Lecture 1101 Sound Waves Review Physics Help Q&A: tutor.leiacademy.org Force on a Charge Moving in a Magnetic Field A charge moving in a magnetic field can have a magnetic force exerted by the B-field. The magnitude F B of the magnetic force exerted on the particle is proportional to the charge, q, and to the speed, v, of the particle. When a charged particle moves parallel to the magnetic field vector, the magnetic force acting on the particle is zero. When the particle s velocity vector makes any angle θ 0 with the field, the force acts in a direction perpendicular to the plane formed by the velocity and the field. The magnetic force exerted on a positive charge is in the direction opposite the direction of the magnetic force exerted on a negative charge moving in the same direction. The magnitude of the magnetic force is proportional to sinθ, where θis the angle the particle s velocity makes with the direction of the magnetic field. 1
Magnetic Force on a Moving Charge = Direction: Right-Hand Rule #1 This rule is based on the right-hand rule for the cross product. Your thumb is in the direction of the force if qis positive. The force is in the opposite direction of your thumb if qis negative. 2
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Motion of a Charged Particle in a Magnetic Field Consider a particle moving in an external magnetic field with its velocity perpendicular to the field. The force is always perpendicular to the direction of velocity. As a result, the path of the motion is a circle. The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle. = 90 = = = = 4
Charged Particles Moving in Electric and Magnetic Fields In many applications, charged particles will move in the presence of both magnetic and electric fields. In that case, the total force is the sum of the forces due to the individual fields. The total force is called the Lorentz force. In general: = + Velocity Selector A uniform electric field is perpendicular to a uniform magnetic field. When the force due to the electric field is equal but opposite to the force due to the magnetic field, the particle moves in a straight line. = = =/ 5
Magnetic Force on a Current Carrying Conductor A force is exerted on a current-carrying wire placed in a magnetic field. The current is a collection of many charged particles in motion. The direction of the force is given by the right-hand rule. The magnetic force is exerted on each moving charge in the wire. = To find the total force acting on the wire, we multiply the force exerted on one charge by the number of charges in the segment. = = = give both the length of the wire segment and also the direction of the current. A current carrying wire is placed in a magnetic field as shown. Which direction best represent the force applied by the B-field on the wire? x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x A B C D E zero F out of the page G into the page 6
Force on a Current Carrying Wire in Arbitrary Shape Using the results of a straight wire: = Consider a small segment of the curved wire,!, the force exerted on this segment is! =! The total force is = "! # $ 7
Torque on a Current Loop The rectangular loop carries a current Iin a uniform magnetic field. There is no magnetic force on sides 1 & 3, since the wires are parallel to the field. = = 0 '(!) 1 +! 3 There is a force on sides 2 & 4 since they are perpendicular to the field. The magnitude of the magnetic force on these sides will be: = = + '(!) 2 +! 4 The direction of 2 is out of the page. The direction of 4 is into the page. Torque on a Current Loop side view The forces are equal and in opposite directions, but not along the same line of action. The forces produce a torque around point O. The maximum torque happens in the position as shown and -.$/ can be found as: -.$/ = 20/2=0 = +0 = +0 = 12) +)+ (' 12) 3((4 This maximum value occurs when the field is parallel to the plane of the loop. 8
Torque on a Current Loop at an Angle with the B-field Assume the magnetic field makes an angle of q < 90 o with the normal direction of the loop. The net torque about point O will be: - 5 = 2 0 2 5 =+0 5 = 5 We can write this in terms of a cross product: -= Note that the direction of the loop vector is in its normal direction with a magnitude of the loop s area. The torque has a maximum value when the field is perpendicular to the normal to the plane of the loop. The torque is zero when the field is parallel to the normal to the plane of the loop. Three identical current-carrying loops are placed in a uniform B-field shown below. Which current loop experiences the greatest torque? 6 7 +) 9
