Preparing Material Properties

Introduction

Use this guide after choosing a target geometry and before building a scatterer. It covers how to record, convert, and check the material values that will be attached to the target. Constructor selection and object assembly are covered in Building Scatterers.

See Scattering Boundary Conditions for how material properties enter the interface conditions.

Record the source values

Record each value with its units, source, and measurement conditions. Density and sound speed can depend on temperature, salinity, pressure, and composition. In dispersive or lossy media, effective wave properties may also depend on frequency. A contrast copied from a publication is usable only when its reference medium is known (Medwin and Clay 1998).

See Notation and Symbols for the shared material-property definitions and units.

Convert units before constructing the scatterer. In particular, elastic moduli should be supplied in pascals and geometric dimensions should follow the units declared in the constructor call.

Make the reference medium explicit

Absolute target properties remain properties of the target, but their acoustic interpretation and all derived ratios depend on the exterior medium used in the calculation. Keep those medium values with the target setup, even when they are not arguments to the scatterer constructor:

library(acousticTS)

medium <- data.frame(
  density = 1026,
  sound_speed = 1500
)

body <- data.frame(
  density = 1045,
  sound_speed = 1520
)

The same exterior density and sound speed should later be supplied to the scattering calculation. Otherwise, ratios derived during target preparation and properties completed during model evaluation can describe different physical systems.

Choose absolute properties or ratios

Fluid-like constructors commonly accept either absolute values such as density_body and sound_speed_body or directional ratios such as g_body and h_body. For a body in the exterior medium, those ratios are g_{21}=\rho_2/\rho_1 and h_{21}=c_2/c_1.

Compute the ratios from the recorded absolute values when both are available:

g_body <- body$density / medium$density
h_body <- body$sound_speed / medium$sound_speed

data.frame(g_body, h_body)

##     g_body   h_body
## 1 1.018519 1.013333

Supply one representation for each property. Providing an absolute density and a contradictory density ratio does not describe a unique target. The same rule applies to sound speed.

Ratios and perturbation contrasts are also different quantities. The package helpers rho() and compressibility() calculate the density and compressibility perturbations used by weak-scattering formulations:

data.frame(
  gamma_rho = acousticTS::rho(medium, body),
  gamma_kappa = compressibility(medium, body)
)

##    gamma_rho gamma_kappa
## 1 0.01818182 -0.04384916

The ratios g_{21} and h_{21} approach one for a weak contrast. The perturbations \gamma_\rho and \gamma_\kappa approach zero. They should not be substituted for one another.

Check combined fluid properties

Density and sound speed should also be checked together. Acoustic impedance and absolute compressibility provide compact diagnostics for an interface:

property_check <- rbind(
  exterior = medium,
  body = body
)

property_check$impedance <-
  property_check$density * property_check$sound_speed
property_check$compressibility_pa_inverse <-
  1 / (property_check$density * property_check$sound_speed^2)

property_check

##          density sound_speed impedance compressibility_pa_inverse
## exterior    1026        1500   1539000               4.331817e-10
## body        1045        1520   1588400               4.141871e-10

A density ratio close to one does not guarantee a small acoustic contrast if the sound speeds differ materially. Likewise, a gas component should not be treated as weakly contrasting merely because the chosen density value has not yet been checked against its sound speed.

Convert elastic properties

Elastic properties are reported in several equivalent parameterizations. The package provides pois(), bulk(), young(), shear(), and lame() to convert between the common isotropic elastic constants (Achenbach 1973).

