Nonlinear Acoustics Flashcards
Why is it called linear acoustics?
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
What real effects are not modelled by linear acoustics?
Wave steepening
Shock formation
Harmonic generation
When are nonlinear effects noticeable?
At MHz frequencies, nonlinear effects are noticeable at surprisingly low acoustic pressures ~0.5 MPa
What are the assumptions for linear acoustics?
Linear relationship between acoustic pressure and density p = c_0^2 ρ
Products of two acoustic variables were neglected
What are the assumptions for the nonlinear acoustic wave equation?
Material nonlinearity: fluid becomes harder to compress the higher the pressure
Convective nonlinearity: wave is carried forward on particle motion, like the wind
What are the two nonlinearities?
material and convective nonlinearity
(which distort the wave)
What are cumulative nonlinearities?
The effects of the nonlinearities accumulates as the wave travels (it gets more and more distorted the further it goes)
What are local nonlinearities?
nonlinear effects that do not carry forward with the wave but only change it locally
Why is the relationship between pressure and particle velocity usually assumed to be linear?
p = ρ_0 c_0 u
as nonlinearities remain local and therefore become negligible as the cumulative effect grows
What is the convective nonlinearity?
c ≈ c_0 + u
c = sound speed
c_0 = sound speed of medium
u = acoustic particle velocity
What happens during convective nonlinearity?
Peaks travel faster than troughs and it gradually accumulates and peak becomes steeper than trough
What happens when a fluid is compressed sufficiently?
Its stiffness will increase and sound speed will increase at higher pressures (causing material nonlinearity)
What is the gradient of linear pressure density relationship?
c_0 ^2
What is the gradient of nonlinear pressure density relationship?
c^2 ρ
What is used to find the material nonlinearity equation?
Using Taylor series to express sound speed as a function of the size of the acoustic fluctuation
What is the material nonlinearity equation?
c ≈ c_0 + 1/2 B/A u
B/A = non linearity parameter
What happens at large amplitudes?
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
What is the nonlinearity coefficient and what does it account for?
β = 1 + B/2A
Accounts for the material and convective nonlinearity (how much wave speed will change)
What is the combined speed affected by convective and material nonlinearities?
c ≈ c_0 + β u
What happens as a result of nonlinearity?
wave steepening: sawtooth wave develops as symmetrical pulse input and overtime asymmetric pulse develops
How does single frequency input affect the signal?
As the wave steepens, harmonics are generated
Frequencies appear at 2f_0, 3f_0, 4f_0, 5f_0
How does double frequency input affect the signal?
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)
In tissue, which nonlinearity has a larger effect?
Material nonlinearity has larger effect than convective nonlinearity
e.g.in blood β = 1 + 3 = 4, material nonlinearity accounts for 75%
What is the equation for change in sound speed?
β u / c_0 x 100
What does the change in sound speed of a particle show?
The change in sound speed is small, but it matters because the change
to the wave shape accumulates (distorts more as it propagates)
When does β u = c_0?
Can’t get anywhere close to particle velocity to the sound speed
would require a huge pressure
What is the shock parameter?
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)
At what distance will the shock occur?
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
When will there not be a shock?
At high frequencies as they are more strongly absorbed
What happens during steepening?
Steepening moves energy to higher harmonics 2f_0, 3f_0
What does clinical ultrasound use?
Sufficiently high frequencies and pressures to generate nonlinear effects
What is generated for a single frequency source?
Harmonics are generated
