MPHY0007 Physics for Biomedical Engineering: Waves - Lecture Notes 1 PDF

MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 1 Course Introduction Lecturer information Peter Munro ([email protected]). Room 1.18, Malet Place Engineering Building. Please feel free to email me if you have questions...

MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 1 Course Introduction Lecturer information Peter Munro ([email protected]). Room 1.18, Malet Place Engineering Building. Please feel free to email me if you have questions. Please do stop me during lectures if you have a question or comment. See the Moodle page for a link to an online version of the textbook: Physics for Scientists and Engineers, tenth edition, by Serway and Jewett. Course content Lecture notes 1: Introduction to waves: – Introduction to optical coherence tomography. – Oscillatory motion, the pre-cursor to waves. – Mathematical preliminaries (vector notation etc.). – Hooke’s law and simple harmonic motion. – Time harmonic oscillations. – Introduction to waves. – Mathematical properties waves. – The wave equation. – Wave speed. – The wave equation, frequency and wavelength. – Interference, beating and standing waves. – Additional resources. Lecture notes 2: Introduction to electromagnetic theory Lecture notes 3: Advanced wave properties Lecture notes 4: Applications of waves Course content Where possible, the course will taught in the context of optical coherence tomography, a relatively recently introduced medical imaging technique. – I will generally shorten “optical coherence tomography” to OCT. – Some OCT related material is not directly examinable, in this case I state this on the slide title. The waves part of the module relies heavily on maths, most of which you will have seen before. The idea of this is that you will be able to understand concepts from first principles. As you work through the notes, try to perform the mathematical manipulations yourself without the notes, this will help you to learn. 1 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Course content I will distribute exercise sheets periodically throughout the course. I recommend working through these as they are released as they give a good indication of the level of detail I expect you to learn the material. Motivation: OCT (not examinable) Most of you will be familiar with ultrasound imaging: Axial resolution is obtained via: – Propagation time of sound waves. Lateral resolution is obtained via: – Beam steering/scanning. Motivation: OCT (not examinable) Optical coherence tomography is very similar to ultrasound, however it uses light instead of sound waves, note the difference in resolution and imaging depth. (Image courtsey of Dr Timothy Hillman, formerly of the University of Western Australia, see http://obel.ee.uwa.edu.au/research/fundamentals/introduction-oct/) 2 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Motivation: OCT (not examinable) A typical OCT image of an atypical retina (Image courtsey of Dr Pearse Keane, Moorfields Eye Hospital) Motivation: OCT (not examinable) Deep learning is being applied to OCT images with a view to performing automatic diagnosis (see https://deepmind.com) 3 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 https://deepmind.com/applied/deepmind- health/working- nhs/health- research- tomorrow Motivation: OCT (not examinable) Update: deep learning has demonstrated “mind blowing” automatic diagnosis of OCT images of the retina (see https://deepmind.com) Nature Medicine, 24: 2018, 1342–1350, 2018 Motivation: OCT (not examinable) Start with a beam and a scatterer such as a gold nano-particle. 4 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Motivation: OCT (not examinable) The nano-particle scatters light. Motivation: OCT (not examinable) We can make a point measurement by detecting how much light is scattered. Note: no depth discrim- ination. 5 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Motivation: OCT (not examinable) We can build up a line scan by recording the amount of scattered light for each lateral beam position. Motivation: OCT (not examinable) We can build up a line scan by recording the amount of scattered light for each lateral beam position. 6 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Motivation: OCT (not examinable) We can build up a line scan by recording the amount of scattered light for each lateral beam position. Motivation: OCT (not examinable) We can build up a line scan by recording the amount of scattered light for each lateral beam position. 