Wave Propagation in Ultrasound: 1D to 3D Equations and Solutions

What is the primary focus of the chapter on wave propagation?

Deriving equations that describe pressure fields in 1-D to 3-D, starting with the 1-D wave equation.

What does the variable 'p' represent in the context of wave propagation?

Acoustic pressure (pa).

What is the first important relation used in acoustics derived from Newton's second law?

The relation between pressure change and displacement, expressed as a function of density and acceleration.

How is the wave equation in 1-D derived?

By considering a fluid rather than a solid and applying Newton's second law to a cylindrical fluid volume.

What does the wave speed 'c0' depend on?

The elastic properties of the tissue (κ) and its density (ε).

What is the general solution to the 1-D wave equation for a wave of pure tone?

p(x, t) = f(x ∓ ct), where f represents the wave function.

What is the superposition principle in wave propagation?

The principle that states the total pressure can be expressed as the sum of two wave functions propagating in opposite directions.

What is the significance of the equation p(x, t) = p0 cos(ωt ∓ kx)?

It describes a time harmonic plane wave in 1-D.

What are the momentum equations used for in wave propagation?

To describe the motion of fluids or solids in a domain, incorporating density, velocity, and stress.

What is the definition of the strain tensor in linear elastic solids?

ε = 1/2[∇u + (∇u)T], representing the relationship between displacements and strains.

What does the term 'isotropic linear elastic (Hookean) solids' refer to?

Materials that exhibit linear elastic behavior and have the same properties in all directions.

What is the relationship between pressure change and relative volume change in acoustics?

∆p = κ∆V, where κ is the compression modulus.

What is the importance of the wave equation in acoustics?

It models wave propagation in a homogeneous medium without accounting for energy loss or scattering.

What does the equation ω²p/ωx² = 1/c²ωt² represent?

It is part of the wave equation, relating pressure and wave speed.

What is the role of the gradient operator (∇) in the momentum equations?

It denotes the spatial rate of change of a quantity, such as velocity or pressure.

How is the material derivative (D/Dt) defined?

D/Dt = ω/ωt + v · ∇, representing the change of a quantity following a fluid particle.

What does the term 'free plane wave' refer to?

A wave that travels in a single direction in space, characterized by uniform phase across wavefronts.

What is the significance of the equation p(x, t) = P0 = constant?

It represents a trivial solution to the wave equation indicating constant pressure.

What is the purpose of deriving the wave equation in three dimensions?

To extend the understanding of wave propagation from 1-D to 3-D scenarios.

What does the notation 'p0' typically represent in wave equations?

The amplitude of the wave pressure.

What is the physical interpretation of the wave speed 'c0'?

The speed of sound in the medium through which the compression wave travels.

What is the relationship between displacement and pressure in wave propagation?

Pressure changes are related to the second derivative of displacement with respect to time and space.

What does the equation p(x, t) = f(x - ct) + g(x + ct) illustrate?

It demonstrates the principle of superposition for waves traveling in both directions.

What is the significance of the term 'bulk compression' in the context of wave propagation?

It refers to the change in volume of a fluid due to pressure changes, relevant in ultrasound applications.

What does the term 'acoustic pressure' refer to?

The local pressure variation from the ambient pressure in a medium due to sound waves.

What is the equation for local relative volume change in wave propagation?

V → V0 = 1 + ∇· ϕu

What does the symbol ε represent in wave propagation equations?

The local relative volume change, defined as ε = 1 + ∇· ϕu.

What is the relationship between the mass conservation and the local relative volume change?

Mass conservation yields ε = ε0(1 + ∇· ϕu)−1.

What does the equation εDϕv/Dt represent?

It represents the change in volume flow rate in wave propagation.

What is the momentum equation for wave propagation assuming small strains?

ω²ϕu/ε0ωt² = ∇· σ.

What is the general constitutive law for small strain linear elastic solids?

σ = λL(tr(ε))I + 2µLε.

What are the Lamé constants and their significance?

λL and µL are constants that define the elastic properties of materials.

What is the wave equation for dilatation waves?

