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# Chapter 15 Notes PHY9B

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This 24 page Class Notes was uploaded by Mae Underwood on Tuesday October 6, 2015. The Class Notes belongs to PHY9B at University of California - Davis taught by Randy Harris in Fall 2015. Since its upload, it has received 38 views. For similar materials see Classical Physics in Physics 2 at University of California - Davis.

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Date Created: 10/06/15

Chapter 15 Mechanical Waves Terms Mechanical Wave A disturbance from the state of equilibrium travelling through a medium causing the particles of the medium to be displaced The medium itself is not displaced but energy is transferred ie energy is transported but not matter Sinusoidal Wave Repeating wave pattern that can be described by functions of sine or cosine Displacement The instantaneous distance in a specified direction from its equilibrium position Amplitude A The maximum displacement from the equilibrium position to the furthest point of travel peaktrough of a wave A greater amplitude means there is more energy in the system Frequency f The number of oscillations in unit time usually in seconds Period T Time it takes for one complete oscillationcycle of motion Phase Difference The angle usually in radians between two similar oscillations Transverse Wave Displacement of particles in the medium are perpendicular to the direction of wave travel propagation Longitudinal Wave Displacement of particles in the medium are parallel to the direction of wave travelpropagation Interference When waves travelling in opposite directions overlap Normal Mode Frequencies Frequencies that enable sinusoidal waves to exist Wave Pulse A single disturbance travelling through a medium rather a repetitive wave pattern Wavelength A Length of one complete wave pattern eg one trough to the next adjacent trough or one compression to another Compression A region of increased density in a longitudinal wave Rarefaction A region of decreased density in a longitudinal wave Wave Speed v Take one point on a wave The wave speed is the speed needed to keep AKA Phase Speed alongside that point as the wave propagates ie speed to keep a phase kx out constant Equations Angular Frequency in Time Period T Wave Speed for a v 1 f Periodic Wave v F Wave Speed for a v E Transverse Wave v Where F restoring force to equilibrium 2n Wave Number k k 7 Angular Frequency 00 a vk Sinusoidal Wave Function 3 05 13 ACOSUOC i wt 32y 1 32y Wave equation 3x2 172 312 xdirection Instantaneous Power of a 33 5y P xt Fv F Wave y 3 3x at P t F 33 3x x39 6x 6 Instantaneous Power of a F kwAzsm2 kx wt Sinusoidal Wave 2 1Fu 12a23inz kx wt Pmax Jim12a Maximum Power Pave P W Wave Intensity I A 47172 Equation for a Standing Wave yx t ASWsmkxsmwt Fundamental Frequency of a Vibrating String and 1 F its Corresponding Sound f1 E E Wave 15 Wavelength Graph of a sine wave with x amp y axes that express x amp y distance from Wikimedia Commons the free media repository 0 Speed of wave propagation is NOT the same as the speed of particle displacement 0 Every particle in the medium undergoes simple harmonic motion SHM with the same frequency ie frequency does not vary across the medium 0 Wave speed is determined by the properties of the medium 0 Waves of the all frequencies travel at the same speed ie if frequency increases wavelength decreases to maintain wave speed 153 Mathematical Description of a Wave 0 Transverse wave The value of y depends on which particle we are observing f xaxis is the equilibrium position then y depends on x and time t yx 0 t Acoswt Acos21tft Since it s periodic the motion of a specific point x at a time t is equivalent to the motion of the point x0 at the time t Note xv is just the equation for speed yx t Acosw t 9 Since cos 6 c036 x yx t Acos21tf t v 1 Substitude f i andf T yx t Acos21tf 27t Substitute k 7 quotwave numberquot yx t Acoskx wt Note These are all variations of the same wave function Graphing Wave Functions There are two kinds of graphs 1 YDisplacement of a particle from the equilibrium position versus the XDisplacement x yx t 0 Acoskx AcosZnL o Represents the shape of the wave at t0 o The distance from one point of the wave to an adjacent point is one wavelength 1 2 YDisplacement of a particle form the equilibrium position versus the time travelled t 3390quot 039 t AC05 wt