Magnetic Field Lines for a Loop Figure (a) shows the magnetic field lines surrounding a current loop. Figure (b) compares the field lines to that of a bar magnet. Notice the similarities in the patterns. 10
Magnetic Field for a Long, Straight Conductor: Direction The magnetic field lines are circles concentric with the wire. The field lines lie in planes perpendicular to the wire. The magnitude of the field is constant on any circle of radius a. The right-hand rule for determining the direction of the field is shown. 11
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Ampere s Law Ampere s law states that the line integral of! around any closed path equals (, where is the total steady current passing through any surface bounded by the closed path: 9! =: ; : ; is the vacuum permeability. : ; =4< 10 =>? / It useful only for cases of a high degree of symmetry, similar to that of Gauss s law in calculating E-field. 9! =: ; = 2< = : ; = : ; 2< For an infinite straight current-carrying wire Put your right-hand thumb in the current s direction and your fingers curl in the direction for the loop of integration. Magnetic Force Between Two Parallel Conductors Two parallel wires each carry a steady current. The field due to the current in wire 2 exerts a force on wire 1 of 21 = 1 l 2. = : ; 2<+ A = A 3 = : ; A 2<+ 3 A = A 3 = : ; A 2<+ 3 Same for A. Newton s Third Law: A = A 14
Magnetic Fields Summary and Synthesis The origin of the magnetic field is moving charges (electric currents). Magnetic fields can be created by various current distributions. Ampère s law is useful in calculating the magnetic field of a highly symmetric configuration carrying a steady current. Magnetic field created by an arbitrary current distribution can be calculated with Biot- SavartLaw Three different contexts of using right hand rules: B-force on moving charges = = B-field due to! = : ;! 4< moving charges =:; /2< B-field due to a current loop Biot-Savart Law Biot and Savartexperimentally determined a mathematical expression that gives the magnetic field at some point in space due to a current.! = : ;! 4< : ; is the vacuum permeability. : ; =4< 10 =>? / The B-field direction can be found with right-hand rule and is perpendicular to both the direction of the current segment (! ) and the vector directed from current segment toward P. The magnitude of B-field is inversely proportional to, where is the distance from! to P. The magnitude of B-field is proportional to the current and to the magnitude! of the length element. The magnitude of B-field is proportional to 5, where 5 is the angle between the vectors! and. 15
Rank the points A, B, and C in terms of the magnitudeof the magnetic field created by the length segment!. A. A=B=C>0 B. A<B<C C. A>B>C D. B=C>A E. B>C>A=0 F. B>C>A>0 Total Magnetic Field! is the field created by the current in the length segment!. To find the total field, sum up (integrate) the contributions from all the current elements # = : ; 4< "! $ The integral is over the entire current distribution. The law is also valid for a current consisting of charges flowing through space. For example, this could apply to a beam of moving charges such as those in an accelerator. 16
Magnetic Field for a Long, Straight Conductor Find the field contribution from a small element of current and then integrate over the current distribution. The thin, straight wire is carrying a constant current! =!C < 2 5 D E =!CF( 5DE F( 5 = + = +/F( 5 Integrating over all the current elements gives # = : ; 4< "! = : ; 4< "!CF( 5 DE + /cos 5 $ = : ; 4<+ C = +1+5!C = +!5 cos 5 $ # J K " F( 5!5 = : ; J L 4<+ 5 A 5 DE Magnetic Field for a Long, Straight Conductor If the conductor is infinitely long but with a right angle turn as shown: The B-field due to current in section 2 is zero. The B-field due to current in section 1 can be found by setting θ 1 = π/2 and θ 2 = 0 = : ; 4<+ = : ; 4<+ D E J K " F( 5!5 = : ; J L 4<+ 5 A 5 DE 1 M + 2 17
Magnetic Field for a Curved Wire Segment Find the field at point Odue to the wire segment. For the arc part = : ; 4< "! $ # For the straight segment! = 0 Using 5 as the primary variable: 5 = /+ = : ; 4<+ = : ; 4<+ 5 θ will be in radians! =! O = : ; 4< "! + = : ; 4<+ P Magnetic Field for a Circular Loop of Wire Consider the previous result, with a full circle θ= 2π = : ; 4<+ 5 = : ; 4<+ 2< =: ; 2+ This is the field at the center of the loop. 18