7 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Motivation: OCT (not examinable) So far imaging is two-dimensional, ie, there is no depth information. Example: microscope image of finger print. Motivation: OCT (not examinable) However, if we use (effectively, we’ll consider this later) the propagation time of light also, we can build a 3D image. 8 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Motivation: OCT (not examinable) Looking at a single xz slice (z = depth), what is resolved in this image? Motivation: OCT (not examinable) OCT image terminology: C-scan, B-scan and A-scan: Motivation: OCT (not examinable) OCT image formation depends upon wave properties. We are going to study a variety of wave prop- erties, and where possible, relate them back to OCT. Hopefully you will learn about waves and OCT at the same time. Now on to the physics of waves! 2 Revision Vector notation Scalar quantities do not depend on... – Direction. 9 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Examples of scalar quantities: – Pressure, temperature, speed, volume, power... Vector quantities do depend on direction. Examples of vector quantities: – Velocity, electromagnetic field, displacement, force... Vector notation Vectors can be denoted in a number of ways. → − a, → E.g. a, → − − a , A and A. → − The textbook by Jewett and Serway uses A. I use a. What are unit vectors? – Vectors with unit length, for example, î, ĵ and k̂ which are the unit vectors parellel to the x, y and z axes in the Cartesian coordinate system. Vector notation a =?, i.e., express a in terms of ax , ay , az , î, ĵ and k̂. a = ax î + ay ĵ + az k̂ Often simplified to a = (ax , ay , az ). 10 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Vector notation Sometimes we can represent a vector using a scalar, where simplifying assumptions are made. For example, imagine a mass moves along the x-axis. In this case a =?. a = (ax , 0, 0) = ax î. We often use ax to represent displacement, as its vector nature is understood from the context. 3 Oscillatory motion Hooke’s law Jewett and Serway, Figure 15.1. Hooke’s law Fs = −kx (1) k is the spring constant. x is displacement of the block relative to its equilibrium position. 11 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Hooke’s law Which equation relates force (Fs ) and acceleration (a)? Fs = ma (2) Which equation relates acceleration (a) to displacement (x) and time (t)? d2 a= x (3) dt2 Simple harmonic motion Combine (1), (2) and (3): d2 x k =− x (4) dt2 m We postulate a solution of the form: x = A cos(ωt + φ) (5) Exercise: find the value of ω. Solution: insert (5) into (4). d2 x = −ω 2 A cos(ωt + φ) (6) dt2 k k − x = − A cos(ωt + φ) (7) m m q k Equating (6) and (7) reveals ω = m. 12 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Simple harmonic motion The value of φ is determined by initial conditions. For example, a t = 0, x = A cos(φ). Is knowledge of position at a certain time (ie, x(t0 )) sufficient to uniquely determine φ? No: consider if we specify x(0) = 0.5 with A = 1. To uniquely specify φ we also need information on the velocity at t = 0. ? = :=3 0.8 ? = !:=3 0.6 0.4 0.2 0 x -0.2 -0.4 -0.6 -0.8 -5 0 5 !t Time-harmonic oscillations Recall the general solution (5) x = A cos(ωt + φ) For a given combination of spring (ie, k) and block (ie, m), ω is fixed. Thus, if we know A and φ, we uniquely know the form of x(t). In other words, if I tell you the values of A and φ, you immediately know to write x = A cos(ωt + φ). It is very convenient to omit ωt and write a time-harmonic form as X = A exp(iφ). Time-harmonic oscillations We denote time-harmonic forms with a capital, in this case X. We use a rule to go from the time-harmonic form (X) to the instantaneous form (x) as: x = < {X exp(iωt)} (8) Eg: < {A exp(iφ) exp(iωt)} = A cos(ωt + φ) Exercise: if x = A sin(ωt), what is X? – cos(ωt + φ) = cos(ωt) cos(φ) − sin(ωt) sin(φ) = sin(ωt) holds when φ = 3π/2 (note: cos(3π/2) = 0, sin(3π/2) = −1). – Thus, X = A exp(i3π/2). 13 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Energy of simple harmonic motion What is the equation relating mass (m), velocity (v) and kinetic energy (K)? 