For example, Young’s modulus and Poisson ratio can be converted before an elastic scatterer is built:

E <- 70e9
nu <- 0.32
density_solid <- 2700

G <- shear(E = E, nu = nu)
K <- bulk(E = E, nu = nu)
lambda <- lame(E = E, nu = nu)

c_L <- sqrt((lambda + 2 * G) / density_solid)
c_T <- sqrt(G / density_solid)

data.frame(E, nu, G, K, lambda, c_L, c_T)

##       E   nu           G           K      lambda      c_L      c_T
## 1 7e+10 0.32 26515151515 64814814815 47138047138 6090.927 3133.756

Supply only the independent values intended for the conversion. Redundant and inconsistent elastic constants can hide which pair was actually used. For a stable isotropic linear-elastic material, check that K>0, G>0, and E>0 as applicable, and that -1<\nu<0.5 before construction. Lamé’s first parameter need not be positive for every stable auxetic material.

Different target classes expose different elastic inputs. An elastic-shell constructor may accept a pair such as E and nu, while a calibration sphere may be specified by density and longitudinal and transverse wave speeds. Use the representation supported by the intended constructor rather than converting values that are already available in the required form.

Keep layered components separate

For a layered target, retain one labeled record per physical region. Do not collapse shell, core, viscous layer, and exterior values into an unlabeled set of generic properties.

Region	Values to retain
exterior fluid	density and sound speed
fluid-like body or core	density and sound speed, or documented ratios
elastic solid or shell	density and an independent elastic parameter set
viscous layer	density, compressional sound speed, shear viscosity, and bulk viscosity when required

The theoretical ratios between adjacent media can always be derived from these labeled absolute values. Constructor argument names may instead express a component relative to the exterior medium, so check the constructor documentation before supplying a precomputed ratio.

Build a constructor-ready target

Once the values are checked, attach them to a compatible shape. Absolute properties can be supplied directly:

body_shape <- prolate_spheroid(
  length_body = 0.04,
  radius_body = 0.004,
  n_segments = 60
)

body_target <- fls_generate(
  shape = body_shape,
  density_body = body$density,
  sound_speed_body = body$sound_speed
)

extract(body_target, "body")