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Magnetic Flux The magnetic flux associated with a magnetic field is defined in a way similar to electric flux. Consider an area element!on an arbitrarily shaped surface. The magnetic field in this element is and!is a vector that is perpendicular to the surface and has a magnitude equal to the area da. The magnetic flux Φ B is Φ = "! The unit of magnetic flux is T. m 2 = Wb Wb is a weber Magnetic Flux A special case is when a plane of area A makes an angle θwith!. The magnetic flux is Φ = cos 5. In this case shown on the right, the field is parallel to the plane and Φ B = 0. The magnetic flux is Φ B = BA cos θ. In this case when the field is perpendicular to the plane and Φ= BA. This is the maximum value of the flux. 20
Induction Magnetic fields can vary in time. The changing magnetic fields can induce emf and current in a circuit: Faraday s Experiment Conclusions An electric current can be induced in a loop by a changing magnetic field. This would be the current in the secondary circuit of this experimental setup. The induced current exists only while the magnetic field through the loop is changing. This is generally expressed as: an induced emfis produced in the loop by the changing magnetic field. The actual existence of the magnetic flux is not sufficient to produce the induced emf, the flux must be changing. 21
Faraday s Law of Induction An emfis induced in a loop when the magnetic flux through the loop changes with time. The emfinduced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit. Mathematically, E =!Φ!1 Φ B is the magnetic flux through the circuit and is found by Φ = "! If the circuit consists of N loops, all of the same area, and if Φ B is the flux through one loop, an emfis induced in every loop and Faraday s law becomes E = S!Φ!1 Faraday s Law Example Assume a loop enclosing an area A lies in a uniform magnetic field. Φ = "! The magnetic flux through the loop is : Φ = cos 5 The induced emfis E =!!1 cos5 Φ can change when: (Ways of Inducing an emf) The magnitude of the magnetic field can change with time. The area enclosed by the loop can change with time. The angle between the magnetic field and the normal to the loop can change with time. And any combination of the above can occur. 22
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Sliding Conducting Bar The induced emfis E =!Φ!1 = 3!C!1 = 3 Since the resistance in the circuit is R, the current is = E T Energy Considerations: U =!C M =!U!1 =!C!1 = 3 M = T = E T = 3 T 24
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Lenz s Law Faraday s law indicates that the induced emfand the change in flux have opposite algebraic signs. E =!Φ!1 This has a physical interpretation that has come to be known as Lenz s law. Developed by German physicist Heinrich Lenz Lenz s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop. The induced current tends to keep the original magnetic flux through the circuit from changing. Lenz Law Example The conducting bar slides on the two fixed conducting rails. The magnetic flux due to the external magnetic field through the enclosed area increases with time. The induced current must produce a magnetic field that opposes the original change of magnetic flux. So the B-field must be out of the page with the induced current in counterclockwise. If the bar moves in the opposite direction, the direction of the induced current will also be reversed. 26
Induced Current Directions Example A magnet is placed near a metal loop. a)find the direction of the induced current in the loop when the magnet is pushed toward the loop (a and b). b)find the direction of the induced current in the loop when the magnet is pulled away from the loop (c and d). 27
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A circular loop of wire falling toward a long straight wire carrying a current to the left. What is the direction of the induced current in the loop of wire? (Viewing from the perspective looking into the paper.) (a) clockwise (b) counterclockwise (c) Zero (d) impossible to determine Induced emf and Electric Fields An electric field is created as a result of the changing magnetic flux. With a conducting loop, a current can be observed. However, without a conducting loop, the E-field must still exist in empty space the conducting loop is simply there to show the results of the E-field, which moves charges and creates a current.!!1 > 0 x x x x x x x x E =!Φ =!W!1!C x x x x x x x x E =9! = W 9! =!Φ!1 # W = "! $ 2< = <!!1 =! 2!1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x E-field is created due to changing B-field not by any charges. 29
Generators Electric generators transform mechanical energy/work into electrical forms. The AC generator consists of a loop of wire rotated by some external means in a magnetic field. E =!Φ!1 Φ = =F( 5 Φ changes as a result of the periodical changes of 5 due to the rotation of the loop. DC Generators The DC (direct current) generator has essentially the same components as the AC generator. The main difference is that the contacts to the rotating loop are made using a split ring called a commutator. Use the active figure to vary the speed of rotation and observe the effect on the emf generated. 30
