MPHY007_waves_lecture_1_slides.pdf

Course Introduction Revision Oscillatory motion Wave motion Physics of waves MPHY0007: Physics for Biomedical Engineering Lecture notes 1 Peter Munro Deptartment of Medical...

Course Introduction Revision Oscillatory motion Wave motion Physics of waves MPHY0007: Physics for Biomedical Engineering Lecture notes 1 Peter Munro Deptartment of Medical Physics and Biomedical Engineering University College London January 2024 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Lecturer information I Peter Munro ([email protected]). I Room 1.18, Malet Place Engineering Building. I Please feel free to email me if you have questions. I Please do stop me during lectures if you have a question or comment. I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Course content I Lecture notes 1: Introduction to waves: I Introduction to optical coherence tomography. I Oscillatory motion, the pre-cursor to waves. I Mathematical preliminaries (vector notation etc.). I Hooke’s law and simple harmonic motion. I Time harmonic oscillations. I Introduction to waves. I Mathematical properties waves. I The wave equation. I Wave speed. I The wave equation, frequency and wavelength. I Interference, beating and standing waves. I Additional resources. I Lecture notes 2: Introduction to electromagnetic theory I Lecture notes 3: Advanced wave properties I Lecture notes 4: Applications of waves Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Course content I Where possible, the course will taught in the context of optical coherence tomography, a relatively recently introduced medical imaging technique. I I will generally shorten “optical coherence tomography” to OCT. I Some OCT related material is not directly examinable, in this case I state this on the slide title. I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Course content I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) Most of you will be familiar with ultrasound imaging: Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) Most of you will be familiar with ultrasound imaging: I Axial resolution is obtained via: Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) Most of you will be familiar with ultrasound imaging: I Axial resolution is obtained via: I Propagation time of sound waves. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) Most of you will be familiar with ultrasound imaging: I Axial resolution is obtained via: I Propagation time of sound waves. I Lateral resolution is obtained via: Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) Most of you will be familiar with ultrasound imaging: I Axial resolution is obtained via: I Propagation time of sound waves. I Lateral resolution is obtained via: I Beam steering/scanning. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) I 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/) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) A typical OCT image of an atypical retina (Image courtsey of Dr Pearse Keane, Moorfields Eye Hospital) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) Deep learning is being applied to OCT images with a view to performing automatic diagnosis (see https: //deepmind.com) https://deepmind.com/applied/deepmind-health/working-nhs/ health-research-tomorrow Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion 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 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) Start with a beam and a scatterer such as a gold nano-particle. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) The nano-particle scatters light. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) We can make a point measurement by detecting how much light is scattered. Note: no depth discrimination. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) We can build up a line scan by recording the amount of scattered light for each lateral beam position. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) We can build up a line scan by recording the amount of scattered light for each lateral beam position. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) We can build up a line scan by recording the amount of scattered light for each lateral beam position. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) We can build up a line scan by recording the amount of scattered light for each lateral beam position. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) So far imaging is two-dimensional, ie, there is no depth information. Example: microscope image of finger print. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) Looking at a single xz slice (z = depth), what is resolved in this image? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) OCT image terminology: C-scan, B-scan and A-scan: Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Motivation: OCT (not examinable) OCT image formation depends upon wave properties. We are going to study a variety of wave properties, 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! Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Scalar quantities do not depend on... Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Scalar quantities do not depend on... I Direction. