First created: Oct. 26, 2003 Updated: January 28, 2008
Acoustic Impedance for Spherical Elastic Waves

......

1. Impedance Concept
   Electrical impedance is voltage over current, and is measured in ohms: Z = V/I. Mechanical impedance is force applied over velocity of the connection point: Z = F_x / v_x. Acoustic impedance for pressure waves in a fluid (propagating in the +x direction) is pressure over velocity: Z = P / v_x. There is no popular general formulation of acoustic impedance for waves in an elastic medium. However, for purely longitudinal waves (along the x-axis) it makes sense to define acoustic impedance as Z = -σ_xx / v_x. (Note that P is positive when σ_xx is negative) It is also popular for purely transverse waves with displacement along y and propagation along the +x direction to define Z = -σ_xy / v_y. This is sometimes called "shear impedance" in seismology.
   The significance of impedance in any of these forms is when considering a travelling wave solution at a boundary, and looking for the condition under which there is no back-reflection. In all cases, there are two boundary conditions which apply at the interface. The definition of impedance is chosen so that when two materials have the same impedance, travelling waves solutions will automatically satisfy the first boundary condition if the second is satisfied. This furnishes the useful concept of "impedance matching" to understand when reflections will occur at interfaces.

2. Elastic Mechanics
   In what follows, we consider only the situation of longitudinal waves in an elastic medium. The components of the strain tensor are e_ij. The stress tensor has components σ_ij. These are related by

σ_ij = 2 μ e_ij + λ Θ δ_ij

[2.1]

where Θ = e_kk is the dilation. In spherical coordinates, Θ = e_rr + e_θθ + e_φφ. λ and μ are the Lam� elastic constants. These are related to the longitudinal and transverse speeds of sound through:

v_L = sqrt( (λ+2 μ)/ρ) and v_T = sqrt( μ/ρ)

[2.2]

Let u be the displacement field, with components u_r, u_θ and u_φ. For the situation of longitudinal waves considered here, u_θ = u_φ = 0 and u_r is a function of r and t only. In this case, e_rr = ∂ u_r / ∂ r, and e_θθ = e_φφ = u_r / r.

Formulae for the components of the strain tensor in spherical coordinates may be found at:
www.rochester.edu/College/ME/gans/ME445/stress.pdf
web.utk.edu/~kit/443/strainrate_cyl_sph.pdf
documents.wolfram.com/applications/structural/GoverningEquationsofElasticity.html
[ at the end of section 7.3, in unusual notation ]
ciks.cbt.nist.gov/~garbocz/paper93/node2.html (special case)

The displacement field has three possible forms:

u = e^-iωt (A/k_L) ∇ ( j_l(k_L r) S_lm(θ,φ) )

[2.3]

u = e^-iωt (B) ∇ × (r j_l(k_T r) S_lm(θ,φ) )

[2.4]

u = e^-iωt (C/k_L) ∇ × ∇ × (r j_l(k_L r) S_lm(θ,φ) )

[2.5]

where j_l(x) are spherical Bessel functions and S_lm(θ,φ) are real-valued spherical harmonics:

	= - sqrt(2) Re( Y_l^m(θ,φ) )	[ m > 0 ]
S_lm(θ,φ)	= Y_l^m(θ,φ) )	[ m = 0 ]
	= - sqrt(2) Im( Y_l^m(θ,φ) )	[ m < 0 ]

[2.6]

where Y_l^m(θ,φ) are the conventionally-defined and normalized complex spherical harmonics. In particular, note that S₀₀ = 0.282095 .

3. Longitudinal Acoustic Impedance
Consider a purely longitudinal spherical travelling wave. In this case l = m = 0, so that only the radial component of u is nonvanishing:

u_r = e^-iωt

A S₀₀
-------
k_L

∂
---
∂r

j₀(k_L r)

[3.1]

Since the wave is travelling outward, the desired spherical Bessel function (or Hankel function) is

j₀(x) = e^ix / x

[3.2]

Next, note that if x = k_L r then (d/dr) = k_L (d/dx). Thus

u_r = e^-iωt (A S₀₀ ) (d/dx) e^ix x^-1

[3.3]

u_r = e^-iωt (A S₀₀ ) e^ix [ i x^-1 - x^-2 ]

