Focal Depth Shifting of Phase Conjugated Wave in PekerisWave GuidePekeris ウエーブガイドにおける位相共役波の焦点深度シフト

Yoshiaki Tsurugaya†(NEC), Toshiaki Kikuchi(National Defense Academy) and KoichiMizutani(Univ. of Tsukuba)

鶴ヶ谷芳昭†(NEC)，菊池年晃(防衛大)，水谷孝一(筑波大）

1. Introduction

The interest to the phase conjugated wave in ashallow water is increasing, and many studies havebeen conducted. Most of those studies are utilizingthe characteristics which the phase conjugated waveconverges on the position of a sound source. Thoseare a different unique characteristics from a normalsound propagation. However, since the convergenceposition is limited, the application is also limited.Recently, the researches which change aconvergence position has started[1,2]. Previously,we have proposed the method of moving aconvergence position in the direction of range[3].Now, we propose about the method of shifting theconvergence position in a depth direction. And theresult examined about the characteristics isreported.

2. Array Element Shifting and the Green'sFunction

When the sound propagation in shallow water isexamined in a ray theory, the sound fields of areceiving point are composed of four kinds of ray,and the sound pressure is expressed with thefollowing formula[4].

0

1,m

mzr

2

2

1

1 expexp

m

m

m

m

R

ikR

R

ikR

4

4

3

3 expexp

m

m

m

m

R

ikR

R

ikR(1)

22mnmn RR z

hsm zzHm 21z , hsm zzmH 122z

hsm zzHm 23z , hsm zzmH 124z

Although mnR is related to the amplitude and the

phase of each sound ray, since change of amplitudecan be ignored, the sound pressure is affected by thephase. If a focal depth change is compensated bydepth change of an array element (receiving point),these formulas provide no phase change. Thesephase characteristics are applied to a phase

conjugate field. Then, in the arrangement shown inFig.1, the sound waves radiated from the soundsource are received with each array element. Thereceived signals are processed by the phaseconjugation. And if the signals are radiated from thesame element after performing phase conjugationprocessing, the sound waves will be converged onthe position of a sound source. Now, in order tochange a focal depth, the signals are radiated fromthe different element from the received elementafter phase conjugation processing. The phaseconjugate field is written as

N

nynsnsc GGG

1

,,, rrrrrr (2)

Where snG rr , is the Green's function for the

thn array element from the sound source, and

ynG rr, is the Green's function for the point rfrom the element which shifted only y .

3. Depth Shift and Sound Field

Shown in Fig.1, the depth of water, soundsource depth, and the frequency of the sound waveare 100 m, 50 m, and 500 Hz, respectively. Therange between the sound source and the array is5.0km. The sound waves radiated from the soundsource is received with the array element arrangedto a 1/2-wavelength spacing from the sea surface tothe sea bottom. After the received signals are phase-conjugated at every element, they are all radiatedfrom the element shifted to an upper part or a lowerpart. Three elements shifting downward are shownin Fig. 1. The Green’s functions in Eq. (2) areobtained by a normal mode method. The soundspeed and the density of seawater are 1500 m/s and1000 kg/m3, respectively. Those of sediment are1600 m/s and 1500 kg/m3, respectively. The mode

Fig.1 Arrangement of a sound source and an array, andthe outline of an element shifting

--------------------------------------------------------------y-tsurugaya@bp.jp.nec.com

Proceedings of Symposium on Ultrasonic Electronics, Vol.28, (2007), pp. 473-47414-16 November, 20073-11P-58

numbers produced in this environment are 23. Thephase conjugation processing is given to the signalreceived by the array element n . The signal isradiated from the yn element shifted downwardfrom the input element. The sound fields of thephase conjugated wave are obtained by using Eq.(2)and the normal mode method.

Figure 2 shows the sound fields near the soundsource when the number of shift elements is nine.The position of the original sound source is range 0,and 50m in depth. The point where sound pressureis high is seen from the source location up anddownward. The difference between the depth of theoriginal sound source and depth in the high soundpressure part is almost equal to the product of thenumber of shift elements and the array spacing. Onthe other hand, the point where sound pressure ishigh has hardly changed horizontally. Next, thenumber of shift elements is changed, and thechange in the part where sound pressure is high,that is, focus is examined. When the number of shiftelement is 7, 10, and 18, the sound pressure depthdistribution in the range of the focus is shown inFig.3. The solid line, dashed line, and dash dot linecorrespond to the number of shift elements, 7, 10,and 18, respectively. Each curve has a symmetricpeak for the original source location, and otherpeaks exist, too. It is thought that these peakscorrespond to four kinds of eigenray. The soundpressure depth distribution in the range of the focuswhen the element is shifted up is shown in Fig.4contrary to Fig.3. As for the solid line, dashed line,and the dash dot line, the number of shift elementcorrespond to 7, 10, and 15, respectively. Eachcurve has a symmetric peak corresponding to thenumber of shift elements, and other peaks exist, too.Finally, when the elements are shifted to both theupper side and the lower side, the sound pressuredistribution near the sound source is shown in Fig.5.The symmetric peaks appear at the position of thecorresponding number of shift element, and anuncertain peak has disappeared.

4. Conclusion

In the phase conjugation processing in shallowwater, changing the depth of the formed focusbecame possible by changing the input element andthe radiating element.

References1. S. C. Walker et al.: JASA, 118 (2005) 13412. S. C. Walker et al,:JASA, 120 (2006) 27553. T. Kikuchi et al.: Ext. Abstr. (Spring Meet.,2007); Acoustical Soc. Japan,2-6-1 [in Japanese]4. L. M. Brekhovskikh: Fundamentals of OceanAcoustics (AIP Press, 2003)

Fig.2 Sound field near a sound source when nine elementsare shifted downward.

Fig. 3 Sound pressure depth distribution in the distance ofthe sound source when the number of element shifting ischanged. element shifting; 7(－), 10(---), 18(-.-)

Fig.4 Sound field near sound source when the elements isshifting up. element shifting; 7(－), 10(---), 15(-.-)

Fig.5 Sound field near a sound source when the elementsare sifting to both the upper side and the lower side.Number of shifting is 15 elements.