency (< 1 khz) sound absorption. h) A bubble emits and scatters sound only at its resonant frequency. i) A sonar emits a 10 ms chirp with bandwidth f = 1 khz centered at 10 khz. If two returns are separated in time by 5 ms, they cannot be resolved. j) A coherent group of bubbles has a lower Q than individual bubbles of the same resonance frequency and, consequently, has a large spectral absorption bandwidth, f.

2 2. Short problems a) Write a solution to the planar wave equation that is traveling in the x and +z direction. b) If a surface mixed layer (isothermal, isohaline) is 300 m deep with c(surface) = 1500 m/s, what is the angle of the limiting ray trapped in the surface layer for a sound source at the surface? What are the time, range, and ray path arc length from the surface to the lower turning point? c) What is the acoustic intensity of a = 500 W acoustic source which radiates omnidirectionally (spherical spreading) at a range 10 m? What is the rms sound pressure level SPL in Pa and in db? d) Which of the following ocean phenomena will adversely affect (destroy) the Lloyd Mirror Effect? A) large ocean swell, B) bottom reflected rays, C) strong refraction, D) cold water. e) A sound path has 5 bottom bounces with R 12 = 0.8. What is the transmission loss (TL) as a percentage of sound energy lost due to the bottom, i.e., what percentage of the sound energy is lost due to the bottom bounces? 3. Navigation A transponder is on the bottom at 4500 m depth in the North Pacific. We want to locate it as accurately as possible. A ship survey is conducted, measuring the roundtrip travel time between the ship and the transponder for various ship positions. Attached is sound speed profile from 31.03N, W, measured at the time of the experiment (the Acoustic Engineering Test (AET), 1994, near Jasper Seamount first at sea test of the ATOC source). How should the ship survey be conducted to minimize x,y errors in the absolute bottom transponder location? I.e., how do you position the ship survey points so as to cancel sound speed errors as much as possible? The ATOC source is suspended from FLIP 500 m below the ocean surface. An acoustic interrogator on it measures round trip travel times to the bottom transponder (actually 4 of them). Assume the bottom transponder position and interrogator have nominal positions such that the horizontal range is 4000 m. What horizontal range and depth errors arise when different estimates of sound speed are used with different acoustic propagation? a) Obtain a single average c by calculating cave = (1/H) * int(c(z))dz over z1,z2, and then assume a single straight line ray. b) a ray trace assuming constant sound speed gradient layers.

3 Use the attached MatLab code for ray tracing (appended at end of this document). Test the code for a known case and show this. Modify as necessary for the specifics of this problem. Obtain the eigenray by varying the launch angle until the final endpoint of the ray at the transponder depth has exactly (within millimeters) 4000 m range. Make sure the number of layers n is adequate, so the results do not change if n is further increased. Plot the straight ray and the actual curved ray path (z vs x) and a measure of the deviation. How and why do the two differ? Is ray tracing necessary to obtain sub-meter location accuracy for navigation and/or geodesy applications? 4. Sonar Equation - 1 Depth m Sound speed m/s a) Consider #3 (survey geometry) above. The omni-directional interrogator and transponder source levels are 195 db re 1 upa at 1 m. They transmit 10 khz, 1 khz bandwidth coded signals/pings lasting 1 ms. Estimate the signal to noise ratio budget assuming coherent processing of the signal (use Matlab). What is the rms travel time precision (= 1/ (bandwidth Hz * SNR in terms of amplitude)? Give the form of the sonar equation you use and the values for each term. Make clear all assumptions made and show all terms of the budget. Use #5d,e below for missing information and guidance. If you sent a coded signal lasting 0.1 s, what is the improvement in SNR and travel time precision. b) Change (a) to an inverted echosounder configuration, i.e., a bottom transducer transmitting and receiving the echo from the sea surface. Repeat calculations above, but account for round trip range. Assume a -6 db loss at the surface. Be sure to handle both cases, 1 and 100 ms signal durations.

4 5. Sonar equation 2 A circular piston-type transducer, radius a = 30 cm, can be operated at 1 khz or 10 khz. a) If 0-1 = 300 W at 1 khz and 0-10 = 100 W at 10 khz, what are the axial source levels (SL) at each frequency? (use = 1024 kg/m 3, c 0 = 1500 m/s) Use directivity index = DI = 10 log 10 Q t where Q t = (ka) 2. b) Assume spherical spreading to a 10 cm radius rigid sphere at range 150 m. What is the one-way transmission loss (TL) at each frequency if attenuation due to molecular absorption is neglected? c) What is the target strength (TS) of the sphere at each frequency? d) If the spectral densities of omni-directional ambient noise are P nn (1 khz) = 55 db re 1 Pa 2 /Hz and P nn (10 khz) = 40 db re 1 Pa 2 /Hz (e.g., from Medwin Figure 5.11), what are the ambient noise (AN) levels for the transducer at each frequency? Assume system bandwidth is f = 100 Hz. (Hint: AN must include P nn, bandwidth, and receiver directivity terms.) e) If the return signal is detected by the same transducer, at which frequency is detection of the sphere more likely? Use DT=SNR = (SL 2 TL + TS) (AN). f) If the water contains 1 mm radius bubbles (the only bubble size present) at an average density of N = 2000 bubbles/m 3, what is the additional TL due to the bubbles at each frequency? Use the figure below to estimate the various damping terms. Show how you get the damping constants from the figure. Make sure TL accounts for the round trip distance. g) Which frequency is more likely to detect the sphere in the bubble water? What is the SNR for the most favorable frequency? (Hint: use DT from e, but include this new bubble loss contribution in TL.)

