Bioe225/
Lecture 2: Ultrasound as a Wave
1 Lecture Objectives
• What is a wave?
• Derive the acoustic wave equation
• Describe an acoustic wave using standard terminology
• Wave properties
2 Waves Defined
• A disturbance or oscillation that travels through space and time, accompanied by a transfer of energy from one point to another, with little or no permanent displacement of matter.
3 Waves Defined
Longitudinal Wave
4 Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State Waves Defined
Shear Wave
5 Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State Waves Defined Surface Wave
6 Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State Waves Defined
We will focus on longitudinal waves in this course
7 Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State Mathematical Representation
• The wave equation: 2 2 1 ∂ p ∇ p = c2 ∂t 2
8 Derivation of the acoustic wave equation
1. Conservation of momentum: difference in pressure between two points in space (pressure gradient) gives rise to a force between those two points: force = mass . acceleration ∝ - pressure gradient
Assume an infinitesimal mass: m = ρAdx dx Velocity of the particle along the x direction: v = dt dv dv dP dF = − dPA → ρAdx = − dPA → ρ = − dt dt dx
And since both velocity and pressure can have space and time dependencies, it is more accurate to write the relationship as partial derivatives:
∂v ∂P 1. ρ = − ∂t ∂x
9 Derivation of the acoustic wave equation
2. Conservation of energy: if velocity at x + Δx is greater than it is at x , the pressure at point x + Δx will be less than the pressure at point x . This pressure drop is related to compressibility as follows: ∂P ∂v − = K ∂t ∂x Take equation 1, divide both sides by ρ and take the derivative with respect to x :
∂v 1 ∂P ∂ ∂v 1 ∂2P ∂ ∂v 1 ∂2P = − ⇒ = − ⇒ = − ∂t ρ ∂x ∂x ∂t ρ ∂x2 ∂t ∂x ρ ∂x2 1 ∂P ∂v substituting − for , we have: K ∂t ∂x
∂2P K ∂2P 2. = ∂t2 ρ ∂x2 10 Derivation of the acoustic wave equation
K Compressibility and the speed of sound are related as: c = ρ ∂2P ∂2P Therefore, equation 2. in one dimension becomes: = c2 ∂t2 ∂x2 ∂2P ∂2 ∂2 ∂2 In three dimensions: = c2 + + P ∂t2 ( ∂x2 ∂y2 ∂z2 )
∂2 ∂2 ∂2 And since + + = ∇ . ∇ = ∇2, the most general form of the acoustic ∂x2 ∂y2 ∂z2 wave equation is:
∂2 P(x, y, z, t) = c2 ∇2P(x, y, z, t) ∂t2
11 Mathematical Representation
• The wave equation: 1 ∂2 ∇2P(x, y, z, t) = P(x, y, z, t) c2 ∂t2 • Many phenomena can be represented by this equation!
12 Mathematical Representation
• The wave equation: 1 ∂2 ∇2P(x, y, z, t) = P(x, y, z, t) c2 ∂t2 • Many phenomena can be represented by this equation! – “The most incomprehensible thing about the universe is that it is comprehensible.” – Albert Einstein
13 Solutions to the Wave Equation
• The most common solution to this equation is:
• For this lecture, we will assume propagation in The z-direction and the solution simplifies to: A eiωt+ikz
14 Some Terminology
T • ω: radial frequency • f: frequency = ω/2π • T: period = 1/f time A space • A: amplitude ˆ A • : wave vector k – | |= ω/c = 2π/λ λ – k ˆ points in direction of propagation • λ: wavelength = c/f
15 Phase, φ
• Phase describes how many cycles a signal has accumulated
• It is always relative
• k describes how quickly phase accumulates in space (φ=kz)
• ω describes how quickly phase accumulates in time (φ=ωt)
16 Wavefronts
• A surface of equal phase
• What shape are the wavefronts of: A eiωt+ikz
17 Wavefronts
• A surface of equal phase
• What shape are the wavefronts of: A eiωt+ikz
18 Wavefronts
• A surface of equal phase
• What shape are the wavefronts resulting from a point source on: A eiωt+ikz
19 Wavefronts
• A surface of equal phase
• What shape are the wavefronts resulting from a point source on: A eiωt+ikz
20 Continuous wave vs. Pulsed Wave
21 Components of a multi-pulse signal
Pulse length (units: seconds) Duty Cycle = PL/(1/ PRF) = PL*PRF = PL/ RP
Ratio of time on to total time
t
Repetition Period, or more often denoted: Pulse repetition frequency (PRF = 1/ repetition period, units: Hz) These three properties of a pulse will be revisited when discussing resolution, driving transducers, and therapeutic effects. 22 Wave Phenomena
• Superposition - The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements.
23 Wave Phenomena
• Interference is the result of superposition when two waves are in the same place at the same time • The result can be constructive or destructive interference.
24 Wave Phenomena
• Diffraction - The apparent bending of waves around small obstacles and the spreading out of waves past small openings.
25 Wave Phenomena
• Reflection - a fraction of the wave “bounces back” at a boundary
• Refraction - change in direction of a wave due to a change in its medium. 26 Standing Waves
Interference of two identical harmonic waves traveling in opposite directions
27 Questions
• E-mail: [email protected]
• Office hours: TBD
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