Motors Motors are devices into which energy is transferred by electrical transmission while energy is transferred out by work. A motor is a generator operating in reverse. A current is supplied to the coil by a battery and the torque acting on the current-carrying coil causes it to rotate. Motors As the coil rotates in a magnetic field, an emfis induced. This induced emfalways acts to reduce the supply current in the coil. The term back emfis commonly used to refer to this induced emf. The back emfincreases in magnitude as the rotational speed of the coil increases, which limits the current in the rotating coil.!)f1( (' 0+FD )' 31
Motors The induced emfexplains why the current is higher when starting a motor than when running it. When starting, there is no back emf, so current is high. When running in higher rpm, the back emfreduces the total current in the motor. W ; + W #$YZ = W ; W #$YZ T T Eddy Currents Changing B-field and flux can induce emf/e-field and cause electrons to move in metal. This creates circulating currents called eddy currents, which occurs when metal moves through a non-uniform magnetic field. The direction of eddy currents will create a B-field that opposes the change of flux. When moving into and out of a small area of B- field, the eddy currents are in opposite directions. Eddy currents create heat and are often undesirable because they transform usable energy into internal energy.!!1 0 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 32
Eddy Currents The magnetic field is directed into the page. As the plate enters the B-field, the magnetic flux going through the plate increases. The induced eddy current will create a B-field opposes the increasing flux and therefore is counterclockwise. The result reverses when the plate leaves the field. The induced eddy currents produce a magnetic retarding force and the swinging plate eventually comes to rest and the kinetic energy is transformed into internal energy heat ( T). Inductance A time-varying current in a circuit produces an induced emfopposing the emf that initially set up the time-varying current. Circuit elements that produces significant induced emfis called inductors Induction caused by a single element is called self-inductance Inductions caused among different elements are mutual inductance W ; + W #$YZ T 33
Self-Inductance When the switch is closed, the current does not immediately reach its maximum value. As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time. This increasing flux creates an induced emfin the circuit. ^ (1) + W ^ W ; + W ^ ( E _ When the switch just closes.! 1!1 >0 T Self-Inductance The direction of the induced emfis such that it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field. The direction of the induced emfis opposite the direction of the emfof the battery. This results in a gradual increase in the current to its final equilibrium value. This effect is called self-inductance. Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself. The emfe is called a self-induced emf. 34
Self-Inductance, Equations An induced emfis always proportional to the time rate of change of the current. Because the emfis proportional to the flux, which is proportional to the field and the field is proportional to the current. E _ =!!1 (1) W ; + E _ is a constant of proportionality called the inductance of the coil. T It depends on the geometry of the coil and other physical characteristics. Inductance of a Coil For a closely spaced coil of N turns carrying current, the induced emfis E _ = S!Φ!1 Connecting to the inductance expression E _ =!!1!!1 = S!Φ!1 = SΦ = E _!/!1 The inductance is a measure of the opposition to a change in current. The SI unit of inductance is the henry (H) 1` = 1W / 35
Inductance of a Solenoid Assume a uniformly wound solenoid having S turns and length 3. Assume 3 is much greater than the radius of the solenoid. The flux through each turn of area is Φ = = : ; = : ;S 3 The inductance is = S 3 =: ; =: ; S/3 = SΦ = : ;S 3 = : ; 3 =: ; W This shows that L depends on the geometry of the object. RL Circuit A circuit element that has a large self-inductance is called an inductor. The circuit symbol is We assume the self-inductance of the rest of the circuit is negligible compared to the inductor. However, even without a coil, a circuit will have some self-inductance. The inductance results in a back emf. Therefore, the inductor in a circuit opposes changes in current in that circuit. The inductor attempts to keep the current the same way it was before the change occurred. The inductor can cause the circuit to be sluggish as it reacts to changes in the voltage. 36