Acoswt Acos2nf o Represents the motion of the particle at x0 o The distance from one point of the wave to an adjacent point is one time period T Note These graphs represent waves in the xdirection The quantity kx i cut is the phase Partial Derivatives 0 First partial derivative of the wave function slope of the wave at point x and time t 3xa 3t k v 0 Second partial derivative of the wave function Curvature of the wave concaveconvex 32y 2 6x2 k yx 6231 62y Z If we dIVIde wby and that a 27tf 27731 vk 32x 32y 32y 12 2 612 T 612 39 6x2 k2 17 The wave equation for a wave travelling in the xdirection 02y 1 02y 0x2 172 0t2 0 Shape of the wave concave up acceleration at this point is positive 0 Shape of the wave concave down acceleration is negative 0 At points of inflection where curvaturesecond derivative wrt x 0 acceleration 0 154 Speed of a Transverse Wave Factors for wave speed on a string 0 Tension T 0 Linear mass density p kgm 0 Tension oc v 0 Greater tension means there is a greater restoring force therefore increasing the wave speed 0 mass of medium oc 0 Greater mass means a greater inertia resisting restoration to equilibrium therefore decreases the wave speed I L IJ TnL 155 Energy in Wave Motion Creation of a wave apply a force to a portion of the medium and hence do work on the system creates a displacement As the wave propagates each portion of the medium does work on the next portion thereby transporting energy 0 In a transverse wave there is no acceleration in the xdirection therefore the xcomponents of F have equal magnitude F o This means that the slope is obtained by F3 0y F 0x Gym t FyX t 2 FT 0 The wave is travelling in the xdirection while the component FX restoring force is applied in the opposite direction thereby canceling out vX and keeping the particle in the same xposition this is part of the definition of a transverse wave Then for the slope to be positive Fy must then be negative l String Wave velocity x Fx L r A Fx 0 Rate at which work is done Instantaneous Power a a F3 3 P xt F xt V xt y m am 0y 0y 0 Energy IS only transferred where a at O and a at 0 Le When transverse force and velocity Fy and Vy are nonzero o For a sinusoidal wave 03 0y P xt F kaAZSin2 kx wt ax at Substitute a vk amp v2 E Px t 1Fu12a2inzkx wt 39AKSmL39 Kx wb 73 Awa39nCRxCUH P F85 E35 Fc A2kmwckx mgt5 FAQquot Kwsin C KK 00 E7 Matte 3511 par5 3 57 VJE 7 it CI 6 fa F alggsli JZC39 quot 3 s Flaw J A3 cl WW Azmlemz rx out ldQ FEE FEEg er quot lairEig To get the maximum power consider the fact that the maximum the sine function can equal is 1 Pmax JFquwz Since sine squared is never negative power is always positive or zero For average power consider the fact that the average of sinzx 12 1 1 Pave EPmax EvF Azwz Since P is never negative energy never flows in the opposite direction to wave travel 0 PenodT For a mechanical wave power is quadrupled iff is doubled P oc f2 For electromagnetic waves Pave DC A2 Wave Intensity For 3D wave intensity I is the average rate at which energy is transported by the wave across a surface that is perpendicular to the direction of wave propagation per unit area I Average Power Pave W Area A m2 If waves are radially spread out equally in all directions then I oc T Z If the power output is P then the average intensity through a sphere of radius r and surface area 4an is P 47TT 2 156 Wave Interference Boundary Conditions and Superposition When a wave strikes the boundaries of a medium all or part of the wave is reflected Interference is when the incident and reflected waves overlap As the 2 pulses overlap and pass the total displacement of the medium the algebraic sum of displacements at that point and time o Fixed End A reflected pulsewave in the reverse direction and displacement is produced Ma 0 Free End A reflected pulsewave with the same direction of displacement but travelling in the opposite direction is produced Interference A o Principle of Linear Superposition ie wave function is linear only to the 1st power When 2 waves overlap the actual displacement of any point on the string at any time is obtained by adding the displacement of the 2 individual wavespulses together yxlt y1xlt y2xlt 157 Standing Waves on a String o If a sinusoidal wave is reflected by a fixed end of a string the travelling and incident waves overlap to form a standing wave 51 quot to DRE Li 5 L3 L A LL L 11 C013 L gt N nodes points on the string that never moves A antinodes points