1 K= mv 2 (9) 2 If the displacement of our harmonically oscillating object is given by x = A cos(ωt + φ), what is its velocity? d v= A cos(ωt + φ) = −ωA sin(ωt + φ) (10) dt So the kinetic energy is: 1 1 K= mv 2 = mω 2 A2 sin2 (ωt + φ) (11) 2 2 Simple harmonic motion The oscillator also has potential energy (U ). We can calculate potential energy using the equation: Z xf Z xf 1 U= Fa dx = kxdx = kx2f (12) 0 0 2 where Fa is the force applied to the spring (= kx) and xf is the position of the oscillator. In general then, we have 1 2 1 U= kxf = kA2 cos2 (ωt + φ) (13) 2 2 So what is the total energy, i.e., K + U =? K + U = 12 kA2 4 Wave motion Introduction to waves Waves transfer energy without transferring matter. Examples of different types of waves: – Electromagnetic. – Mechanical, e.g., waves on a string, sound waves. Waves require the propagation of a physical disturbance such as: – Displacement. ∗ Parallel (longitudinal wave); or ∗ Perpendicular (transverse wave) to wave propagation direction. – Electromagnetic field. 14 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Introduction to waves Figures from Jewett and Serway. Introduction to waves Figures from Jewett and Serway. 15 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Mathematical properties of waves We know that a physical property such as displacement is propagated by a wave. Consider a wave on a string, plot displacement along the string at two instants in time. – The wave moves with velocity v = ∆x/∆t. 16 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 – So if a point on the wave is at position x0 at t = 0, it moves to point x = x0 + vt at some later time. – So x − vt = x0 defines the position of a particular point on the wave at any time. Mathematical properties of waves We could use a function y(x, t) to denote the height of any element on this wave as a function of x and t. However, note that we have y(x, t) = y(x − vt, 0), since the wave propagates without changing its shape. We simplify this to y(x, t) = f (x − vt). Mathematical properties of waves Consider the same wave on a string, can you plot the displacement of a particular point on the string as a function of time? To plot this we might as well set x = 0, meaning y(0, t) = f (−vt). Mathematical properties of waves We have shown that a wave has the same shape as both a function of time and space, i.e., x−vt = x0 defines the location of a particular point on the wave with time. We can make this defintion more formal by defining a function, f , which defines each “point” on the wave. f is a function of x and t and takes the form: f (x, t) = f (x ± vt) (14) (14) is sometimes called the wave function. Wave speed A particular “point” on a wave is at position x and time t defined by x − vt = constant The speed the wave is thus given by d d x = (vt + constant) = v (15) dt dt 17 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 The wave equation We won’t cover the derivation of the wave equation, see Jewett and Serway if you are interested. The derivation of the wave equation is not examinable. The wave equation in one-dimension is: d2 u 1 d2 u = (16) dx2 v 2 dt2 where u can be displacement, electric field etc., depending upon the type of wave. Example: a function of the form u = f (x + vt) satisfies the wave equation, verify this! There are many possible types of solutions to the wave equation that depend upon: – The nature of v, i.e., real or imaginary. – Boundary conditions, i.e., is the value of u fixed or known at any point in space or time? The wave equation The solution of the wave equation is an entire subject in itself. We will look at just one solution, a travelling wave, on an infinite string:   2π s(x, t) = A sin x − 2πf t (17) λ First check that (17) satisfies the wave equation:  2 d2 s 2π LHS = = − s(x, t) (18) dx2 λ 1 d2 s 1 RHS = 2 2 = − 2 (2πf )2 s(x, t) (19) v dt v LHS=RHS requires v = f λ (20) Wavelength 2π
We haven’t yet explained what λ and f are. λ is the wavelength. Recall (17) s(x, t) = A sin λ x − 2πf t 18 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Frequency 2π