[3.4]

so the radial velocity is v_r = ∂/∂t u_r which equals

v_r = (-i ω) e^-iωt (A S₀₀ ) e^ix [ i x^-1 - x^-2 ]

[3.5]

Next, we need to calculate the components of the strain tensor e_ij:

e_rr = (∂/∂r) u_r = k_L d/dx u_r

[3.6]

e_rr = e^-iωt (k_L A S₀₀ ) (d/dx) { e^ix [ i x^-1 - x^-2 ] }

[3.7]

e_rr = e^-iωt (k_L A S₀₀ ) e^ix [ i( i x^-1 - x^-2) + (-ix^-2 + 2 x^-3 ) ]

[3.8]

e_rr = e^-iωt (k_L A S₀₀ ) e^ix [ -x^-1 - i x^-2 - i x^-2 + 2 x^-3 ]

[3.9]

e_rr = e^-iωt (k_L A S₀₀ ) e^ix [ -x^-1 - 2 i x^-2 + 2 x^-3 ]

[3.10]

e_θθ = u_r / r = k_L u_r / x

[3.11]

e_θθ = e^-iωt (k_L A S₀₀ ) e^ix [ i x^-2 - x^-3 ]

[3.12]

e_φφ = u_r / r = k_L u_r / x

[3.13]

e_φφ = e^-iωt (k_L A S₀₀ ) e^ix [ i x^-2 - x^-3 ]

[3.14]

The dilation Θ is e_rr + e_θθ + e_φφ :

Θ = e^-iωt (k_L A S₀₀ ) e^ix [ - x^-1 ]

[3.15]

so we can now evaluate the stress tensor component σ_rr = λ Θ + 2 μ e_rr

σ_rr = e^-iωt (k_L A S₀₀ ) e^ix [ λ (-1) x^-1 + 2μ [ -x^-1 - 2 i x^-2 + 2 x^-3 ] ]

[3.16]

The acoustic impedance can now be calculated from Z = - σ_rr / v_r:

Z = - [k_L/ (-iω)][ λ (-1) x^-1 + 2μ [ -x^-1 - 2 i x^-2 + 2 x^-3 ] ] / [ i x^-1 - x^-2]

[3.17]

Z = - [ i / v_L][ λ (-1) + 2μ [ -1 - 2 i x^-1 + 2 x^-2 ] ] / [ i - x^-1]

[3.18]

Z = [ 1 / v_L][ λ + 2μ [ 1 + 2 i x^-1 - 2 x^-2 ] ] / [ 1 + i x^-1]

[3.19]

For convenience, let s = 1/x:

Z = [ 1 / v_L][ (λ+2μ) + 4μ [ i s - s² ] ] / [ 1 + i s ]

[3.20]

Note that λ+2μ = ρ v_L² and μ = ρ v_T². The acoustic impedance of a purely longitudinal travelling spherical wave is:

Z(s) =

ρ [ v_L² + 4 v_T² ( i s - s² ) ]
-------------------------------
C_L ( 1 + i s )

[3.21]

where s = 1 / ( k_L r ). The discussion assumed that the wave is travelling outward, in the +r direction. But this formula also accomodates inward travelling waves (negative k_L) by making s negative.

4. Acoustic Impedance Formulation of CFM (SPH,0) Eigenvalue Equation
Consider longitudinal waves in one dimension, propagating either in the +r direction (outward) or the -r direction (inward). The rr component of stress in the medium is σ_rr. The longitudinal displacement field is u_r(r,t), given by:

u_r(r,t) = e^-iωt

A S₀₀
-------
k_L

∂
---
∂r

(

exp(ik_Lr)
-----------
k_L r

)

[4.1]

where k_L > 0 for outward waves and k_L < 0 for inward waves. A is the amplitude with dimensions of metres. S₀₀ is the angular factor, equal to 0.282095 for (SPH,l=0) modes. Or alternatively in terms of dimensionless variable x = k_L r:

u_r(x,t) = e^-iωt

A S₀₀

∂
---
∂x

(

e^ix
----
x

)

[4.2]

u_r(x,t) = e^-iωt

A S₀₀

e^ix

(

i
---
x

1
---
x²

)

[4.3]

The longitudinal velocity is v_r(x,t) = ∂ u_r(x,t) / ∂ t, = (-iω) u_r(x,t) which can be expressed as

v_r(x) = e^-iωt A S₀₀ ω e^ix ( (1/x) + i/x²).