where the displacement is greatest o The string appears to be subdivided into a number of segments 0 In a wave that travels along a string 0 Amplitude constant 0 Wave pattern moves with a speed wave speed 0 As opposed to the travelling wave the wave pattern of a standing wave remains in the same position along the string and its amplitude fluctuates standing wave 0 The principle of superposition explains how incident and reflected waves combine to form a standing wave mphm WW A H M U V V commie 15 th in maximum mg Pl priming for mewWM in NM Destructive interference the displacements of overlapping waves are always equal and opposite 180 out of phase and cancel each other out Constructive interference displacements of overlapping waves are always identical in phase resulting in a large resultant displacement The distance between successive nodes or antinodes is AZ A wave function for a standing wave can be determined by adding the individual wave functions y1xt and y2xt of the overlapping waves y1x t Acoskx wt incident wave going left yz x t Acoskx wt re f lected wave going right A sign change denotes a phase shift of 180 orn radians At x 0 the reflected wave yz x t Acosoot and the incident wave y1x t Acosoot or Acosoot 11 The wave function for the standing wave in the figure above is thus yx t y1x t yz x t A coskx wt coskx wt Using the sum and difference identities of two angles cosa i b cosacosb i sinasinb yx t y1x t yz x t 2A3inkxsinwt or yx t ASWSinkxSinwt The standing wave amplitude is twice the amplitude of either of the original waves Standing waves vs travelling waves 0 All points between successive pair of nodes oscillate in phase for standing waves 0 Nodes can be found by setting sinkx 0 in the equation above so the displacement is always zero This occurs when kx 0 TE 2 TE 3 TE 0 Note that there is a node at x 0 A standing wave does not transfer energy The 2 waves that form it individually carry equal amounts of power in opposite directions There is a local flow of energy from each node to adjacent antinodes and back Average rate of energy transfer O at every points 158 Normal Modes of a String When a string held rigidly at both ends is plucked a wave is produced which is reflected and re reflected from the ends of the string making a standing wave The standing wave produces a sound wave in the air with a frequency determined by properties of the string A sinusoidal wave on a string with fixed ends must have a node on both ends The length of the string L A L 2 n2 fixed at both ends 2L An 7 fixed at both ends Where n 1 2 3 o For a wave to be a standing wave it must satisfies both of the equations above 0 The smallest frequency f1 corresponds to the largest wavelength n 1 11 2L 17 f1 Z fixed at both ends This is called the fundamental frequency 17 n2 2 nfl fixed at both ends Where n 1 2 3 fn These frequencies are called harmonics and the series is called a harmonic series Musicians sometimes ca frequencies above n 1 overtones with the second harmonic being the first overtone For a string with fixed ends at 0 and x L the wave function of the nth standing wave is given by ynx t ASWSinknxSinwnt 27i39 wn znf n kn 1 n A normal mode of an oscillating system is a motion in which all particles of the system move sinusoidally with the same frequency Each normal mode has its unique frequency and vibration pattern A harmonic oscillator which has 1 oscillating particle has only one normal mode and one frequency The string fixed at both ends has infinitely many normal modes since its made up of many particles Complex Standing Waves If we displace a string so that its shape is the same as one of the normalmode patterns and then release it it would vibrate with the frequency of that mode The shape of the displaced string is actually a superposition of many normal modes several simple harmonic motions of different frequencies are present at the same time The displacement of any point on the string is the sum of the displacements associated with the individual modes The standing wave on the string and the travelling sound wave in the air have similar harmonic content Harmonic analysis finding the representation for a specific vibration patter Fourier Series the sum of sinusoidal functions that represents a complex wave Combining the equations for the fundamental frequency of a string and the speed of the wave on the string 1 F f1 I This equation also represents the fundamental