LIST OF SYMBOLS.

f = frequency

fpeak = frequency corresponding to the spectral peak at each inclination angle (i.e., $\mid V_{plane}(\omega )\mid /{max}_{\forall \omega }\mid V_{plane}(\omega )\mid \hspace{0.17em}\cong exp{[-(f-f_{peak})}^2/2{\sigma }_{\omega p}^2]$).

g(r⃗d, r⃗′) = effective Green’s function valid from the scattering region to the detector

Go = geometric gain value for velocity potential field at focus when Wsource is approximated by a Gaussian (m)

GT = dimensionless aperture gain function that accounts for the focusing of the ultrasound source

H = dimensionless filtering characteristics for the ultrasound source

k = effective wave number along the propagation path

k̃ = effective complex wave number along the propagation path (i.e., k̃ = k + iα)

KuV = conversion constant relating voltage to particle velocity for ultrasound source (m/s V−1)

Mimage = matrix used to generate image point

n⃗f = the outward normal for the plane at arbitrary angle to beam axis

pplane = pressure field from rigid plane placed near focal plane

r⃗f = locations on inclined plane

r⃗1 = points on aperture plane of image source

r⃗T , r⃗d = locations on aperture plane of transmitter/detector

Sf = surface of inclined plane near focal plane

SI = aperture area of image source

ST = aperture area of ultrasound transmitter

Vinc = voltage applied to the ultrasound source during transmit

Vplane = voltage from ultrasound source due to the backscatter from rigid plane near focus

wx, wy, wz = equivalent Gaussian dimensions on receive of pressure field in focal region

xI, yI, zI = coordinate location of image point

zf = distance of rigid plane to the focal plane

zp = distance along the beam axis from the focus to the intersection of the plane with the beam axis

zT, zd = distance of the aperture plane of the ultra-sound transmitter/detector to the focal plane

α = effective attenuation along the propagation path

αerror = error in attenuation associated with inclination angle of plane

αo = slope of attenuation assuming strict linear frequency dependence (i.e., α =αof)

Γplane = reflection coefficient of plane

θf , φf = angles describing orientation of plane with beam axis

ξx , ξy , ξz = coordinate system for the image source

ξ̂x , ξ̂y , ξ̂z = unit normal vectors defining coordinate system for the image source

ρ = density

σωp = Gaussian bandwidth for reflected voltage from inclined plane (i.e., $\mid V_{plane}(\omega )\mid /{max}_{\forall \omega }\mid V_{plane}(\omega )\mid \hspace{0.17em}\cong exp{[-((f-f_{peak})}^2/2{\sigma }_{\omega p}^2])$).

ω = radian frequency

ωmax = radian frequency corresponding to spectral peak

• = dot product