[4.4]

The medium has two regions: "p" and "m". Region "p" is defined by r < R_p. Region "m" is defined by r > R_p where R_p is the nanoparticle radius.
Next, suppose the the disturbance in region "p" consists of two parts, travelling in different directions, with amplitudes A_po ("o" for outward) and A_pi ("i" for inward).

v_rpo(x) = e^-iωt A_po S₀₀ ω e^ix ( (1/x) + i/x²).

[4.5]

where x > 0 and

v_rpi(y) = e^-iωt A_pi S₀₀ ω e^iy ( (1/y) + i/y²).

[4.6]

where y < 0.
Likewise, suppose the the disturbance in region "m" consists of two parts, travelling in different directions, with amplitudes A_mo and A_mi:

v_rmo(x) = e^-iωt A_mo S₀₀ ω e^ix ( (1/x) + i/x²).

[4.7]

where x > 0 and

v_rmi(y) = e^-iωt A_mi S₀₀ ω e^iy ( (1/y) + i/y²).

[4.8]

where y < 0.
Consider a wave propagating in the +r direction in region "p": It has acoustic impedance Z_p(1/ξ) where ξ is defined (conventionally - e.g.: eq. (1) in Voisin 2002; eq. (2) in Verma 1999) as:

ξ = |k_Lp| R_p = ω R_p / v_Lp.

[4.9]

where v_Lp is the longitudinal speed of sound in region "p". Thus, for outgoing waves in region "p":

Z_p(1/ξ) = - σ_rr / v_rpo(ξ), [outward travelling]

[4.10]

where both σ_rr and v_rpo(ξ) are evaluated at r = R_p just inside the "p" region.
Next, consider a wave propagating in the -r direction in region "p": In this case k_Lp = - ω / v_Lp and so k_Lp R_p = -ξ. The acoustic impedance of the inward moving wave in region "p" is:

Z_p(-1/ξ) = + σ_rr / v_rpi(-ξ). [inward travelling]

[4.11]

where once again both σ_rr and v_rpi(-ξ) are evaluated at r = R_p just inside the "p" region.
Next, consider travelling waves in the "m" region. The longitudinal speed of sound is v_Lm. Following Voisin et al.'s notation [Physica B 2002], let α = v_Lm / v_Lp. (Note that Verma et al. 1999 define "κ" = v_Lp/v_Lm ) The wavevector in this region is:

|k_Lm| = ω / v_Lm = (1/α) ω / v_Lp = k_Lp / α.

[4.12]

Thus, |k_Lm| R_p = ξ / α. The acoustic impedance relations for outward and inward travelling waves in the "m" region are, respectively:

Z_m(α/ξ) = - σ_rr / v_rmo(ξ/α) [outward travelling]

[4.13]

Z_m(-α/ξ) = + σ_rr / v_rmi(-ξ/α) [inward travelling]

[4.14]

   We assume the r → ∞ boundary condition that the wave in region "m" is purely outward so that A_mi = 0. At r = R_p, there are boundary conditions both of continuity of σ_rr and v_r.
   The velocity at r = R_p for the outgoing wave in the "p" region is v_rpo(ξ). The velocity at r = R_p for the ingoing wave in the "p" region is v_rpi(-ξ). The total velocity at r = R_p is v_rptot = v_rpo(ξ) + v_rpi(-ξ).
   The total stress at r = R_p just inside the "p" region is:

σ_rrptot = Z_p(-1/ξ) v_rpi(-ξ) - Z_p(1/ξ) v_rpo(ξ)

[4.15]

To match the boundary conditions with the outward going wave in the "m" region, it must be the case that -σ_rrptot / v_rptot = Z_m(α/ξ). Therefore:

Z_p(1/ξ) v_rpo(ξ) - Z_p(-1/ξ) v_rpi(-ξ)
--------------------------------------	=	Z_m(α/ξ)
v_rpo(ξ) + v_rpi(-ξ)

[4.16]

Z_p(1/ξ) v_rpo(ξ) - Z_p(-1/ξ) v_rpi(-ξ)

= Z_m(α/ξ) ( v_rpo(ξ) + v_rpi(-ξ) )

