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 28.