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Scalar quantities do not depend on... I Direction. I Examples of scalar quantities: Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Scalar quantities do not depend on... I Direction. I Examples of scalar quantities: I Pressure, temperature, speed, volume, power... Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Scalar quantities do not depend on... I Direction. I Examples of scalar quantities: I Pressure, temperature, speed, volume, power... I Vector quantities do depend on direction. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Scalar quantities do not depend on... I Direction. I Examples of scalar quantities: I Pressure, temperature, speed, volume, power... I Vector quantities do depend on direction. I Examples of vector quantities: Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Scalar quantities do not depend on... I Direction. I Examples of scalar quantities: I Pressure, temperature, speed, volume, power... I Vector quantities do depend on direction. I Examples of vector quantities: I Velocity, electromagnetic field, displacement, force... Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Vectors can be denoted in a number of ways. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Vectors can be denoted in a number of ways. → − a,→ I E.g. a, → − −a , A and A. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Vectors can be denoted in a number of ways. → − a,→ I E.g. a, → − −a , A and A. I The textbook by Jewett and → − Serway uses A. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Vectors can be denoted in a number of ways. → − a,→ I E.g. a, → − −a , A and A. I The textbook by Jewett and → − Serway uses A. I I use a. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Vectors can be denoted in a number of ways. → − a,→ I E.g. a, → − −a , A and A. I The textbook by Jewett and → − Serway uses A. I I use a. I What are unit vectors? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Vectors can be denoted in a number of ways. → − a,→ I E.g. a, → − −a , A and A. I The textbook by Jewett and → − Serway uses A. I I use a. I What are unit vectors? I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I a =?, i.e., express a in terms of ax , ay , az , iˆ, jˆ and k̂. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I a =?, i.e., express a in terms of ax , ay , az , iˆ, jˆ and k̂. I a = ax iˆ + ay jˆ + az k̂ Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I a =?, i.e., express a in terms of ax , ay , az , iˆ, jˆ and k̂. I a = ax iˆ + ay jˆ + az k̂ I Often simplified to a = (ax , ay , az ). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Sometimes we can represent a vector using a scalar, where simplifying assumptions are made. For example, imagine a mass moves along the x-axis. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Sometimes we can represent a vector using a scalar, where simplifying assumptions are made. For example, imagine a mass moves along the x-axis. I In this case a =?. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Sometimes we can represent a vector using a scalar, where simplifying assumptions are made. For example, imagine a mass moves along the x-axis. I In this case a =?. I a = (ax , 0, 0) = ax î. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Vector notation I Sometimes we can represent a vector using a scalar, where simplifying assumptions are made. For example, imagine a mass moves along the x-axis. I In this case a =?. I a = (ax , 0, 0) = ax î. I We often use ax to represent displacement, as its vector nature is understood from the context. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Hooke’s law I Jewett and Serway, Figure 15.1. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Hooke’s law I Fs = −kx (1) I k is the spring constant. I x is displacement of the block relative to its equilibrium position. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Hooke’s law I Which equation relates force (Fs ) and acceleration (a)? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Hooke’s law I Which equation relates force (Fs ) and acceleration (a)? I Fs = ma (2) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Hooke’s law I Which equation relates force (Fs ) and acceleration (a)? I Fs = ma (2) I Which equation relates acceleration (a) to displacement (x) and time (t)? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Hooke’s law I Which equation relates force (Fs ) and acceleration (a)? I Fs = ma (2) I Which equation relates acceleration (a) to displacement (x) and time (t)? I d2 a = 2x (3) dt Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I Combine (1), (2) and (3): d 2x k 2 =− x (4) dt m Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I Combine (1), (2) and (3): d 2x k 2 =− x (4) dt m I We postulate a solution of the form: x = A cos(ωt + φ) (5) Exercise: find the value of ω. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I Combine (1), (2) and (3): d 2x k 2 =− x (4) dt m I We postulate a solution of the form: x = A cos(ωt + φ) (5) Exercise: find the value of ω. I Solution: insert (5) into (4). d 2x = −ω 2 A cos(ωt + φ) (6) dt 2 k k − x = − A cos(ωt + φ) (7) m m Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I Combine (1), (2) and (3): d 2x k 2 =− x (4) dt m I We postulate a solution of the form: x = A cos(ωt + φ) (5) Exercise: find the value of ω. I Solution: insert (5) into (4). d 2x = −ω 2 A cos(ωt + φ) (6) dt 2 k k − x = −A cos(ωt + φ) (7) m m q I Equating (6) and (7) reveals ω = m k. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I The value of φ is determined by initial conditions. For example, a t = 0, x = A cos(φ). 0.8 ? = :=3 ? = !:=3 0.6 0.4 0.2 0 x -0.2 -0.4 -0.6 -0.8 -5 0 5 !t Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I The value of φ is determined by initial conditions. For example, a t = 0, x = A cos(φ). 0.8 ? = :=3 ? = !:=3 0.6 I Is knowledge of position at a 0.4 certain time (ie, x(t0 )) sufficient 0.2 0 x to uniquely determine φ? -0.2 -0.4 -0.6 -0.8 -5 0 5 !t Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I The value of φ is determined by initial conditions. For example, a t = 0, x = A cos(φ). 0.8 ? = :=3 ? = !:=3 0.6 I Is knowledge of position at a 0.4 certain time (ie, x(t0 )) sufficient 0.2 0 x to uniquely determine φ? -0.2 -0.4 I No: consider if we specify -0.6 -0.8 x(0) = 0.5 with A = 1. To -5 0 5 uniquely specify φ we also need !t information on the velocity at t = 0. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I Recall the general solution (5) x = A cos(ωt + φ) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I Recall the general solution (5) x = A cos(ωt + φ) I For a given combination of spring (ie, k) and block (ie, m), ω is fixed. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I Recall the general solution (5) x = A cos(ωt + φ) I For a given combination of spring (ie, k) and block (ie, m), ω is fixed. I Thus, if we know A and φ, we uniquely know the form of x(t). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I Recall the general solution (5) x = A cos(ωt + φ) I For a given combination of spring (ie, k) and block (ie, m), ω is fixed. I Thus, if we know A and φ, we uniquely know the form of x(t). I In other words, if I tell you the values of A and φ, you immediately know to write x = A cos(ωt + φ). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I Recall the general solution (5) x = A cos(ωt + φ) I For a given combination of spring (ie, k) and block (ie, m), ω is fixed. I Thus, if we know A and φ, we uniquely know the form of x(t). I In other words, if I tell you the values of A and φ, you immediately know to write x = A cos(ωt + φ). I It is very convenient to omit ωt and write a time-harmonic form as X = A exp(iφ). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I We denote time-harmonic forms with a capital, in this case X. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I We denote time-harmonic forms with a capital, in this case X. I We use a rule to go from the time-harmonic form (X ) to the instantaneous form (x) as: x = < {X exp(iωt)} (8) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I We denote time-harmonic forms with a capital, in this case X. I We use a rule to go from the time-harmonic form (X ) to the instantaneous form (x) as: x = < {X exp(iωt)} (8) I Eg: < {A exp(iφ) exp(iωt)} = A cos(ωt + φ) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I We denote time-harmonic forms with a capital, in this case X. I We use a rule to go from the time-harmonic form (X ) to the instantaneous form (x) as: x = < {X exp(iωt)} (8) I Eg: < {A exp(iφ) exp(iωt)} = A cos(ωt + φ) I Exercise: if x = A sin(ωt), what is X ? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I We denote time-harmonic forms with a capital, in this case X. I We use a rule to go from the time-harmonic form (X ) to the instantaneous form (x) as: x = < {X exp(iωt)} (8) I Eg: < {A exp(iφ) exp(iωt)} = A cos(ωt + φ) I Exercise: if x = A sin(ωt), what is X ? I 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). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Time-harmonic oscillations I We denote time-harmonic forms with a capital, in this case X. I We use a rule to go from the time-harmonic form (X ) to the instantaneous form (x) as: x = < {X exp(iωt)} (8) I Eg: < {A exp(iφ) exp(iωt)} = A cos(ωt + φ) I Exercise: if x = A sin(ωt), what is X ? I 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). I Thus, X = A exp(i3π/2). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Energy of simple harmonic motion I What is the equation relating mass (m), velocity (v ) and kinetic energy (K )? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Energy of simple harmonic motion I What is the equation relating mass (m), velocity (v ) and kinetic energy (K )? I 1 K = mv 2 (9) 2 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Energy of simple harmonic motion I What is the equation relating mass (m), velocity (v ) and kinetic energy (K )? I 1 K = mv 2 (9) 2 I If the displacement of our harmonically oscillating object is given by x = A cos(ωt + φ), what is its velocity? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Energy of simple harmonic motion I What is the equation relating mass (m), velocity (v ) and kinetic energy (K )? I 1 K = mv 2 (9) 2 I If the displacement of our harmonically oscillating object is given by x = A cos(ωt + φ), what is its velocity? I d v = A cos(ωt + φ) = −ωA sin(ωt + φ) (10) dt Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Energy of simple harmonic motion I What is the equation relating mass (m), velocity (v ) and kinetic energy (K )? I 1 K = mv 2 (9) 2 I If the displacement of our harmonically oscillating object is given by x = A cos(ωt + φ), what is its velocity? I d v = A cos(ωt + φ) = −ωA sin(ωt + φ) (10) dt I So the kinetic energy is: 1 1 K = mv 2 = mω 2 A2 sin2 (ωt + φ) (11) 2 2 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I The oscillator also has potential energy (U). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I The oscillator also has potential energy (U). I We can calculate potential energy using the equation: Z xf Z xf 1 U= Fa dx = kxdx = kxf2 (12) 0 0 2 where Fa is the force applied to the spring (= kx) and xf is the position of the oscillator. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I The oscillator also has potential energy (U). I We can calculate potential energy using the equation: Z xf Z xf 1 U= Fa dx = kxdx = kxf2 (12) 0 0 2 where Fa is the force applied to the spring (= kx) and xf is the position of the oscillator. I In general then, we have 1 1 U = kxf2 = kA2 cos2 (ωt + φ) (13) 2 2 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I The oscillator also has potential energy (U). I We can calculate potential energy using the equation: Z xf Z xf 1 U= Fa dx = kxdx = kxf2 (12) 0 0 2 where Fa is the force applied to the spring (= kx) and xf is the position of the oscillator. I In general then, we have 1 1 U = kxf2 = kA2 cos2 (ωt + φ) (13) 2 2 I So what is the total energy, i.e., K + U =? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Simple harmonic motion I The oscillator also has potential energy (U). I We can calculate potential energy using the equation: Z xf Z xf 1 U= Fa dx = kxdx = kxf2 (12) 0 0 2 where Fa is the force applied to the spring (= kx) and xf is the position of the oscillator. I In general then, we have 1 1 U = kxf2 = kA2 cos2 (ωt + φ) (13) 2 2 I So what is the total energy, i.e., K + U =? I K + U = 21 kA2 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Introduction to waves I Waves transfer energy without transferring matter. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Introduction to waves I Waves transfer energy without transferring matter. I Examples of different types of waves: I Electromagnetic. I Mechanical, e.g., waves on a string, sound waves. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Introduction to waves I Waves transfer energy without transferring matter. I Examples of different types of waves: I Electromagnetic. I Mechanical, e.g., waves on a string, sound waves. I Waves require the propagation of a physical disturbance such as: I Displacement. I Parallel (longitudinal wave); or I Perpendicular (transverse wave) to wave propagation direction. I Electromagnetic field. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Introduction to waves Figures from Jewett and Serway. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Introduction to waves Figures from Jewett and Serway. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I We know that a physical property such as displacement is propagated by a wave. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I We know that a physical property such as displacement is propagated by a wave. I Consider a wave on a string, plot displacement along the string at two instants in time. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I We know that a physical property such as displacement is propagated by a wave. I Consider a wave on a string, plot displacement along the string at two instants in time. I The wave moves with velocity v = ∆x/∆t. I 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. I So x − vt = x0 defines the position of a particular point on the wave at any time. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I 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. I However, note that we have y (x, t) = y (x − vt, 0), since the wave propagates without changing its shape. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I 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. I However, note that we have y (x, t) = y (x − vt, 0), since the wave propagates without changing its shape. I We simplify this to y (x, t) = f (x − vt). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I Consider the same wave on a string, can you plot the displacement of a particular point on the string as a function of time? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I 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). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I 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. I 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) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Mathematical properties of waves I 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. I 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) I (14) is sometimes called the wave function. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Wave speed I A particular “point” on a wave is at position x and time t defined by x − vt = constant Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Wave speed I A particular “point” on a wave is at position x and time t defined by x − vt = constant I The speed the wave is thus given by d d x = (vt + constant) = v (15) dt dt Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion The wave equation I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion The wave equation I 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. I The wave equation in one-dimension is: d 2u 1 d 2u = (16) dx 2 v 2 dt 2 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! Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion The wave equation I 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. I The wave equation in one-dimension is: d 2u 1 d 2u = (16) dx 2 v 2 dt 2 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! I There are many possible types of solutions to the wave equation that depend upon: I The nature of v , i.e., real or imaginary. I Boundary conditions, i.e., is the value of u fixed or known at any point in space or time? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion The wave equation I 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 − 2πft (17) λ Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion The wave equation I 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 − 2πft (17) λ I First check that (17) satisfies the wave equation:  2 d 2s 2π LHS = 2 = − s(x, t) (18) dx λ 1 d 2s 1 RHS = 2 2 = − 2 (2πf )2 s(x, t) (19) v dt v LHS=RHS requires v = f λ (20) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Wavelength I We haven’t yet explained what λ and f are. λ is the wavelength. Recall (17) 2π
s(x, t) = A sin λ x − 2πft Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Frequency I And, f is the frequency, it’s reciprocol is the period T. Recall (17) 2π
s(x, t) = A sin λ x − 2πft Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference 2 1 d 2u I The wave equation ( ddxu2 = v 2 dt 2 ) is linear. Revision: what does linearity mean? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference 2 2 I The wave equation ( ddxu2 = v12 ddt u2 ) is linear. Revision: what does linearity mean? I 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? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference 2 2 I The wave equation ( ddxu2 = v12 ddt u2 ) is linear. Revision: what does linearity mean? I 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 (u 1 (x, t) + u2 (x, t)) = u1 (x, t) + u2 (x, t) dx 2 dx 2 dx 2 1 d2 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 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference I The sum of two (or more) solutions to the wave equation is also a solution. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference I The sum of two (or more) solutions to the wave equation is also a solution. I This gives rise to what is usually called interference. Interference is responsible for a number of phenomena, for example, “beating” and standing waves. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference I The sum of two (or more) solutions to the wave equation is also a solution. I This gives rise to what is usually called interference. Interference is responsible for a number of phenomena, for example, “beating” and standing waves. I Standing waves: I Consider u1 (x, t) = sin(2πx/λ − 2πft) and u2 (x, t) = sin(−2πx/λ − 2πft). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference I The sum of two (or more) solutions to the wave equation is also a solution. I This gives rise to what is usually called interference. Interference is responsible for a number of phenomena, for example, “beating” and standing waves. I Standing waves: I Consider u1 (x, t) = sin(2πx/λ − 2πft) and u2 (x, t) = sin(−2πx/λ − 2πft). I Use sin(a) + sin(b) = 2 cos( a−b a+b 2 ) sin( 2 ) to simplify u1 (x, t) + u2 (x, t): Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference I The sum of two (or more) solutions to the wave equation is also a solution. I This gives rise to what is usually called interference. Interference is responsible for a number of phenomena, for example, “beating” and standing waves. I Standing waves: I Consider u1 (x, t) = sin(2πx/λ − 2πft) and u2 (x, t) = sin(−2πx/λ − 2πft). I Use sin(a) + sin(b) = 2 cos( a−b a+b 2 ) sin( 2 ) to simplify u1 (x, t) + u2 (x, t): I u1 (x, t) + u2 (x, t) = −2 cos(2πx/λ) sin(2πft). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Interference I The sum of two (or more) solutions to the wave equation is also a solution. I This gives rise to what is usually called interference. Interference is responsible for a number of phenomena, for example, “beating” and standing waves. I Standing waves: I Consider u1 (x, t) = sin(2πx/λ − 2πft) and u2 (x, t) = sin(−2πx/λ − 2πft). I Use sin(a) + sin(b) = 2 cos( a−b a+b 2 ) sin( 2 ) to simplify u1 (x, t) + u2 (x, t): I u1 (x, t) + u2 (x, t) = −2 cos(2πx/λ) sin(2πft). I This is not a travelling wave... Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 2 : f t = 0.50 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 2 : f t = 0.62 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 2 : f t = 0.75 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 2 : f t = 0.88 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 2 : f t = 1.00 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 2 : f t = 1.12 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 2 : f t = 1.25 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 2 : f t = 1.37 : 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -20 -10 0 10 20 2: x/ 6 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves A standing wave has nodes (displacement always 0) and antinodes (displacement oscillates between extremes). The peaks and troughs remain at the same location in space. 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 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves I Recall u1 (x, t) + u2 (x, t) = −2 cos(2πx/λ) sin(2πft). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves I Recall u1 (x, t) + u2 (x, t) = −2 cos(2πx/λ) sin(2πft). I 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). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves I 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). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Standing waves I 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). I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Application to OCT: A laser cavity I Some OCT systems use lasers that have a tunable wavelength. Lasers emit electromagnetic waves of (nearly) a single wavelength λ. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Application to OCT: A laser cavity I Some OCT systems use lasers that have a tunable wavelength. Lasers emit electromagnetic waves of (nearly) a single wavelength λ. I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Application to OCT: A laser cavity I Some OCT systems use lasers that have a tunable wavelength. Lasers emit electromagnetic waves of (nearly) a single wavelength λ. I 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. I 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? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Application to OCT: A laser cavity I Some OCT systems use lasers that have a tunable wavelength. Lasers emit electromagnetic waves of (nearly) a single wavelength λ. I 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. I 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? I The shortest separation between nodes is λ/2 so the mirrors could be separated by 1300nm/2 = 650nm. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I Piano tuning was traditionally done using tuning forks. E.g.: https://www.youtube.com/watch?v=qoDAue56LXM. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I Piano tuning was traditionally done using tuning forks. E.g.: https://www.youtube.com/watch?v=qoDAue56LXM. I How did the tuner know when the frequency of the string matched that of the tuning fork? Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I Piano tuning was traditionally done using tuning forks. E.g.: https://www.youtube.com/watch?v=qoDAue56LXM. I How did the tuner know when the frequency of the string matched that of the tuning fork? I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I Piano tuning was traditionally done using tuning forks. E.g.: https://www.youtube.com/watch?v=qoDAue56LXM. I How did the tuner know when the frequency of the string matched that of the tuning fork? I 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. I The period of oscillation is much greater than that of either of the two individual sound waves. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I 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. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I 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. I Or, can write u1 (x, t) = sin(2πf1 (x/v − t)) and u2 (x, t) = sin(2πf2 (x/v − t)). Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I 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. I Or, can write u1 (x, t) = sin(2πf1 (x/v − t)) and u2 (x, t) = sin(2πf2 (x/v − t)). I 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) Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I Beating causes the wave to have two distinct temporal modulations. 2 1.5 1 0.5 u1 (x,t) + u2 (x, t) ≈ 0 2 sin 2π f¯(x/v − t) cos [2π(∆f /2)t]  -0.5 -1 -1.5 -2 0 10 20 t 30 40 50 f Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I The term representing beating is: cos [2π(∆f /2)t] which reaches its maximum in amplitude when cos [2π(∆f /2)t] = ±1 Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Beating I The term representing beating is: cos [2π(∆f /2)t] which reaches its maximum in amplitude when cos [2π(∆f /2)t] = ±1 I That is when 2π(∆f /2)t = nπ, or t = 0, 1/∆f , 2/∆f ,.... So the beat frequency is given by fbeat = ∆f. Peter Munro Physics of waves  Course Introduction Revision Oscillatory motion Wave motion Additional resources I An introduction to OCT (non-examinable): http://obel.ee.uwa.edu.au/research/fundamentals/ introduction-oct/. I 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. Peter Munro Physics of waves