[4.17]

v_rpo(ξ) Z_p(1/ξ) - v_rpi(-ξ) Z_p(-1/ξ)

= v_rpo(ξ) Z_m(α/ξ) + v_rpi(-ξ) Z_m(α/ξ)

[4.18]

v_rpo(ξ) ( Z_p(1/ξ) - Z_m(α/ξ) )

= v_rpi(-ξ) ( Z_p(-1/ξ) + Z_m(α/ξ) )

[4.19]

Z_p(1/ξ) - Z_m(α/ξ)
---------------------
Z_p(-1/ξ) + Z_m(α/ξ)

v_rpi(-ξ)
---------
v_rpo(ξ)

[4.20]

For the situation where "p" represents the interior of a nanoparticle and "m" is the surrounding matrix, the displacement must be non-singular at the origin. This requires that A_po = - A_pi. For this case,

v_rpi(-ξ)
---------
v_rpo(ξ)

e^-iωt A_pi S₀₀ ω e^-iξ ( (-1/ξ) + i/ξ²)
---------------------------------------<
e^-iωt A_po S₀₀ ω e^iξ ( (1/ξ) + i/ξ²)

[4.21]

v_rpi(-ξ)
---------
v_rpo(ξ)

= -

e^-iξ ( (-1/ξ) + i/ξ²)
---------------------
e^iξ ( (1/ξ) + i/ξ²)

[4.22]

v_rpi(-ξ)
---------
v_rpo(ξ)

e^-2iξ ( ξ - i )
--------------
ξ + i;

[4.23]

This then leads to the eigenvalue equation for CFM for (SPH,0) modes:

Z_p(1/ξ) - Z_m(α/ξ)
---------------------
Z_p(-1/ξ) + Z_m(α/ξ)

e^-2iξ ( ξ - i )
--------------
ξ + i

[4.24]

which may be solved to obtain the dimensionless eigenvalues ξ. Note that the frequencies depend directly on the acoustic impedances of the two materials at the interface, as well as the value of ξ.

5. Check on Correctness: FSM Limit
In the "free sphere model" (FSM) zero traction boundary conditions apply at the surface of the sphere. A particular special limiting case of equation [4.24] is where the matrix is very soft and light, so that Z_m → 0:

Z_p(1/ξ)
---------
Z_p(-1/ξ)

e^-2iξ ( ξ - i )
--------------
ξ + i

[5.1]

e^iξ( ξ + i ) Z_p(1/ξ)

e^-iξ ( ξ - i ) Z_p(-1/ξ)

[5.2]

e^iξ(ξ+i) ( vLp²+4 vTp²(i/ξ-1/ξ²))		e^-iξ (ξ-i) (vLp²+4 vTp²(-i/ξ-1/ξ²))
--------------------------------------	=	---------------------------------------<
( 1 + i / ξ )		( 1 - i / ξ )

[5.3]

e^iξ ( vLp²+4 vTp²(i/ξ-1/ξ²))

e^-iξ (vLp²+4 vTp²(-i/ξ-1/ξ²))

[5.4]

(e^iξ-e^-iξ) ( vLp²+4 vTp²(-1/ξ²))

(e^iξ + e^-iξ) (4 vTp²(-i/ξ))

[5.5]

(e^iξ - e^-iξ)
------------
2 i

( 4 vTp² - ξ²vLp² )

(e^iξ + e^-iξ)
------------
2

(4 vTp² ξ)

[5.6]

sin(ξ)

( 4 vTp² - ξ²vLp² )

cos(ξ)

(4 vTp² ξ)

[5.7]

(4 vTp² ξ)
----------------------
(4 vTp² - ξ²vLp²)

tan(ξ)

[5.8]

ξ
-------------------

1 -

1
---
4

ξ²

vLp²
------
vTp²

tan(ξ)

[5.9]

This correctly corresponds to the equation for FSM (SPH,0) frequencies, as for example in Cerullo, De Silvestri and Banin, PRB60 1928 1999, equation (2).

6. Contour Plots
Here are links to contour plots of Q and ξ as a function of speed ratio and density ratio:

1. Equal Poisson ratio
2. Poisson ratio of nanoparticle greater than matrix
3. Poisson ratio of matrix greater than nanoparticle
4. Actual material combinations

Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

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