frequency of the sound wave created by the vibrating string h 39 Harmonic EL 151 1 2 2m 2L 2A 2 3rd 2L 31 I f ha 3 2 F d 4th 212 4L 5 H d t 2 I Jim l The first four normal modes of a string taken from httpwwwco bwcomAudio string harmonicshtm Chapter 15 Mechanical Waves Terms Mechanical Wave A disturbance from the state of equilibrium travelling through a medium causing the particles of the medium to be displaced The medium itself is not displaced but energy is transferred ie energy is transported but not matter Sinusoidal Wave Repeating wave pattern that can be described by functions of sine or cosine Displacement The instantaneous distance in a specified direction from its equilibrium position Amplitude A The maximum displacement from the equilibrium position to the furthest point of travel peaktrough of a wave A greater amplitude means there is more energy in the system Frequency f The number of oscillations in unit time usually in seconds Period T Time it takes for one complete oscillationcycle of motion Phase Difference The angle usually in radians between two similar oscillations Transverse Wave Displacement of particles in the medium are perpendicular to the direction of wave travel propagation Longitudinal Wave Displacement of particles in the medium are parallel to the direction of wave travelpropagation Interference When waves travelling in opposite directions overlap Normal Mode Frequencies Frequencies that enable sinusoidal waves to exist Wave Pulse A single disturbance travelling through a medium rather a repetitive wave pattern Wavelength A Length of one complete wave pattern eg one trough to the next adjacent trough or one compression to another Compression A region of increased density in a longitudinal wave Rarefaction A region of decreased density in a longitudinal wave Wave Speed v Take one point on a wave The wave speed is the speed needed to keep AKA Phase Speed alongside that point as the wave propagates ie speed to keep a phase kx out constant Equations Angular Frequency 00 Time Period T Wave Speed for a v 1 f Periodic Wave v F Wave Speed for a v E Transverse Wave v Where F restoring force to equilibrium 2n Wave Number k k 7 Angular Frequency 00 a vk Sinusoidal Wave Function 3 05 13 ACOSUOC i wt 32y 1 32y Wave equation 3x2 172 312 xdirection Instantaneous Power of a 33 5y P xt Fv F Wave y 3 3x at P t F 33 3x x39 6x 6 Instantaneous Power of a F kwAzsm2 kx wt Sinusoidal Wave 2 1Fu 12a23inz kx wt Pmax Jim12a Maximum Power Pave P W Wave Intensity I A 47172 Equation for a Standing Wave yx t ASWsmkxsmwt Fundamental Frequency of a Vibrating String and 1 F its Corresponding Sound f1 E E Wave 15 Wavelength Graph of a sine wave with x amp y axes that express x amp y distance from Wikimedia Commons the free media repository 0 Speed of wave propagation is NOT the same as the speed of particle displacement 0 Every particle in the medium undergoes simple harmonic motion SHM with the same frequency ie frequency does not vary across the medium 0 Wave speed is determined by the properties of the medium 0 Waves of the all frequencies travel at the same speed ie if frequency increases wavelength decreases to maintain wave speed 153 Mathematical Description of a Wave 0 Transverse wave The value of y depends on which particle we are observing f xaxis is the equilibrium position then y depends on x and time t yx 0 t Acoswt Acos21tft Since it s periodic the motion of a specific point x at a time t is equivalent to the motion of the point x0 at the time t Note xv is just the equation for speed yx t Acosw t 9 Since cos 6 c036 x yx t Acos21tf t v 1 Substitude f i andf T yx t Acos21tf 27t Substitute k 7 quotwave numberquot yx t Acoskx wt Note These are all variations of the same wave function Graphing Wave Functions There are two kinds of graphs 1 YDisplacement of a particle from the equilibrium position versus the XDisplacement x yx t 0 Acoskx AcosZnL o Represents the shape of the wave at t0 o The distance from one point of the wave to an adjacent point is one wavelength 1 2 YDisplacement of a particle form the equilibrium position versus the time travelled t 3390quot 039 t AC05 wt Acoswt Acos2nf o Represents the motion of the particle at x0 o The distance from one point of the wave to an adjacent point is one time period T Note These graphs represent waves in the xdirection The quantity kx i cut is the phase Partial Derivatives 0 First partial derivative of the wave function