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Fundamentals of Ultrasonic Wave P Ti Propagation Fundamentals of Ultrasonic Wave PtiPropagation Victor Humphrey Institute of Sound and Vibration Research, University of Southampton, UK. UIA 24th May 2011 Aim • To giv e a general ov erv iew of ultrasound propagation covering: – Basics –Liiinear propagation – Interfaces – Diffraction – Nonlinear propagation Basics Propagation of Acoustic Disturbances PISTON TUBE FLUID PISTON MOVES WITH VELOCITY u FLUID COMPRESSED LOCALLY MOVES WITH VELOCITY u FRONT EDGE OF DISTURBANCE MOVES WITH VELOCITY c0 -THE SPEED OF SOUND Propagation of Acoustic Disturbances PISTON STOPS DISTURBANCE CONTINUES TO MOVE AT c0 LOCAL FLUID VELOCITY BECOMES u AS DISTURBANCE PASSES Basic Physical Characteristics • In fluids only longitudinal ultrasound waves propagate • Displacement of media is in direct ion of propagation •Sofift tissues have very low shear modulus Basic Parameters 3 basic parameters: •Particle velocity u ux xˆ u y yˆ uz zˆ •Total pressure PT •Total density T When no sound is present: u 0 T 0 PT P0 With an acoustic wave: u T 0 PT P0 p Acoustic component Basic Equations The acoustic wave equation is obtained by combining three equations: 1. The Continuity Equation (Conservation of Mass); 2. The Force Equation (Conservation of Momentum); 3.The Equation of State. tot (totu) ptot u u 0 u tot t x x x t A B 2 ptot f (tot ) P0 T 0 2 T 0 2 0 0 The One Dimensional Wave Equation Linearize these equations by assuming that u << c0 and ρ << ρ0. Also use fact that ρ0 doesn’t vary in space or time Can then derive the Wave Equation in 1-D: 2 p 1 2 p 2 2 2 0 x c0 t For a plane wave: p 0c0u The product of equilibrium density 0 and speed of sound c0 is known at the characteristic acoustic impedance Z. Values • Consider two cases of a singgqyle frequency wave: – An airborne ultrasonic source producing a signal at 137 dB re 20 μPa; – A 3. 5 MHz ultrasonic imaging system producing 5 MPa at focus. Pressure Frequency Medium Displacement Velocity Acceleration Amplitude /MHz Amplitude Amplitude Amplitude /MPa / m /ms-1 /ms-2 0. 0002 0040.04 AiAir 191.9 x10-6 494.9x 10-1 121.2x 10+5 5 3.5 Water 1.5x10-7 3.3 7.3x10+7 Acoustic Intensity • The average rate of flow of energy through a unit area normal to direction of propagation. 1 P2 I 2 0c0 Strictly for plane waves. P is acoustic pressure amplitude. Di fferences will occur in fldfields where the particle velocity and pressure are not in phase, but this is not normally significant. For pulsed waveforms need to allow for the mark-to-space ratio in calllculating the time averaged intensity ITA Interfaces Interfaces • At a boundary between two 1 T fluid media some of the R energy will be transmitted Z Z and some reflecte d 1 2 • The extent of transmission • e.g. for normal incidence the depends on Amplitude Transmission and – Relative characteristic Reflection coefficients are given acoustic impedance of by media 2 (Z2 Z1) – Angle of incidence T R (1 Z1 Z2 ) (Z2 Z1) Note that in going from a low impedance material to high impedance the pressure may increase (but the displacement decrease)! Interfaces • At a boundary between high and low impedance materials (Z1 >> Z2) the reflected pressure wave is inverted. • For asymmetric waveforms (such as those generated by nonlinearity) this can turn a high amplitude compression in to a high amplitude tension. t t Mode Conversion • At a boundary between a fluid and solid it is possible to excite shear waves and surface waves AiAnimat ions courtesy of Dr. Dan Russell , Ketteri ng UUiniversi ty Transducer Fields Near Field and Far Field of a Planar Source Can consider that the field of a planar ultrasonic Transition can be source can be divided in to two regions: considered to occur at the Rayygleigh Distance R0 where for a 2a Near field Far field circular source of radius a: a2 ka2 R Simplest model: in the near field the 0 2 propagation is planar (and collimated) while in the far field it is spherically spreading. Diffraction The Radiated Field: The Rayleigh Integral • The total complex pressure generated by a vibrating source can be evaluated by integrating over the contributions due to all the surface elements. j Ue jkr •The RAYLEIGH INTEGRAL p(x) dp(x) 0 dS 2 r • which in principle enables the S S calculation of the sound field