4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES Student Notes
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1 4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES Student Notes I. DIFFERENT TYPES OF WAVES A. TRANSVERSE AND LONGITUDINAL WAVES B. WAVE PULSES AND TRAVELLING WAVES C. SOUND AND WATER WAVES II. DEFINING TERMS III. GRAPHS OF WAVES A. DISPLACEMENT-TIME GRAPHS B. DISPLACEMENT-POSITION GRAPHS IV. WAVEFRONTS AND RAYS V. ELECTROMAGNETIC (EM) WAVES A. THE WAVE EQUATION FOR EM WAVES B. THE ELECTROMAGNETIC SPECTRUM VI. WHEN WAVES HIT BOUNDARIES BETWEEEN TWO MEDIA A. REFLECTION AND TRANSMISSION B. SNELL S LAW VII. WHEN WAVES INTERACT WITH EACH OTHER A. THE SUPERPOSITION PRINCIPLE B. CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE C. INTERFERENCE: PATH AND PHASE DIFFERENCE VIII. DIFFRACTION A. WHEN WAVES GO AROUND OBSTACLES B. WHEN WAVES GO THROUGH APERTURES I. DIFFERENT TYPES OF WAVES A. TRANSVERSE AND LONGITUDINAL WAVES DEFINE: Wave = a disturbance travelling through something that carries energy and momentum from one place to another There are 3 main characteristics of waves: 1. Carry energy from one place to another through solids, liquids, gases (mechanical waves) and vacuums (electromagnetic waves) 2. No large-scale net motion of the medium through which waves travel 3. Involve oscillations (vibrations) which exhibit simple harmonic motion (periodic) 2 different types of waves: 1
2 TRANSVERSE When the disturbance is perpendicular to the direction of wave travel EXAMPLES? NOTE: Individual coils on a slinky do NOT move along the slinky s length!!! DEMO - Slinkies LONGITUDINAL When the disturbance occurs in the same direction as that of the wave travel EXAMPLES? NOTE: Individual coils on a slinky do NOT move along the slinky s length!!! DEMO - Slinkies B. WAVE PULSES AND TRAVELLING WAVES wave pulse = one isolated vibration DEMO: PhET Waves on a String DEMO: Rope against wall Show how reflected wave is INVERTED. NOTE: Shape of wave depends on initial displacement of slinky. DEMO: String tied on a tuning fork vs. Source: Physics for the IB Diploma, 5 th Ed, Tsokos 2
3 Wave pulse travels at a speed v down the slinky/rope v determined by tension T and density (mass per unit length) Turns out v is determined by properties of medium, NOT how wave was created! continuous wave = series of repeating vibrations creates travelling waves DEMO If whatever is forcing the rope/string executes SHM, then pulses are sent continuously String will look like a moving sine wave (also called harmonic wave) EXAMPLE 1 The diagram shows a transverse wave on a string. The wave is moving from right to left. In the position shown, point X has zero displacement and point Y is at a position of maximum displacement. Describe the subsequent direction of motion of point X and of point Y (upward, downward, right, left ). left upwards right downwards X direction of wave Y EXAMPLE 2 A wave travels along a stretched string. The diagram shows the variation with distance along the string of the displacement of the string at a particular instant in time. A small marker is attached to the string at the point labelled M. The undisturbed position of the string is shown as a dotted line. Directions of wave travel M a) On the diagram: (i) draw an arrow to indicate the direction in which the marker is moving. (ii) indicate, with the letter A, the amplitude of the wave. (iii) indicate, with the letter λ, the wavelength of the wave. (iv) draw the displacement of the string a time T/4 later, where T is the period of oscillation of the wave. Indicate, with the letter N, the new position of the marker. b) The wavelength of the wave is 5.0 cm and its speed is 10 cm s -1. Determine (i) the frequency of the wave. [2.0 Hz] (ii) how far the wave has moved in T/4 s. [1.3 cm] 3
4 Source: Physics, 8 th Ed, Cutnell & Johnson C. SOUND AND WATER WAVES Sound = a series of successive high/low air pressure zones (longitudinal wave) also called compressions (high pressure) and rarefactions (low pressure). Speed of waves depends on the properties of the medium (material) through which they are travelling. Lighter spring = faster wave speed Hot air carries sound more quickly than cold! DEMOS: Musical instruments. Tuning forks though air and through table/solids Water waves : neither transverse nor longitudinal. Water particles near the surface move in a circular path, giving the water particles both transverse and longitudinal components. But we treat water waves as transverse in nature. DEMO: Water in a tank with a light. Source: Physics, 8 th Ed, Cutnell & Johnson II. DEFINING TERMS DEMO: PhET Waves on a String Source: Physics for the IB Diploma, 5 th Ed, Tsokos CREST CREST TROUGH TROUGH 4
