Nonlinear Acoustics

Question 1

Why is it called linear acoustics?

Answer 1

Wave propagation speed (sound speed) depends on the medium

Doubling the source amplitude doubles the pressure everywhere

Total acoustic pressure = sum of individual waves

Two pulses can cross without interacting

A propagating plane wave pulse stays the same shape

Question 2

What real effects are not modelled by linear acoustics?

Answer 2

Wave steepening

Shock formation

Harmonic generation

Question 3

When are nonlinear effects noticeable?

Answer 3

At MHz frequencies, nonlinear effects are noticeable at surprisingly low acoustic pressures ~0.5 MPa

Question 4

What are the assumptions for linear acoustics?

Answer 4

Linear relationship between acoustic pressure and density p = c_0^2 ρ

Products of two acoustic variables were neglected

Question 5

What are the assumptions for the nonlinear acoustic wave equation?

Answer 5

Material nonlinearity: fluid becomes harder to compress the higher the pressure

Convective nonlinearity: wave is carried forward on particle motion, like the wind

Question 6

What are the two nonlinearities?

Answer 6

material and convective nonlinearity

(which distort the wave)

Question 7

What are cumulative nonlinearities?

Answer 7

The effects of the nonlinearities accumulates as the wave travels (it gets more and more distorted the further it goes)

Question 8

What are local nonlinearities?

Answer 8

nonlinear effects that do not carry forward with the wave but only change it locally

Question 9

Why is the relationship between pressure and particle velocity usually assumed to be linear?

Answer 9

p = ρ_0 c_0 u

as nonlinearities remain local and therefore become negligible as the cumulative effect grows

Question 10

What is the convective nonlinearity?

Answer 10

c ≈ c_0 + u

c = sound speed

c_0 = sound speed of medium

u = acoustic particle velocity

Question 11

What happens during convective nonlinearity?

Answer 11

Peaks travel faster than troughs and it gradually accumulates and peak becomes steeper than trough

Question 12

What happens when a fluid is compressed sufficiently?

Answer 12

Its stiffness will increase and sound speed will increase at higher pressures (causing material nonlinearity)

Question 13

What is the gradient of linear pressure density relationship?

Answer 13

c_0 ^2

Question 14

What is the gradient of nonlinear pressure density relationship?

Answer 14

c^2 ρ

Question 15

What is used to find the material nonlinearity equation?

Answer 15

Using Taylor series to express sound speed as a function of the size of the acoustic fluctuation

Question 16

What is the material nonlinearity equation?

Answer 16

c ≈ c_0 + 1/2 B/A u

B/A = non linearity parameter

Question 17

What happens at large amplitudes?

Answer 17

Wave speed no longer just a property of the medium

Both the convective and material nonlinearities act to:

– increase the propagation speed of positive parts of the wave

– decrease propagation speed of negative parts of wave

Question 18

What is the nonlinearity coefficient and what does it account for?

Answer 18

β = 1 + B/2A

Accounts for the material and convective nonlinearity (how much wave speed will change)

Question 19

What is the combined speed affected by convective and material nonlinearities?

Answer 19

c ≈ c_0 + β u

Question 20

What happens as a result of nonlinearity?

Answer 20

wave steepening: sawtooth wave develops as symmetrical pulse input and overtime asymmetric pulse develops

Question 21

How does single frequency input affect the signal?

Answer 21

As the wave steepens, harmonics are generated

Frequencies appear at 2f_0, 3f_0, 4f_0, 5f_0

Question 22

How does double frequency input affect the signal?

Answer 22

Harmonics of each occur at 2f_1, 3f_1 and 2f_2, 3f_2

Also difference and sum frequencies appear (f_1 + f_2), (f_1 - f_2)

Question 23

In tissue, which nonlinearity has a larger effect?

Answer 23

Material nonlinearity has larger effect than convective nonlinearity

e.g.in blood β = 1 + 3 = 4, material nonlinearity accounts for 75%

Question 24

What is the equation for change in sound speed?

Answer 24

β u / c_0 x 100

Question 25

What does the change in sound speed of a particle show?

Answer 25

The change in sound speed is small, but it matters because the change
to the wave shape accumulates (distorts more as it propagates)

Question 26

When does β u = c_0?

Answer 26

Can’t get anywhere close to particle velocity to the sound speed

would require a huge pressure

Question 27

What is the shock parameter?

Answer 27

As the wave travels and steepens, this is an expression for when the gradient in the wave first becomes infinite

(When the gradient is large enough for the pressure change to occur on an intermolecular scale)

Question 28

At what distance will the shock occur?

Answer 28

At a smaller distance than the distance travelled before peak catches trough

This is due to the fact that here we are finding the first point of infinity which happens sooner than the peak meeting the trough

Question 29

When will there not be a shock?

Answer 29

At high frequencies as they are more strongly absorbed

Question 30

What happens during steepening?

Answer 30

Steepening moves energy to higher harmonics 2f_0, 3f_0

Question 31

What does clinical ultrasound use?

Answer 31

Sufficiently high frequencies and pressures to generate nonlinear effects

Question 32

What is generated for a single frequency source?

Answer 32

Harmonics are generated