Chapter 14: Nonlinearity & Harmonic Imaging

Question 1

The linear propagation regime has the following characteristics

Answer 1

• the shape of a sound wave remains the same as it propagates - two sound waves can pass through eachother without interacting - the total acoustic pressure due to two sound waves is the sum of the individual acoustic pressures - the sound speed is a property of the medium and does not depend on the sound itself.

Question 2

For ultrasound waves with amplitudes larger than a few hundred kPa none of these useful properties are quite true any more and the sound propagation is nonlinear. What are some non-linear effects?

Answer 2

1. steepening of waves and shock formation during propagation 2. harmonic generation in propagation waves 3. radiation pressure 4. cavitation and microstreaming

Question 3

Under what conditions of the wave does nonlinear propagation occur?

Answer 3

large amplitudes because pt=p0+p. The sound speed increases with acoustic pressure.

Question 4

How do we affect the original wave equation?

Answer 4

Made the linearising assumptions of 1) the small acoustic mach number i.e. the particle velocity much smaller than the sound of speed. 2) the equation of state, the pressure-density relation linear.

Question 5

both these assumptions of the nonlinear equation are cumulative what does that mean?

Answer 5

they accumulate as the wave propagates

Question 6

What is convective nonlinearity

Answer 6

The wave is carried forward by the motion of the medium just as it was in the case of the wind.

Question 7

What is the material nonlinearity?

Answer 7

In practice the pressure-density relation is nonlinear. When the fluid is compressed sufficiently its stiffness (bulk modulus) will increase. The sound speed will, therefore, be faster at higher pressures. (think of chain link with straight arms/bent arms).

Question 8

How can we determine the sound speed from this graph?

Answer 8

The sound speed is the gradient

Question 9

How do the material and convective material nonlinearities affect the property of the wave?

Answer 9

1. Increase the propagation speed of positive parts of the wave
2. Decrease propagation speed of negative parts of the wave

Question 10

what are two ways in which to measure the nonlinearity parameter?

Answer 10

The Thermodynamic Method and the Finite Amplitude Method

Question 11

combining the convective and material nonlinearities gives an expression for the wave speed as? where beta is the coefficient of nonlinearity…

Answer 11

Question 12

Outline the thermodynamic method in obtaining the B/A.

Answer 12

The Thermodynamic method - a sample of the material of interest is placed in a pressure vessel and its sound speed is measured as a function of the total pressure p_T to give an estimate of dc/dp_T.

The B/A can then be obtained by this equation:

The derivation is shown aswell.

Question 13

Outline the finite amplitude method in obtaining B/A.

Answer 13

The Finite Amplitude method - propagate a sinusoidal wave a known distance at an acoustic pressure that will ensure it is in a nonlinear regime, and measure the amplitude of the second harmonic that is generated which can then be related to B/A.

Question 14

Explain the parts of the wave steepening and harmonic generation.

Answer 14

1. A sinusoidal wave starts to distort as the peaks catch the troughs.
2. The wave develops shocks, forming sawtooth waves.
3. The wave distortion has shifted energy from lower to higher frequencies.
4. The higher frequencies components are more strongly absorbed.
5. The shock is therefore dissipated and the wave ends up as a lower amplitude sine wave (‘old age’).

Question 15

In biomedical acoustics we are rarely interested in the latter, ‘old age’, stage of the propagation, but this highlights the point that there is a balance between the nonlinearity and the absorption. This balance is characterised by the ____________?

Answer 15

This balance is characterised by the Gol’dberg number.

Question 16

What is tissue harmonic imaging

Answer 16

the fundamental component - the transmitted component - is filtered out leaving the harmonics. The strongest of these is the second harmonic, ie. the first overtone, at twice the source frequency. This is then used to form an image. Because the wavelength at the second harmonic is half that at the fundamental frequency, the beamwidth will be narrower, and the image will therefore have higher lateral spatial resolution.

Question 17

Why is it not possible to use a filter to separate the frequencies in the transmit pulse and those shifted nonlinearly to the second harmonic?

Answer 17

because they have a spread of frequencies.

Question 18

For filtering to be good at separating the fundamental and harmonic components in the reflections, it is therefore necessary to make the transmit pulse sufficiently _______

What is the downside to this affect?

Answer 18

narrowband that the harmonic content due to the lower frequencies does not overlap with the higher frequencies in the transmit pulse itself.

The downside of reducing the bandwidth to achieve this increases the pulse duration thereby worsening axial spatial resolution.

Question 19

What is pulse inversion?

Answer 19

It is a technique to remove the fundamental component without lengthening the transmit pulse and therefore retains the axial resolution. This works by sending two transmit pulses, one an inverted version of the other. The reflections of the fundamental will be linear, and so they will also appear inverted, but the nonlinear components in the two cases will not simply be inversions of each other.

Question 20

For what reason was differential tissue harmonic imaging introduced?

Answer 20

Half the bandwidth is unused on transmit and half is unused on receive.

Question 21

what is differential tissue harmonic imaging

Answer 21

In differential tissue harmonic imaging a two-frequency pulse is used to maximise the use of the transmit bandwidth, and the nonlinear components at the second harmonic and difference frequency are used for the image, maximising the use of the receive bandwidth.