slope of the wave at point x and time t 3xa 3t k v 0 Second partial derivative of the wave function Curvature of the wave concaveconvex 32y 2 6x2 k yx 6231 62y Z If we dIVIde wby and that a 27tf 27731 vk 32x 32y 32y 12 2 612 T 612 39 6x2 k2 17 The wave equation for a wave travelling in the xdirection 02y 1 02y 0x2 172 0t2 0 Shape of the wave concave up acceleration at this point is positive 0 Shape of the wave concave down acceleration is negative 0 At points of inflection where curvaturesecond derivative wrt x 0 acceleration 0 154 Speed of a Transverse Wave Factors for wave speed on a string 0 Tension T 0 Linear mass density p kgm 0 Tension oc v 0 Greater tension means there is a greater restoring force therefore increasing the wave speed 0 mass of medium oc 0 Greater mass means a greater inertia resisting restoration to equilibrium therefore decreases the wave speed I L IJ TnL 155 Energy in Wave Motion Creation of a wave apply a force to a portion of the medium and hence do work on the system creates a displacement As the wave propagates each portion of the medium does work on the next portion thereby transporting energy 0 In a transverse wave there is no acceleration in the xdirection therefore the xcomponents of F have equal magnitude F o This means that the slope is obtained by F3 0y F 0x Gym t FyX t 2 FT 0 The wave is travelling in the xdirection while the component FX restoring force is applied in the opposite direction thereby canceling out vX and keeping the particle in the same xposition this is part of the definition of a transverse wave Then for the slope to be positive Fy must then be negative l String Wave velocity x Fx L r A Fx 0 Rate at which work is done Instantaneous Power a a F3 3 P xt F xt V xt y m am 0y 0y 0 Energy IS only transferred where a at O and a at 0 Le When transverse force and velocity Fy and Vy are nonzero o For a sinusoidal wave 03 0y P xt F kaAZSin2 kx wt ax at Substitute a vk amp v2 E Px t 1Fu12a2inzkx wt 39AKSmL39 Kx wb 73 Awa39nCRxCUH P F85 E35 Fc A2kmwckx mgt5 FAQquot Kwsin C KK 00 E7 Matte 3511 par5 3 57 VJE 7 it CI 6 fa F alggsli JZC39 quot 3 s Flaw J A3 cl WW Azmlemz rx out ldQ FEE FEEg er quot lairEig To get the maximum power consider the fact that the maximum the sine function can equal is 1 Pmax JFquwz Since sine squared is never negative power is always positive or zero For average power consider the fact that the average of sinzx 12 1 1 Pave EPmax EvF Azwz Since P is never negative energy never flows in the opposite direction to wave travel 0 PenodT For a mechanical wave power is quadrupled iff is doubled P oc f2 For electromagnetic waves Pave DC A2 Wave Intensity For 3D wave intensity I is the average rate at which energy is transported by the wave across a surface that is perpendicular to the direction of wave propagation per unit area I Average Power Pave W Area A m2 If waves are radially spread out equally in all directions then I oc T Z If the power output is P then the average intensity through a sphere of radius r and surface area 4an is P 47TT 2 156 Wave Interference Boundary Conditions and Superposition When a wave strikes the boundaries of a medium all or part of the wave is reflected Interference is when the incident and reflected waves overlap As the 2 pulses overlap and pass the total displacement of the medium the algebraic sum of displacements at that point and time o Fixed End A reflected pulsewave in the reverse direction and displacement is produced Ma 0 Free End A reflected pulsewave with the same direction of displacement but travelling in the opposite direction is produced Interference A o Principle of Linear Superposition ie wave function is linear only to the 1st power When 2 waves overlap the actual displacement of any point on the string at any time is obtained by adding the displacement of the 2 individual wavespulses together yxlt y1xlt y2xlt 157 Standing Waves on a String o If a sinusoidal wave is reflected by a fixed end of a string the travelling and incident waves overlap to form a standing wave 51 quot to DRE Li 5 L3 L A LL L 11 C013 L gt N nodes points on the string that never moves A antinodes points where the displacement is greatest o The string appears to be subdivided into a number of segments 0 In a wave that travels along a string 0 Amplitude constant 0 Wave pattern moves with a speed wave speed 0 As opposed to the travelling wave the wave pattern of a standing wave remains in the