And, f is the frequency, it’s reciprocol is the period T. Recall (17) s(x, t) = A sin λ x − 2πf t Interference 2 1 d2 u The wave equation ( ddxu2 = v 2 dt2 ) is linear. Revision: what does linearity mean? Hint: If, u1 (x, t) and u2 (x, t) are both solutions of the wave equation, is u1 (x, t) + u2 (x, t) also a solution? d2 d2 d2 (u1 (x, t) + u 2 (x, t)) = u1 (x, t) + u2 (x, t) dx2 dx2 dx2 2 1 d 1 d2 = 2 2 u1 (x, t) + 2 2 u2 (x, t) v dt v dt 1 d2 = 2 2 (u1 (x, t) + u2 (x, t)) v dt Interference The sum of two (or more) solutions to the wave equation is also a solution. This gives rise to what is usually called interference. Interference is responsible for a number of phenomena, for example, “beating” and standing waves. Standing waves: – Consider u1 (x, t) = sin(2πx/λ − 2πf t) and u2 (x, t) = sin(−2πx/λ − 2πf t). – Use sin(a) + sin(b) = 2 cos( a−b a+b 2 ) sin( 2 ) to simplify u1 (x, t) + u2 (x, t): ∗ u1 (x, t) + u2 (x, t) = −2 cos(2πx/λ) sin(2πf t). – This is not a travelling wave... 19 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs do not move in space. 2 : f t = 0.50 : 2 : f t = 0.62 : 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -20 -10 0 10 20 -20 -10 0 10 20 2: x/ 6 2: x/ 6 2 : f t = 0.75 : 2 : f t = 0.88 : 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -20 -10 0 10 20 -20 -10 0 10 20 2: x/ 6 2: x/ 6 2 : f t = 1.00 : 2 : f t = 1.12 : 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -20 -10 0 10 20 -20 -10 0 10 20 2: x/ 6 2: x/ 6 20 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 2 : f t = 1.25 : 2 : f t = 1.37 : 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -20 -10 0 10 20 -20 -10 0 10 20 2: x/ 6 2: x/ 6 2 : f t = 1.50 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Standing waves Recall u1 (x, t) + u2 (x, t) = −2 cos(2πx/λ) sin(2πf t). The standing waves always has zero displacement at cos(2πx/λ) = 0, giving: λ 3λ 5λ x= , , ,... (21) 4 4 4 as the location of nodes (points of zero displacement). Standing waves The standing waves always has maximum displacement at cos(2πx/λ) = ±1, giving: λ 3λ x = 0, , λ, ,... (22) 2 2 as the location of antinodes (points of maximum displacement). Adjacent nodes are separated by λ/2, adjacent antinodes are separated by λ/2 and the separation between each adjacent node and antinode is λ/4. 21 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 Application to OCT: A laser cavity Some OCT systems use lasers that have a tunable wavelength. Lasers emit electromagnetic waves of (nearly) a single wavelength λ. Lasers have a cavity formed by two parallel mirrors. One condition for lasing is that the parallel mirrors of the cavity cause a standing wave to oscillate between the two mirrors. Assuming that nodes of the standing wave are located at the two mirrors, what is the shortest separation between the mirrors that will allow the laser to lase at λ = 1300nm? The shortest separation between nodes is λ/2 so the mirrors could be separated by 1300nm/2 = 650nm. Beating Piano tuning was traditionally done using tuning forks. E.g.: https://www.youtube.com/watch? v=qoDAue56LXM. How did the tuner know when the frequency of the string matched that of the tuning fork? When there is a small difference in frequency between the string and tuning fork, interference between the two sound waves results in a periodic (in time) variation in the amplitude of the resultant sound wave. The period of oscillation is much greater than that of either of the two individual sound waves. Beating Now consider u1 (x, t) = sin(2πx(f1 /v) − 2πf1 t) and u2 (x, t) = sin(2πx(f2 /v) − 2πf2 t). Recall: f λ = v. Or, can write u1 (x, t) = sin(2πf1 (x/v − t)) and u2 (x, t) = sin(2πf2 (x/v − t)). We use the trigonometric identity sin(a) + sin(b) = 2 sin((a + b)/2) cos((b − a)/2) (23) to show that u1 (x, t) + u2 (x, t) = 2 sin 2π f¯(x/v − t) cos [2π(∆f /2)(x/v − t)]   (24) where f¯ = (f1 + f2 )/2 and ∆f = (f2 − f1 ). When ∆f and x are small we can write: u1 (x, t) + u2 (x, t) ≈ 2 sin 2π f¯(x/v − t) cos [2π(∆f /2)t]   (25) Beating Beating  causes the wave to have two distinct temporal modulations. u1 (x, t) + u2 (x, t) ≈ 2 sin 2π f¯(x/v − t) cos [2π(∆f /2)t] 22 MPHY0007: Physics for Biomedical Engineering: Physics of waves Lecture notes 1 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0 10 20 t 30 40 50 f Beating The term representing beating is: cos [2π(∆f /2)t] which reaches its maximum in amplitude when cos [2π(∆f /2)t] = ±1 That is when 2π(∆f /2)t = nπ, or t = 0, 1/∆f, 2/∆f,.... So the beat frequency is given by fbeat = ∆f. Additional resources An introduction to OCT (non-examinable): http://obel.ee.uwa.edu.au/research/fundamentals/ introduction-oct/. Jewett and Serway: chapter 3 (revision on scalars and vectors), chapter 15 up to and including 15.3, sections 16.1, 16.2 and 16.6 (you don’t need to derive the wave equation), sections 18.1, 18.2 and 18.7. 23