Lecture Sound Waves EM Waves. Physics Help Q&A: tutor.leiacademy.org. The Doppler Effect 11/11/2014

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1 Lecture 1102 Sound Waves EM Waves Physics Help Q&A: tutor.leiacademy.org The Doppler Effect The Doppler effect (or Doppler shift) is the change in frequency (or wavelength) of a wave for an observer moving relative to its source. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. Compared to the emitted frequency, the received frequency is higher during the approach, identical at the instant of passing by, and lower during the recession. When the source of the waves is moving toward the observer, each wave takes slightly less time to reach the observer than the previous wave, causing an increase in the frequency. If the source of waves is moving away from the observer, each wave is emitted from a position farther from the observer than the previous wave, so the arrival time between successive waves is increased, reducing the frequency. 1

2 Doppler Effect, Observer Moving The observer moves with a speed of v o. Assume a point source that remains stationary relative to the air. It is convenient to represent the waves as wave fronts. These surfaces are called wave fronts. The distance between adjacent wave fronts is the wavelength. Doppler Effect, Observer Moving The speed of the sound is, the frequency is, and the wavelength is. = / If the observer is moving towards the source, the time for the adjacent wave crest to pass the moving observer is The frequency heard by the observer,, is. = 1 = + = /( + ) = + = + 1 If the observer is moving away from the source, the time for the adjacent wave crest to pass the moving observer is = /( ) = + = 1 = = = 1 = 2

3 Doppler Effect, Source Moving Consider the source being in motion while the observer is at rest. As the source moves toward the observer, the wavelength appears shorter. = = / As the source moves away, the wavelength appears longer. = + = + / Doppler Effect, Source Moving The speed of wave is constant. Therefore, = = When the source is moving toward the observer: = / = / = = = = = When the source is moving away from the observer: = + = + = + = = + 3

4 Doppler Effect, General Combining the motions of the observer and the source = + The signs of the velocity depend on the relative direction of the velocity. A positive value is used for motion of the observer or the source toward the other. A negative sign is used for motion of one away from the other. Doppler Effect, final Convenient rule for signs. The word toward is associated with an increase in the observed frequency. The words away from are associated with a decrease in the observed frequency. The Doppler effect is common to all waves. The Doppler effect does not depend on distance. 4

5 Doppler Effect, Submarine Example Two submarines are traveling directly toward each other. Sub A (source) travels at 8.00 m/s emitting at a frequency of 1400 Hz. The speed of sound in the water is 1533 m/s. Sub B (observer) travels at 9.00 m/s. 1. What is the apparent frequency heard by the observer as the subs approach each other? Then as they recede from each other? 2. The subs barely miss each other and pass. What frequency is detected by an observer riding on sub B as the subs recede from each other? Doppler Effect, Submarine Example Approaching each other: ( ) ( ) v + v 1533 m s m s o ƒ' = ƒ = (1400 Hz) v v s 1533 m s 8.00 m s + = 1416Hz Receding from each other: ( ) ( ) v + v 1533 m s m s o ƒ' = ƒ = (1400 Hz) v v s 1533 m s 8.00 m s = 1385Hz 5

6 Shock Waves and Mach Number The speed of the source can exceed the speed of the wave (not with EM waves such as light). The envelope of these wave fronts is a cone whose apex half-angle is called the Mach angle. The ratio / is referred to as the Mach number. The relationship between the Mach angle and the Mach number is = = Shock Wave The conical wave front produced when > is known as a shock wave. The shock wave carries a great deal of energy concentrated on the surface of the cone, with correspondingly great pressure variations. Such shock waves are unpleasant to hear and can cause damage to buildings when aircraft fly supersonically at low altitudes. Jet airplanes traveling at supersonic speeds produce shock waves, which are responsible for the loud sonic boom one hears. 6

7 Electromagnetic Waves Mechanical waves require the presence of a medium (vacuum). Electromagnetic waves can propagate through empty space. Maxwell s equations form the theoretical basis of all electromagnetic waves that propagate through space at the speed of light. Hertz confirmed Maxwell s prediction when he generated and detected electromagnetic waves in Electromagnetic waves are generated by oscillating electric charges. The waves radiated from the oscillating charges can be detected at great distances. Electromagnetic waves carry energy and momentum. Electromagnetic waves cover many frequencies. Ampère s Law and Induction Ampère s Law considers the magnetic fields created by currents: = Maxwell modified the equation to include time-varying electric fields. The idea is that changing E-field (flux) will also induce/create B-field just like changing B- field (flux) induces E-field. = Φ = Φ Displacement Current = + Φ 7

