Acoustic Scattering from a Sphere - Steve Turley

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Soft Sphere, a=10.010 6 theta, degrees10 5d sigma/d omega10 410 310 210 10 20 40 60 80 100 120 140 160 180Figure 4: Dierential cross section for scattering from an acoustically soft sphere of radius 10.0 wavelengths.16
to be equal to the cross sectional area of the sphereA = πa 2 . (107)If I integrate | 2 /π numerically for spheres of radius 10 and 20, I get 10.28 2 and 20.02 2 .> > [theta10,dsigma10]=soft(10);> > [theta20,dsigma20]=soft(20);> > sqrt(sum(dsigma10.*sin(theta10))*(theta(2)-theta(1)))ans =10.2574> > sqrt(sum(dsigma20.*sin(theta20))*(theta(2)-theta(1)))ans =20.0186I can also do these integrals analytically by taking the large argument expansions for the Bessel and Hankelfunctions. Abramowitz and Stegun[5] formula 10.1.16 gives a series expansion for h (1)n (z) ash (1)∑ nn (z) = i −n−1 z −1 e iz (n + 1/2, k)(−2iz) −k . (108)k=0For large x, the dominant term in this series will be the k=0 term. Therefore, the asymptotic values for j nand h (1)n for large x areh n (x) = ei(x−π/2−nπ/2)(109)xj n (x) = cos ( x − π 2 − )nπ2. (110)xNote that the asymptotic expansion for h n in Equation 109 can be subsituted into Equation 63 to giveEquation 66 and into Equation 82 to give Equation 84, providing a good cross check on those sets offormulas.The integrated cross section can be computed using the orthogonality relation for the Legendre polynomials∫ 1The integrated square of Equation 66 is then∫σ =∫ dσdΩ dΩ =P n (x)P m (x) =−1|f| 2 dΩ = 2 k 222n + 1 δ n,m . (111)∞∑n=0(2n + 1) (j n(ka)) 2∣∣h (1)(ka) ∣ 2 . (112)I'd like to make a further reduction of this analytically, but haven't been able to gure it out yet.5.1.2 Near FieldI also implemented a computation of the near eld intensity in the le softnear.m. It takes a very longtime to run in matlab and is much more practical in a compiled language.function [x,y,intensity]=softnear(a)% *** soft(a) ***% compute scattering from a soft sphere of radius a% at the angle theta using nmax as the maximum value% of n in the sum.% The terms in the series oscillate up to about n=ka and then% decrease rapidly. Taking 2ka terms, should be more than sufficient.n17
Page 2 and 3: 8 Dierential cross section for scat
Page 4 and 5: 1.4 Solutions in Spherical Coordina
Page 6 and 7: 3 Scattering TheoryOne of the rst t
Page 8 and 9: Substituting the value for s nm fro
Page 10 and 11: Series Approach As the the soft sph
Page 12 and 13: function j=sphj(n,x)% compute the s
Page 14 and 15: 20Soft Sphere, a=1.0181614d sigma/d
Page 18 and 19: % It always uses a minimum of 5 ter
Page 20 and 21: 5.2.1 Far FieldThe function hard co
Page 22 and 23: 6Hard Sphere, a=1.05d sigma/d omega
Page 24 and 25: Hard Sphere, a=10.010 5 theta, degr
Page 26 and 27: % decrease rapidly. Taking 2ka term
Page 28 and 29: 20Penetrable Sphere, a=1.0,kf=1.5,r
Page 30 and 31: Penetrable Sphere, a=10.0,kf=1.5,rh

Acoustic Scattering from a Sphere - Steve Turley

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