Vary parameters and plot on one graph: depth 10, 100 m; velocity 1, 10 m/s over an appropriate spatial range (e.g., +/- 500 m). 7.

6 6. Doppler effect The submarine makes noise with steady tonals that is received on the ship at the surface. Derive the expression for Doppler shift as a function of speed v, frequency f, depth H and sound speed c. Use it to replicate the curve above. Vary parameters and plot on one graph: depth 10, 100 m; velocity 1, 10 m/s over an appropriate spatial range (e.g., +/- 500 m). 7. Acoustic normal modes In high wind conditions breaking waves produce a layer of bubbles just under the ocean surface. While these bubbles have a minimal effect on the water density, they greatly increase compressibility and thus reduce the speed of sound in the bubble layer. This bubble layer can be treated as an acoustic waveguide. Assume the bubble layer (subscript 0) is h = 2 m thick and 0 = 1 = 1030 kg/m 3, c 0 = 1480 m/s and c 1 = 1510 m/s. Assume an ideal waveguide R u = -1, R l = +1. a) Write down the characteristic equation for this waveguide. b) What is the critical angle for this waveguide? c) What are the cutoff frequencies for the first four modes? Note the maximum vertical wavenumber/eignevalue ( m ) is now (k cos c), rather than just k, where k = 2 f/c 0. d) Write down the expressions for the first four eigenfunctions. e) Sketch the shape of the first two eigenfunctions. f) What is the value of the second eigenfunction at z = 1 m? g) A sound source at 5000 Hz will excite which of the first four modes?

7 h) This is not an ideal waveguide. For a 5000 Hz source the mode propagation angles for the first two modes are 1 = and 2 = What are the corresponding phase propagation numbers ( ) for these two modes? i) What is the interference wavelength for these two modes? j) What are the phase propagation speeds for these two modes? m) Given the solution for the pressure field below the waveguide has the vertical structure P 1 = A exp[- b m k 1 (z-h)] = A Z m (z) for z > h where b m = [(c o /c 1 ) 2 sin 2 m 1] 1/2 and k 1 = 2 f/c 1 What are the skin depths of the first two modes (i.e., where the pressure = 1/e of the value at z = h)? n) If you had a hydrophone 10 m below the surface (8 m below the waveguide), from which mode would most of the detected energy be from? o) Assume most of the attenuation in the waveguide is due to resonant bubbles. How big are these bubbles (5000 Hz)? Use surface values. p) Given R = and r = , what are the total scattering cross section and total extinction cross section? q) Given a measured attenuation rate of 1 db/m, what is the bubble concentration?

8 Matlab code to help with Problem 2b (caution line wrap around) text0 = 'start HW 2.2 ********************************************' % Bruce Howe clear all format compact cd /Volumes/B_HOWE_B/Documents/Acoustics_Class/Homework/HW_2 %cd /Users/howe/Documents/UH/Acoustics_Class/Homework/HW_2 rad2deg = 180 / pi deg2rad = pi / 180 % 1 text1 = 'start HW 2, #2b ********************************************' zcprof = [ ] cprof = [ ] %test zci = 500 czi = interp1(zcprof,cprof,zci,'linear') rnom = 4000 z1 = 500 z2 = 4500 n = 20 dz = (z2 - z1) / (n-1) zray = z1:dz:z2 nray = length(zray) cray = interp1(zcprof,cprof,zray,'linear') rray = 0 * (1:nray); sray = 0 * (1:nray); tray = 0 * (1:nray); thetaray = 0 * (1:nray); cav1 = sum(cray)/nray thetanom = atan2((z2-z1),rnom) % 1000th radian step delta = pi/180 / 1000 % adjust - convenient factor to vary (by hand) to find eigen ray

9 % adjust = adjust = 0 dtheta = adjust * delta thetanom = thetanom + dtheta thetanomd = thetanom * rad2deg % set up straight line at same z points for future reference drline = rnom / (n-1) rline = 0:drline:rnom % nominal slant range range_nom = sqrt(rnom^2+(z2-z1)^2) % ray paramter a = sin(thetanom)/cray(1) thetaray(1) = asin( a * cray(1) ); % for each z layer calculate b, r, s, t, and cumulatives for i=2: nray; i; thetaray(i) = asin( a * cray(i) ); b = (cray(i)-cray(i-1)) / (zray(i)-zray(i-1)); radius = 1/ (a * b); r = abs( radius * (cos(thetaray(i-1)) - cos(thetaray(i))) ); s = abs( radius * (thetaray(i-1) - thetaray(i)) ); ratio1 = (cos(thetaray(i-1))+1)/(cos(thetaray(i))+1); t = abs( (1 / b)* log( (cray(i)/cray(i-1)) * ratio1)); rray(i) = rray(i-1) + r; sray(i) = sray(i-1) + s; tray(i) = tray(i-1) + t; end thetarayd = thetaray.* rad2deg % list zray cray thetarayd rray sray tray plot(rray,-zray) hold on

10 rplot= 0 * (1:nray) +z1 plot([0 rnom],[-z1 -z2], 'r-') plot(rline,-zray,'g-') % 20 (to more easily see) difference in horizontal range between niminal staright line and ray at % each zray ztemp = 20*(rray-rline) plot(rline,ztemp-z1)

The spatial structure of an acoustic wave propagating through a layer with high sound speed gradient

The spatial structure of an acoustic wave propagating through a layer with high sound speed gradient Alex ZINOVIEV 1 ; David W. BARTEL 2 1,2 Defence Science and Technology Organisation, Australia ABSTRACT