8 Ampère -Maxwell Law B-fields can be created by both a current and a changing electric flux: Φ = + = = Φ = Φ = = Φ = 1 = 1 Displacement Current = Φ Maxwell s Equations Maxwell s Equations provide the basis of all electrical and magnetic phenomena: = Gauss s Law = 0 Gauss s Law = Φ = + Φ Faraday s Law Ampère -Maxwell Law = + Lorentz Force Law 8

9 Maxwell s Equation 1 Gauss Law The total electric flux through any closed surface equals the net charge inside that surface divided by = This relates an electric field to the charge distribution that creates it. Maxwell s Equation 2 Gauss Law in Magnetism The net magnetic flux through a closed surface is zero. = 0 The number of magnetic field lines that enter a closed volume must equal the number that leave that volume. That is, we haven t found any magnetic monopoles. 9

10 Maxwell s Equation 3 Faraday s Law of Induction Describes the creation of an electric field by a time-varying magnetic field. The emf, which is the line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path. = Φ One example is that we can place a conducting loop in a time-varying magnetic field, and a current will be induced in the loop. Maxwell s Equation 4 Ampère-Maxwell Law Describes the creation of a magnetic field by a changing electric field and by electric current. The line integral of the magnetic field around any closed path is the sum of times the net current through that path and times the rate of change of electric flux through any surface bounded by that path. = + Φ 10

11 Lorentz Force Law Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be found. = + Maxwell s equations with the Lorentz Force Law completely describe all classical electromagnetic interactions. Maxwell s Equations and Light In empty space, q = 0 and I = 0 = = 0 = 0 = Φ = Φ The last two equations can be solved to show that the speed at which electromagnetic waves travel is the speed of light. This result led Maxwell to predict that light waves were a form of electromagnetic radiation. Hertz performed experiments that verified Maxwell s prediction. 11

12 Hertz s Experiment (Heinrich Rudolf Hertz) An induction coil is connected to a transmitter. The transmitter consists of two spherical electrodes separated by a narrow gap. The coil provides short voltage surges to the electrodes. As the air in the gap is ionized, it becomes a better conductor. The discharge between the electrodes exhibits an oscillatory behavior at a very high frequency, equivalent to an LC circuit. A receiver loop was placed several meters away. Sparks were induced across the gap of the receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitter. In this way, Hertz demonstrated that the oscillating current induced in the receiver was produced by electromagnetic waves radiated by the transmitter. Hertz s Experiment, cont. In a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave properties. Interference, diffraction, reflection, refraction and polarization He also measured the speed of the radiation. It was close to the known value of the speed of light. 12

13 Plane Electromagnetic Waves Assume an electromagnetic wave travels in the x direction. The configurations of and are shown in the diagram. The x-direction is the direction of propagation. The electric field is assumed to be in the y direction and the magnetic field in the z direction. Waves in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves. We also assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only. 13

14 Rays A ray is a line along which the wave travels. All the rays for the type of linearly polarized waves that have been discussed are parallel. The collection of waves is called a plane wave. A surface connecting points of equal phase on all waves, called the wave front, is a geometric plane. A spherical wave creates a wave front as a spherical surface connecting points of radiation sends waves out radially in all directions. 14

15 Properties of EM Waves The solutions of Maxwell s third and fourth equations are wave-like, with both E and B satisfying a wave equation. = Φ = Φ Properties of EM Waves Faraday s Law Equation: = Φ = = 15

16 Properties of EM Waves Ampère -Maxwell Law Equation: = Φ = Recall = = = = Properties of em Waves Therese satisfy a wave equation: = = = cos( ) = cos( ) = = 1 Electromagnetic waves travel at the speed of light: = = 1 Substituting the values for and gives c = x 10 8 m/s 16

17 Properties of em Waves The components of the electric and magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation. This can be summarized by saying that electromagnetic waves are transverse waves. The figure represents a sinusoidal em wave moving in the x direction with a speed c. Properties of em Waves The magnitudes of the electric and magnetic fields in empty space are related: = = cos( ) = cos( ) = = 1 = = = = = = 17