same position along the string and its amplitude fluctuates standing wave 0 The principle of superposition explains how incident and reflected waves combine to form a standing wave mphm WW A H M U V V commie 15 th in maximum mg Pl priming for mewWM in NM Destructive interference the displacements of overlapping waves are always equal and opposite 180 out of phase and cancel each other out Constructive interference displacements of overlapping waves are always identical in phase resulting in a large resultant displacement The distance between successive nodes or antinodes is AZ A wave function for a standing wave can be determined by adding the individual wave functions y1xt and y2xt of the overlapping waves y1x t Acoskx wt incident wave going left yz x t Acoskx wt re f lected wave going right A sign change denotes a phase shift of 180 orn radians At x 0 the reflected wave yz x t Acosoot and the incident wave y1x t Acosoot or Acosoot 11 The wave function for the standing wave in the figure above is thus yx t y1x t yz x t A coskx wt coskx wt Using the sum and difference identities of two angles cosa i b cosacosb i sinasinb yx t y1x t yz x t 2A3inkxsinwt or yx t ASWSinkxSinwt The standing wave amplitude is twice the amplitude of either of the original waves Standing waves vs travelling waves 0 All points between successive pair of nodes oscillate in phase for standing waves 0 Nodes can be found by setting sinkx 0 in the equation above so the displacement is always zero This occurs when kx 0 TE 2 TE 3 TE 0 Note that there is a node at x 0 A standing wave does not transfer energy The 2 waves that form it individually carry equal amounts of power in opposite directions There is a local flow of energy from each node to adjacent antinodes and back Average rate of energy transfer O at every points 158 Normal Modes of a String When a string held rigidly at both ends is plucked a wave is produced which is reflected and re reflected from the ends of the string making a standing wave The standing wave produces a sound wave in the air with a frequency determined by properties of the string A sinusoidal wave on a string with fixed ends must have a node on both ends The length of the string L A L 2 n2 fixed at both ends 2L An 7 fixed at both ends Where n 1 2 3 o For a wave to be a standing wave it must satisfies both of the equations above 0 The smallest frequency f1 corresponds to the largest wavelength n 1 11 2L 17 f1 Z fixed at both ends This is called the fundamental frequency 17 n2 2 nfl fixed at both ends Where n 1 2 3 fn These frequencies are called harmonics and the series is called a harmonic series Musicians sometimes ca frequencies above n 1 overtones with the second harmonic being the first overtone For a string with fixed ends at 0 and x L the wave function of the nth standing wave is given by ynx t ASWSinknxSinwnt 27i39 wn znf n kn 1 n A normal mode of an oscillating system is a motion in which all particles of the system move sinusoidally with the same frequency Each normal mode has its unique frequency and vibration pattern A harmonic oscillator which has 1 oscillating particle has only one normal mode and one frequency The string fixed at both ends has infinitely many normal modes since its made up of many particles Complex Standing Waves If we displace a string so that its shape is the same as one of the normalmode patterns and then release it it would vibrate with the frequency of that mode The shape of the displaced string is actually a superposition of many normal modes several simple harmonic motions of different frequencies are present at the same time The displacement of any point on the string is the sum of the displacements associated with the individual modes The standing wave on the string and the travelling sound wave in the air have similar harmonic content Harmonic analysis finding the representation for a specific vibration patter Fourier Series the sum of sinusoidal functions that represents a complex wave Combining the equations for the fundamental frequency of a string and the speed of the wave on the string 1 F f1 I This equation also represents the fundamental frequency of the sound wave created by the vibrating string h 39 Harmonic EL 151 1 2 2m 2L 2A 2 3rd 2L 31 I f ha 3 2 F d 4th 212 4L 5 H d t 2 I Jim l The first four normal modes of a string taken from httpwwwco bwcomAudio string harmonicshtm

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