ω²ϕu/ε0ωt² = (λL + 2µL)∇(∇· ϕu).

How are dilatation waves characterized?

Dilatation waves are compressional waves where particle movement is parallel to the direction of wave propagation.

What is the condition for shear waves in terms of volume change?

Shear waves do not change the volume, hence εv = 0.

What is the wave speed for shear waves?

The wave speed for shear waves is given by cshear = √(µL/ρ0).

What is the equation for pressure waves in fluids?

The wave equation for pressure waves is derived from the momentum equation and continuity equation.

What does the continuity equation for fluids state?

ωε/ωt + ∇· (εϕv) = 0.

What is the state equation for isothermal processes in fluids?

p = p(ε).

What is the bulk modulus for fluids defined as?

κ = ε0(ωε/ρ0)(%ωp).

What is the wave equation for dilatation disturbances in fluids?

ω²(∇· ϕv)/c² = ∇²(∇· ϕv) = 0.

What is the significance of the equation ∇· (∇²ϕu) = ∇²(∇· ϕu)?

It shows the relationship between divergence and wave propagation in solids.

What characterizes the particle movement in shear waves?

Particle movement is perpendicular to the direction of wave propagation.

What is the equation for the wave speed of dilatation waves?

c0 = √(κ + 4(λL + 2µL)/ε0).

What does the term ∇· ϕu represent in wave propagation?

It represents the divergence of the displacement field.

What is the significance of neglecting higher order terms in wave equations?

It simplifies the equations for small displacements.

What is the relationship between dilatation waves and pressure waves?

Dilatation waves can be seen as pressure waves due to their compressional nature.

What does the equation ω²ϕu/ε0ωt² = 0 imply?

It indicates a wave equation where the wave propagation is not affected by external forces.

What is the role of the identity matrix I in the constitutive law?

It is used to represent the isotropic nature of the material in the stress-strain relationship.

What is the wave equation for dilatation disturbances?

c0 = √(ε0 / κ)

What type of waves cannot be supported by a non-viscous fluid?

Shear waves

What is the relationship between pressure and the divergence of velocity in fluids?

p(ϕr, t) = κ∇·ϕu(ϕr, t)

What does the momentum equation yield in the context of acoustic field equations?

∇·σ = →∇p

What is the velocity potential in wave propagation?

ϕv = →∇φ

What is the first wave equation derived from the acoustic field equations?

∇p = →ε0(ωϕv / ωt)

What does the term 'source potential function' refer to in wave equations?

It is used to calculate acoustic quantities like pressure and velocity potential.

What is the general solution for a plane wave in 3-D?

p(ϕr, t) = f(ct ∓ ϕn · ϕr)

How does the amplitude of a spherical wave change with distance?

It decreases by 1/r due to conservation of energy.

What is the equation for a spherical wave emerging from a point source?

p(ϕr, t) = (1/|ϕr|)f(ct - ϕn · ϕr) + (1/|ϕr|)g(ct + ϕn · ϕr)

What is the significance of the term 1/|ϕr| in the spherical wave equation?

It indicates the decrease in amplitude as the wave spreads over a larger area.

What does the wave vector ϕk represent in wave equations?

It defines the direction of wave propagation.

What is the relationship between frequency (ω) and spatial frequency (k) in wave propagation?

c0 = ω / k

What is the complex form of the plane wave equation?

˜p(ϕr, t) = p0e^(i(ωt - ϕk · ϕr + φ))

What does the term 'time harmonic 3-D plane wave' refer to?

It describes waves that oscillate sinusoidally in time.

What is the equation for the pressure in terms of the velocity potential?

p(ϕr, t) = ε0(ωφ / ωt)

What is the significance of the integration constant C in wave equations?

It can often be set to zero or neglected.

What does the equation ∇²p = (1/c²)ω²p signify?

It represents the wave equation in terms of pressure.

What is the condition for the existence of a solution in the spherical wave equation?

No solution is found before time t = 0.

What is the relationship between pressure and the divergence of velocity in the context of acoustic waves?