ppyyroduced by any distribution of complex normal velocity U of an otherwise rigid infinite plane boundary. Transducer 50 100 r ee 150 ansduc rr 200 T 250 300 Near Field of a Circular Piston Radiator 1.5 Transducer radius 1 x / a 050 .5 TdTransducer axis of symmetry 0 0 0.5 1 1.5 2 2.5 3 z / (a2 / ) The minimum beam diameter is only ~1/4 transducer diameter (d(measured at -3 dB leve l). Focusing In order to produce a narrower beam focus the ultrasound field: Focal plane 2a Geometric focus Focal length D 2 R0 a The amplitu de gain G is given by: G D D Focused Circular Piston • Weak: 0<0 < G <2< 2 Axial variation: • Medium: 2 < G < 2π D sin X p(z) p0G •Strong: 2π < G z X G D X 1 2 z • At geometrific focus: p(D) p0G • -3 dB beam full width in 1.62a focal plane: x 3dB G Focused Field Axial Variation Axial pressure variation for focused transducer 6 Gain = 4.0 5 Gain = 2.0 4 0 3 |p| / p / |p| 2 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 z / D Focused Field Axial Variation Axial pressure variation for focused transducer 25 Gain = 20.0 20 Gain = 10.0 Gain = 6.0 15 00 |p| / p / |p| 10 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 z / D Rectangular Transducers ()(a) (b) Axial variation of normalised acoustic pressure amplitude (p/p0) for: (a) a square transducer; (b) a rectangular transducer with 1:2 aspect ratio. The Far Field Pressure Distribution 2J1kasin p(x) ~ kasin Firs t zero occurs when v ka sinθ = 3.83. Non-linear Propagation Non-linear Propagation •If the acoustic amplitu de is suffic ient ly hig h then non-linear effects will become significant. • A point on the wave with particle velocity u will travel with velocity c0 u B where the coefficient of non-linearity : 1 2A • Values for B/A vary from 5.0 for water to 6.3 for blood, ~ 6 - 7 for liver and ~ 10 for fatty tissue. Propagation of Plane Wave u c0 + uc c0 +½uc c c0 c0 0 x Waveform in space c0 - ur| Non-linear Plane Wave Propagation sigma = 0.00.20.40.60.81.01.21.41.81.62.02.22.6 Non-linear Propagation Results in: • Generation of harmonics • Generation of shock fronts • Enhanced attenuation – Enhanced heating – Enhanced streaming • Saturation The Shock Parameter A measure of the extent of non-linear propagation For a plane wave: = k z = u0 / c Acoustic Mach number u0 Peak particle velocity at source k = 2 Wavenumber z Distance travelled = 1 corresponds to shock front just forming = /2 corresponds to a full shock Shock Distance for a Plane Wave • The shock parameter = 1 corresponds to a shock front just forming. • At high frequencies the plane wave shock distance can be small. • So for example in water: 3.5 f 0 3.5 MHz p 0 1MPa Shock distance 43 mm Plane Wave: Non-linear Propagation 500 400 e / kPa e / 1 300 amplitud 200 2 ee 3 100 Pressur 5 0 0 100 200 300 400 500 Range / mm Fundamental and second to fifth harmonics for a nonlinear plane wave in water. (f0 = 3.5 MHz, P0 = 500 kPa, G = 38). Initial and Distorted Pulses 0.20 0.15 0.10 / MPa 0050.05 ee 0.00 0.01.02.03.04.05.0 Amplitud -0.05 -0.10 -0.15 Time / s 0.30 0.25 0.20 a PP 0150.15 0.10 0.05 0.00 -0.05 Amplitude / M 0.01.02.03.04.05.0 -0.10 -0.15 -0.20 Time / s Measured Output and Model Prediction 6 Measured waveform (H 0-100) 5 Model waveform (H 0-250) 4 Model waveform (H 0-100) 3 Pa 2 1 ressure / M 0 P -1 -2 -3 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Time / us On axis time waveforms at z = 54mmfora31MHzarrayinwater54 mm for a 3.1 MHz array in water. Modelwith250MHzModel with 250 MHz filter (red squares); model with 100 MHz filter (green triangles) and experiment with 100 MHz filter (blue circles). Focused Field (G = 8) 1.00E+00 FUNDAMENTAL SECOND HARMONIC THIRD HARMONIC 1.00E-01 MPa // 1.00E-02 amplitude ee 1.00E-03 Pressur 1.00E-04 0 100 200 300 400 500 Axial distance / mm Axial variation for p0 = 72 kPa, f0 = 2.25 MHz and G = 8.0. Beam Profiles for Demonstration Harmonic Imagggying System Fundamental 2nd Harmonic Predicted Non-linear Distortion 2.5 2.0 1.5 Water 1.0 0.5 ure / MPaure 0.0 Press -0.5 -1.0 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Time / (s) 2 1.5 Tissue Pre dicted waveform at a range of 54 mm 1 in water and tissue (attenuation 0.3 0.5 0 dB/cm/MHz) produced by an array 15 / MPa Pressure mm byy,g 10 mm in size, with focal lengths -0.5 of 80 mm and 50 mm.
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