## $rpos
##    [,1]         [,2]         [,3]        [,4]         [,5]         [,6]    [,7]
## x  0.04  0.039333333  0.038666667  0.03800000  0.037333333  0.036666667  0.0360
## y  0.00  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000  0.0000
## z  0.00  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000  0.0000
## zU 0.00  0.001024153  0.001436044  0.00174356  0.001995551  0.002211083  0.0024
## zL 0.00 -0.001024153 -0.001436044 -0.00174356 -0.001995551 -0.002211083 -0.0024
##            [,8]         [,9]        [,10]        [,11]        [,12]   [,13]
## x   0.035333333  0.034666667  0.034000000  0.033333333  0.032666667  0.0320
## y   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000  0.0000
## z   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000  0.0000
## zU  0.002568181  0.002719477  0.002856571  0.002981424  0.003095516  0.0032
## zL -0.002568181 -0.002719477 -0.002856571 -0.002981424 -0.003095516 -0.0032
##           [,14]        [,15]        [,16]        [,17]        [,18]
## x   0.031333333  0.030666667  0.030000000  0.029333333  0.028666667
## y   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000
## z   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000
## zU  0.003295789  0.003383621  0.003464102  0.003537733  0.003604935
## zL -0.003295789 -0.003383621 -0.003464102 -0.003537733 -0.003604935
##           [,19]       [,20]        [,21]        [,22]        [,23]        [,24]
## x   0.028000000  0.02733333  0.026666667  0.026000000  0.025333333  0.024666667
## y   0.000000000  0.00000000  0.000000000  0.000000000  0.000000000  0.000000000
## z   0.000000000  0.00000000  0.000000000  0.000000000  0.000000000  0.000000000
## zU  0.003666061  0.00372141  0.003771236  0.003815757  0.003855155  0.003889587
## zL -0.003666061 -0.00372141 -0.003771236 -0.003815757 -0.003855155 -0.003889587
##           [,25]        [,26]        [,27]       [,28]        [,29]        [,30]
## x   0.024000000  0.023333333  0.022666667  0.02200000  0.021333333  0.020666667
## y   0.000000000  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000
## z   0.000000000  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000
## zU  0.003919184  0.003944053  0.003964285  0.00397995  0.003991101  0.003997777
## zL -0.003919184 -0.003944053 -0.003964285 -0.00397995 -0.003991101 -0.003997777
##     [,31]        [,32]        [,33]       [,34]        [,35]        [,36]
## x   0.020  0.019333333  0.018666667  0.01800000  0.017333333  0.016666667
## y   0.000  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000
## z   0.000  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000
## zU  0.004  0.003997777  0.003991101  0.00397995  0.003964285  0.003944053
## zL -0.004 -0.003997777 -0.003991101 -0.00397995 -0.003964285 -0.003944053
##           [,37]        [,38]        [,39]        [,40]        [,41]       [,42]
## x   0.016000000  0.015333333  0.014666667  0.014000000  0.013333333  0.01266667
## y   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000  0.00000000
## z   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000  0.00000000
## zU  0.003919184  0.003889587  0.003855155  0.003815757  0.003771236  0.00372141
## zL -0.003919184 -0.003889587 -0.003855155 -0.003815757 -0.003771236 -0.00372141
##           [,43]        [,44]        [,45]        [,46]        [,47]
## x   0.012000000  0.011333333  0.010666667  0.010000000  0.009333333
## y   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000
## z   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000
## zU  0.003666061  0.003604935  0.003537733  0.003464102  0.003383621
## zL -0.003666061 -0.003604935 -0.003537733 -0.003464102 -0.003383621
##           [,48]   [,49]        [,50]        [,51]        [,52]        [,53]
## x   0.008666667  0.0080  0.007333333  0.006666667  0.006000000  0.005333333
## y   0.000000000  0.0000  0.000000000  0.000000000  0.000000000  0.000000000
## z   0.000000000  0.0000  0.000000000  0.000000000  0.000000000  0.000000000
## zU  0.003295789  0.0032  0.003095516  0.002981424  0.002856571  0.002719477
## zL -0.003295789 -0.0032 -0.003095516 -0.002981424 -0.002856571 -0.002719477
##           [,54]   [,55]        [,56]        [,57]       [,58]        [,59]
## x   0.004666667  0.0040  0.003333333  0.002666667  0.00200000  0.001333333
## y   0.000000000  0.0000  0.000000000  0.000000000  0.00000000  0.000000000
## z   0.000000000  0.0000  0.000000000  0.000000000  0.00000000  0.000000000
## zU  0.002568181  0.0024  0.002211083  0.001995551  0.00174356  0.001436044
## zL -0.002568181 -0.0024 -0.002211083 -0.001995551 -0.00174356 -0.001436044
##            [,60] [,61]
## x   0.0006666667     0
## y   0.0000000000     0
## z   0.0000000000     0
## zU  0.0010241528     0
## zL -0.0010241528     0
## 
## $radius
##  [1] 0.000000000 0.001024153 0.001436044 0.001743560 0.001995551 0.002211083
##  [7] 0.002400000 0.002568181 0.002719477 0.002856571 0.002981424 0.003095516
## [13] 0.003200000 0.003295789 0.003383621 0.003464102 0.003537733 0.003604935
## [19] 0.003666061 0.003721410 0.003771236 0.003815757 0.003855155 0.003889587
## [25] 0.003919184 0.003944053 0.003964285 0.003979950 0.003991101 0.003997777
## [31] 0.004000000 0.003997777 0.003991101 0.003979950 0.003964285 0.003944053
## [37] 0.003919184 0.003889587 0.003855155 0.003815757 0.003771236 0.003721410
## [43] 0.003666061 0.003604935 0.003537733 0.003464102 0.003383621 0.003295789
## [49] 0.003200000 0.003095516 0.002981424 0.002856571 0.002719477 0.002568181
## [55] 0.002400000 0.002211083 0.001995551 0.001743560 0.001436044 0.001024153
## [61] 0.000000000
## 
## $theta
## [1] 1.570796
## 
## $g
## NULL
## 
## $h
## NULL
## 
## $density
## [1] 1045
## 
## $sound_speed
## [1] 1520
## 
## $radius_curvature_ratio
## NULL