ωp(ϕr, t) = κ∇·ϕv(ϕr, t)

What does the term 'non-diagonal stresses and strains' refer to in the context of acoustic field equations?

They are zero in the case of no shape change and only volume change.

What is the general form of the wave equation derived from the acoustic field equations?

ω²p/ωt² = (1/k²)∇²p

What does the term 'dilatation wave' refer to?

A wave that results from volume changes rather than shear.

What is the significance of the unit vector ϕn in wave equations?

It defines the direction of wave propagation.

What type of wave is described by the equation p(ϕr, t) = f(ωt - ϕk · ϕr)?

A plane wave.

What is the Laplacian operator in spherical coordinates for no dependence on θ and φ?

∇² = (1/ω²)(∂²p/∂r² + (2/r)(∂p/∂r))

What does the wave equation represent in the context of pressure waves?

The wave equation describes how pressure changes over time and space, given by ω²p/c²(∇²p) = ∂²p/∂t².

What is the general solution for the wave equation in spherical coordinates?

p(ϕr, t) = 1/|ϕr| f(ωt ± k|ϕr|)

What is the form of the time-harmonic spherical wave solution?

p(ϕr, ϕr0, t) = A/|ϕr - ϕr0| cos(ωt - k|ϕr - ϕr0|)

What is the complex form of a 3-D pressure wave?

˜p = ˆpe^{-i⃗k·⃗r} e^{iωt}

What is the Helmholtz equation derived from the wave equation?

ω²/c² ∇²˜p + (1/ωt²)˜p = 0

What does the dispersion relation k = ±ω/c0 signify?

It relates spatial frequency (wavenumber k) to temporal frequency (ω).

What is the Fourier transform of a pressure wave?

P(ϕr, ω) = ∫ p(ϕr, t)e^{-iωt} dt

What is the inverse Fourier transform of a pressure wave?

p(ϕr, t) = (1/2π) ∫ P(ϕr, ω)e^{iωt} dω

What is a point source in the context of wave propagation?

A point source is modeled as a Dirac delta function, representing an infinite response at a specific point in space and time.

What is the Green's function used for in wave equations?

It is used to solve linear problems involving ordinary or partial differential equations with initial conditions.

What is the relationship between the source function and Green's function?

The solution of the wave equation for any source function is the convolution of the Green's function with the source function.

What is the significance of the Dirac delta function in wave equations?

It models an instantaneous point source, allowing for the analysis of wave propagation from a specific location.

What does the equation ∇²G(ϕr → ϕr0) = δ(ϕr → ϕr0) represent?

It defines the Green's function for the Laplace operator, indicating how waves propagate from a point source.

What is the expression for the velocity potential in fluid dynamics?

ϕv(ϕr) = Φ/(4πr²)ϕer, where Φ is the volume rate of fluid entering.

What is the amplitude factor in the time-harmonic spherical wave equation?

A represents the constant or amplitude factor in the wave equation.

What does the term ω represent in wave equations?

ω represents the angular frequency of the wave.

What does the term k represent in wave equations?

k represents the wavenumber, which is related to the spatial frequency of the wave.

What is the significance of the equation ∇² + k² = 0?

It represents the Helmholtz equation, indicating the relationship between spatial derivatives and wave propagation.

What is the physical interpretation of the term ω²p/c² in the wave equation?

It represents the relationship between pressure changes and wave propagation speed in a medium.

What does the term ∂²p/∂t² represent in the wave equation?

It represents the second time derivative of pressure, indicating how pressure changes over time.

What is the role of the divergence theorem in deriving Green's functions?

It allows the transformation of volume integrals into surface integrals, facilitating the solution of differential equations.

What does the term ϕr represent in the context of wave equations?

ϕr represents the position vector in spherical coordinates.

What is the Green's function for the Laplace operator derived from the equation?

G = 1 / (4πr)

What does the symbol δ represent in the context of the wave equation?

The Dirac delta function, which represents a point source.

What is the general solution for a spherical wave in the context of Green's function?

ω²G(ϕr →ϕr0, t) / c² ωt² = δ(ϕr →ϕr0)δ(t)