RL Circuit, Analysis An RLcircuit contains an inductor and a resistor. Assume S 2 is connected to a When switch S 1 is closed (at time t= 0), the current begins to increase. At the same time, a back emfis induced in the inductor that opposes the original increasing current. RL Circuit Applying Kirchhoff s loop rule + + E T!!1 = 0 Solving for the differential equation, we find 1 = E T (1 )=ab _) - = T The current does not instantly increase to its final equilibrium value. If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value. 37
RL Circuit Time Constant The expression for the current can also be expressed in terms of the time constant, τ, of the circuit. 1 = E T (1 )=ab _) - = T Physically, τis the time required for the current to reach 63.2% of its maximum value. The equilibrium value of the current is E/T and is reached as t approaches infinity. RL Circuit Without A Battery Now set S 2 to position b The circuit now contains just the right hand loop. The battery has been eliminated. We can set up the differential equation and solve for (1):!!1 = T! = T!1 1 = ) =b d = E/T (1) + + E _ T 38
Energy in a Magnetic Field The previous circuit shows that the established B-field in a inductor can supply emfto sustain a current in a circuit for some time. Therefore, there must be energy stored in the B-field of the inductor. Let U be the energy stored in the inductor: f e = "!e = " T!1 ; f e = " T ;!1 = f T" ) =b d!1 ; f e = T - 2 ()=b d) g ; = 1 2 1 = ) =b d (1) T - = T + + E _ Energy Density of a Magnetic Field For a solenoid, =: ; W and = : ; =/: ; e = 1 2 = 1 2 : ; W : ; = 2: ; W Here, V is the volume of the solenoid, we can find the magnetic field energy density, h as h = e W = 2: ; This applies to any region in which a magnetic field exists (not just the solenoid). 39
General Features of Waves In wave motion and energy is transferred over a distance. Matter (medium of the wave) is not transferred over a distance. Mechanical Wave Some source of disturbance A medium containing elements that can be disturbed Some physical mechanism through which elements of the medium can influence each other Pulse on a String The wave is generated by a flick on one end of the string. The string is under tension. A single bump is formed and travels along the string. The bump is called a pulse. The diagram shows snapshots of the creation and propagation of the traveling pulse. 40
Pulse on a String The hand is the source of the disturbance. The string is the medium through which the pulse travels. Individual elements of the string are disturbed from their equilibrium position. The elements are connected together so they influence each other. The pulse has a definite height. The pulse has a definite speed of propagation along the medium. The shape of the pulse changes very little as it travels along the string. Transverse Wave A continuous wave is a periodic disturbance traveling through a medium. A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave. To create the wave, you would move the end of the string up and down repeatedly. The particle motion is shown by the blue arrow. The direction of propagation is shown by the red arrow. 41
Longitudinal Wave A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave. Sound waves are another example of longitudinal waves. The displacement of the coils is parallel to the propagation. Traveling Pulse For a pulse traveling to the right i (C,1) = ' (C 1) For a pulse traveling to the left i (C,1) = ' (C + 1) The function yis also called the wave function: y (x, t). The wave function represents the ycoordinate of any element located at position xat any time t. The y coordinate is the transverse position. l = C 1 l = C+1 If tis fixed then the wave function is called the waveform. i = '(l) i = '(l) It defines a curve representing the geometric shape of the pulse at that time. 42
Sinusoidal Waves The wave represented by the curve shown is a sinusoidal wave. It is the same curve as sin θplotted against θ. This is the simplest example of a periodic continuous wave. It can be used to build more complex waves. The wave moves toward the right. In the diagram, the brown wave represents the initial position. As the wave moves toward the right, it will eventually be at the position of the blue curve. Each element moves up and down in simple harmonic motion. This is the motion of the elements of the medium. It is important to distinguish between the motion of the wave and the motion of the elements of the medium. Amplitude and Wavelength The crestof the wave is the location of the maximum displacement of the element from its normal position. This distance is called the amplitude,. The wavelength, m, is the distance from one crest to the next. More generally, the wavelength is the minimum distance between any two identical points on adjacent waves. 43
