ultrasound application in biomedical engineering

7/28/2019 ultrasound application in biomedical engineering

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Mathematical equation and behavior ofacoustic waves in solid material


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In solids

sound waves propagate in four principle modes that are based on the way the particles

oscillate

Sound can propagate as longitudinal waves, shear waves , surface waves and in thinmaterials plate waves.

Longitudinal wave Shear wave

Surface wave Plate wave ( symmetric)Plate wave (asymmetric)


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The longitudinal wave is compressional wave in which the particle motion

is in the same direction as the propagation of the wave.

The shear wave is the wave motion in which the particle motion is

perpendicular to the direction of propagation.

Surface wave have an elliptical particle motion and travel across the

surface of a material. Their velocity is approximately 90% of the shear

wave velocity of the material and their depth of penetration is

approximately equal to one wavelength.

Plate (lamb) wave have a complex vibration occurring in materials where

thickness is less than the wavelength of ultrasound introduced into it.


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What properties of material affect its of acoustic waves ?

Sound does travel at different speeds in different materials.

This is because the mass of the atomic particles and the elastic constants are different

for different materials.

The mass of the particles is related to the density of the material, and the elastic

constant is related to the elastic constants of a material.

The general relationship between the speed of sound in a solid and its density and

elastic constants is given by the following equation.

Where V is the speed of sound, C is the elastic constant, and pis the material density.This equation may take a number of different forms depending on the type of wave

(longitudinal or shear) and which of the elastic constants that are used.


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Elastic constants:

The elastic moduli of a material are important for the understanding of

mechanical behaviour.

If VL and VS are the measured ultrasonic velocities of longitudinal and shear wave then

longitudinal modulus (L)

Shear modulus (G)

Bulk modulus (B)

Poissons ratio ()

Young modulus (Y)

lames modulus ( and )


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The velocity of ultrasonic wave of any kind can be determined from the elastic moduli

(Y: Youngs modulus, G: modulus of rigidity, and : poissons ratio) and

density (d) of the material

The longitudinal and shear wave velocities (VL and VS) can be determined with following

expressions

In terms of lames moduli ( and ) , the ultrasonic velocities can be expressed as;


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Reflection

Reflection coefficient

= Pr/Pi

= (Z2cost-Z1cosi) / (Z2cost+Z1cosi)

When Z2 and Z1 difference is large, large echo is

generated. (Muscle and bone interaction)


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Transmission

Transmission coefficient

= Pt/Pi

= 2Z2cosi/(Z2cosi +Z1cost)

Higher impedance matching if Z2 and Z1 are

approximately the same.

ultrasound application in biomedical engineering

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