The same shape can instead be constructed from ratios derived against the recorded exterior medium:

body_target_ratios <- fls_generate(
  shape = body_shape,
  g_body = g_body,
  h_body = h_body
)

extract(body_target_ratios, "body")

## $rpos
##    [,1]         [,2]         [,3]        [,4]         [,5]         [,6]    [,7]
## x  0.04  0.039333333  0.038666667  0.03800000  0.037333333  0.036666667  0.0360
## y  0.00  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000  0.0000
## z  0.00  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000  0.0000
## zU 0.00  0.001024153  0.001436044  0.00174356  0.001995551  0.002211083  0.0024
## zL 0.00 -0.001024153 -0.001436044 -0.00174356 -0.001995551 -0.002211083 -0.0024
##            [,8]         [,9]        [,10]        [,11]        [,12]   [,13]
## x   0.035333333  0.034666667  0.034000000  0.033333333  0.032666667  0.0320
## y   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000  0.0000
## z   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000  0.0000
## zU  0.002568181  0.002719477  0.002856571  0.002981424  0.003095516  0.0032
## zL -0.002568181 -0.002719477 -0.002856571 -0.002981424 -0.003095516 -0.0032
##           [,14]        [,15]        [,16]        [,17]        [,18]
## x   0.031333333  0.030666667  0.030000000  0.029333333  0.028666667
## y   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000
## z   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000
## zU  0.003295789  0.003383621  0.003464102  0.003537733  0.003604935
## zL -0.003295789 -0.003383621 -0.003464102 -0.003537733 -0.003604935
##           [,19]       [,20]        [,21]        [,22]        [,23]        [,24]
## x   0.028000000  0.02733333  0.026666667  0.026000000  0.025333333  0.024666667
## y   0.000000000  0.00000000  0.000000000  0.000000000  0.000000000  0.000000000
## z   0.000000000  0.00000000  0.000000000  0.000000000  0.000000000  0.000000000
## zU  0.003666061  0.00372141  0.003771236  0.003815757  0.003855155  0.003889587
## zL -0.003666061 -0.00372141 -0.003771236 -0.003815757 -0.003855155 -0.003889587
##           [,25]        [,26]        [,27]       [,28]        [,29]        [,30]
## x   0.024000000  0.023333333  0.022666667  0.02200000  0.021333333  0.020666667
## y   0.000000000  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000
## z   0.000000000  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000
## zU  0.003919184  0.003944053  0.003964285  0.00397995  0.003991101  0.003997777
## zL -0.003919184 -0.003944053 -0.003964285 -0.00397995 -0.003991101 -0.003997777
##     [,31]        [,32]        [,33]       [,34]        [,35]        [,36]
## x   0.020  0.019333333  0.018666667  0.01800000  0.017333333  0.016666667
## y   0.000  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000
## z   0.000  0.000000000  0.000000000  0.00000000  0.000000000  0.000000000
## zU  0.004  0.003997777  0.003991101  0.00397995  0.003964285  0.003944053
## zL -0.004 -0.003997777 -0.003991101 -0.00397995 -0.003964285 -0.003944053
##           [,37]        [,38]        [,39]        [,40]        [,41]       [,42]
## x   0.016000000  0.015333333  0.014666667  0.014000000  0.013333333  0.01266667
## y   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000  0.00000000
## z   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000  0.00000000
## zU  0.003919184  0.003889587  0.003855155  0.003815757  0.003771236  0.00372141
## zL -0.003919184 -0.003889587 -0.003855155 -0.003815757 -0.003771236 -0.00372141
##           [,43]        [,44]        [,45]        [,46]        [,47]
## x   0.012000000  0.011333333  0.010666667  0.010000000  0.009333333