Period and Frequency The period,?, is the time interval required for two identical points of adjacent waves to pass by a point. The period of the wave is the same as the period of the simple harmonic oscillation of one element of the medium. The frequency, ', is the number of crests (or any point on the wave) that pass a given point in a unit time interval. The time interval is most commonly the second. The frequency of the wave is the same as the frequency of the simple harmonic motion of one element of the medium. ' = 1? =A ( `n Units? = 1 ' Wave Model Consider a continuous sinusoidal traveling wave that has a single frequency, and is infinitely long. Wave function of a sinusoidal wave moving to the right (+x direction): i C,1 = '(C 1) i C,1 = 2< m C 1 i C,1 = 2< C m 1? = C 1 = m? = 'm For wave moving to the left (-x direction): i C,1 = 2< C m + 1? 44
Wave Functions With the wave-function i C,1 = 2< C m ± 1? We can also define the angular wave number (or just wave number), D, and the angular frequency, o: D = 2< m, o =2<? = 2<' The wave function can be expressed as: i C,1 = DC±o1 The speed of the wave can be written in several forms: = m' = m? =o ' = 1 D? If y 0 at x=0, t= 0, the wave function will have a phase constant, l. i i C,1 = DC±o1+l C Example A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The vertical position of an element of the medium at t 5 0 and x 5 0 is also 15.0 cm Find?,D,o,,l The wave function is i C,1 = (DC o1+l) = 0.15? = 1 ' = 1 8.00 = 0.125 D =2< m = 2< =15.7 +!/ 0.4 o = 2<' = 2< 8=50.3 +!/ = m' = 0.4 8 =3.2 / i 0,0 = l = l = < 2 i C,1 = (DC o1+</2) =F( (DC o1) 45
i C,1 = 2< C m 1? i C,1 = DC±o1+l = m' = m? =o D 46
Speed of a Transverse Pulse/Wave on a String The speed of the pulse/wave depends on the physical characteristics of the string and the tension to which the string is subjected. =? :? 1) ( 12) 1v : = 3) +!) 1i (' 12) 1v This assumes that the tension is not affected by the pulse. This does not assume any particular shape for the pulse. This speed is the traveling speed of the mechanical disturbance (pulse or wave) through the medium (the string). It is not the transverse speed of the motion of an element of the string ( u in the previous page). Reflection of a Wave, Fixed End When the pulse reaches the support, the pulse moves back along the string in the opposite direction. This is the reflectionof the pulse. The pulse is inverted. Due to Newton s third law When the pulse reaches the fixed end of the string, the string produces an upward force on the support. The support must exert an equalmagnitude and oppositely directed reaction force on the string. 47
Reflection of a Wave, Free End With a free end, the string is free to move vertically. The pulse is reflected. The pulse is not inverted. The reflected pulse has the same amplitude as the initial pulse. Wave at Boundary: Transmission and Reflection When the boundary is intermediate between the last two extremes. Part of the energy in the incident pulse is reflected and part undergoes transmission. Some energy passes through the boundary. 48
Wave at Boundary: Transmission and Reflection Assume a light string is attached to a heavier string. The pulse travels through the light string and reaches the boundary. The part of the pulse that is reflected is inverted. The reflected pulse has a smaller amplitude. Wave at Boundary: Transmission and Reflection Assume a heavier string is attached to a light string. Part of the pulse is reflected and part is transmitted. The reflected part is not inverted. 49
Transmission and Reflection of a Wave Summary The total energy is conserved in the process of reflection and transmission of waves at boundaries. At the boundary, an incident wave s energy equals the sum of the energies of the reflected and transmitted waves The speed of the wave on a string is determined by tension and mass density. Assume constant tension, the speed of the wave in the lighter string is higher An incident wave from a light string (less dense) to a heavy string (higher density) The reflected wave inverts An incident wave from a heavy string (higher density) to a light string (less dense) The reflected wave doesn t inverts End of Midterm 2 Content 50