## y   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000
## z   0.000000000  0.000000000  0.000000000  0.000000000  0.000000000
## zU  0.003666061  0.003604935  0.003537733  0.003464102  0.003383621
## zL -0.003666061 -0.003604935 -0.003537733 -0.003464102 -0.003383621
##           [,48]   [,49]        [,50]        [,51]        [,52]        [,53]
## x   0.008666667  0.0080  0.007333333  0.006666667  0.006000000  0.005333333
## y   0.000000000  0.0000  0.000000000  0.000000000  0.000000000  0.000000000
## z   0.000000000  0.0000  0.000000000  0.000000000  0.000000000  0.000000000
## zU  0.003295789  0.0032  0.003095516  0.002981424  0.002856571  0.002719477
## zL -0.003295789 -0.0032 -0.003095516 -0.002981424 -0.002856571 -0.002719477
##           [,54]   [,55]        [,56]        [,57]       [,58]        [,59]
## x   0.004666667  0.0040  0.003333333  0.002666667  0.00200000  0.001333333
## y   0.000000000  0.0000  0.000000000  0.000000000  0.00000000  0.000000000
## z   0.000000000  0.0000  0.000000000  0.000000000  0.00000000  0.000000000
## zU  0.002568181  0.0024  0.002211083  0.001995551  0.00174356  0.001436044
## zL -0.002568181 -0.0024 -0.002211083 -0.001995551 -0.00174356 -0.001436044
##            [,60] [,61]
## x   0.0006666667     0
## y   0.0000000000     0
## z   0.0000000000     0
## zU  0.0010241528     0
## zL -0.0010241528     0
## 
## $radius
##  [1] 0.000000000 0.001024153 0.001436044 0.001743560 0.001995551 0.002211083
##  [7] 0.002400000 0.002568181 0.002719477 0.002856571 0.002981424 0.003095516
## [13] 0.003200000 0.003295789 0.003383621 0.003464102 0.003537733 0.003604935
## [19] 0.003666061 0.003721410 0.003771236 0.003815757 0.003855155 0.003889587
## [25] 0.003919184 0.003944053 0.003964285 0.003979950 0.003991101 0.003997777
## [31] 0.004000000 0.003997777 0.003991101 0.003979950 0.003964285 0.003944053
## [37] 0.003919184 0.003889587 0.003855155 0.003815757 0.003771236 0.003721410
## [43] 0.003666061 0.003604935 0.003537733 0.003464102 0.003383621 0.003295789
## [49] 0.003200000 0.003095516 0.002981424 0.002856571 0.002719477 0.002568181
## [55] 0.002400000 0.002211083 0.001995551 0.001743560 0.001436044 0.001024153
## [61] 0.000000000
## 
## $theta
## [1] 1.570796
## 
## $g
## [1] 1.018519
## 
## $h
## [1] 1.013333
## 
## $density
## NULL
## 
## $sound_speed
## NULL
## 
## $radius_curvature_ratio
## NULL

These objects describe the same material only when the later model run uses the exterior values stored in medium.

Validation checklist

Before moving to scatterer construction or model evaluation, confirm that:

1. every dimensional value has been converted to the units expected by the package,
2. the exterior medium used to derive ratios is recorded,
3. absolute values and ratios do not contradict one another,
4. g and h ratios have not been confused with perturbation contrasts,
5. each region of a layered target has its own labeled property set,
6. elastic constants form a physically admissible independent set, and
7. frequency-dependent or complex properties have not been replaced silently by unrelated static values.

Next step

Continue to Building Scatterers to select the target class and attach these values to the geometry.

Related pages

• Building Shapes
• Building Scatterers
• Scattering Boundary Conditions
• SOEMS calibration spheres

References

Achenbach, J. D. 1973. Wave Propagation in Elastic Solids. North-Holland Series in Applied Mathematics and Mechanics, v. 16. North-Holland Pub. Co. American Elsevier Pub. Co.

Medwin, Herman, and Clarence S. Clay. 1998. Fundamentals of Acoustical Oceanography. Applications of Modern Acoustics. Academic Press.