5 Source: Physics, 8 th Ed, Cutnell & Johnson There are several key features of waves you need to know: 1. Amplitude (A) : The maximum displacement of the particle in the wave from the equilibrium/mean position (which is not always zero). 2. Displacement (s) : The change in distance (+ or -) that takes place on a particle from the wave passing through it. 3. Frequency (f) : The number of oscillations that take place in one second (the number of complete waves that pass through a fixed point in one second); f = 1/T 4. Period (T) : The time taken for one complete oscillation (time taken for one complete wave to pass a fixed point); T = 1/f 5. Wavelength (λ) : The horizontal length of one cycle of the wave (distance between successive crests/troughs). 6. Wave speed (v) : Speed at which the wave fronts pass a fixed point. 7. Intensity (I) : The power per unit area that is received through the wave ( I = Power/Area). Intensity is proportional to A 2. It should be clear that: EXAMPLE 3 The crests on a long surface water wave are 20.0 m apart, and in 1 minute 10 crests pass by. What is the speed of these waves? [3.4 ms -1 ] EXAMPLE 4 In the diagram below, the distance between points A and B on water waves is 5.0 meters. a) What is the wavelength of these water waves? [2.0 ms -1 ] b) If 5 waves pass point B every 15 seconds, what are the frequencies of these waves? [0.3 Hz] c) What is the period of one wave? [3.0 s] d) What is the speed of one wave? [0.6 ms -1 ] 5
6 Source: Physics for the IB Diploma, 5 th Ed, Tsokos EXAMPLE 5 A radio wave, a form of electromagnetic wave, has a frequency of 99.5 MHz (99.5 x 10 6 Hz). What is its wavelength? [3.0 ms] EXAMPLE 6 Water waves in a lake travel 4.4 m in 1.8 s. The period of oscillation is 1.2 s. What is the speed of the water waves? What is their wavelength? [2.4 ms -1, 2.9 m] III. GRAPHS OF WAVES A. DISPLACEMENT-TIME GRAPHS x-axis = UNITS OF TIME peak to peak or trough to trough = T From this graph: T =? f =? A =? v =? B. DISPLACEMENT-POSITION GRAPHS x-axis = UNITS OF DISTANCE Like a photo taken from the side, a snapshot at a particular time peak to peak or trough to trough = λ From this graph: T =? f =? A =? v =? DANGER: displacement-time graphs and displacement-position graphs look the same at a glance, but are VERY different! 6
7 Note that the equilibrium position is not always at y = 0. Why not? DEMO: Vernier probe d-t graph for pendulum, mass on a spring DO HW #3 Source: Physics for the IB Diploma, 5 th Ed, Tsokos IV. WAVEFRONTS AND RAYS For a 2D wave: Wavefronts = vertical planes going through waves Rays = lines at right angles to wavefronts in the direction of wave travel Source: Physics for the IB Diploma, 5 th Ed, Tsokos For waves made by a point source (stone dropping in water): Circles are wavefronts, radial vectors are rays DEMO in ripple tank Source: Physics, 8 th Ed, Cutnell & Johnson 7
8 V. ELECTROMAGNETIC (EM) WAVES A. THE WAVE EQUATION FOR EM WAVES Electromagnetic waves (EM waves) behave like any other waves. Radio waves and light are two examples of EM waves. Visible light is simply one small part of a larger electromagnetic spectrum of EM waves. The frequency of the waves in this small part determines the color we see. Source: Physics, 8 th Ed, Cutnell & Johnson EM waves are created by an oscillating electric charge which produces varying electric and magnetic field at right angles to each other. Propagate as a transverse wave and are able to move through a vacuum, since no particles are involved The speed of EM waves is x 10 8 m/s = c. B. THE ELECTROMAGNETIC SPECTRUM Source: Physics, 8 th Ed, Cutnell & Johnson For visible light, calculate also the wavelength using wave equation. 8
9 Source: Physics, White light = light having all the colors of the spectrum Sun emits in all frequencies. Spectrum curve for sun: DEMO: White light through a prism EXAMPLE 7 The speed of radio waves in vacuum (equal to the speed of light) is 3.00 x 10 8 m/s. Find the wavelength for a) An AM radio station with frequency 1070 khz [281 m] b) An FM radio station with frequency 91.7 Mhz [3.27 m] VI. WHEN WAVES HIT BOUNDARIES BETWEEEN TWO MEDIA A. REFLECTION AND TRANSMISSION A wave hits the boundary between two different media. Two things can happen: 1. wave is reflected (bounces off) surface must be smooth: λ light > size of surface bumps 2. wave is transmitted (goes through) Often both occur. The amount of each depends on the nature of the media. Wave can also be absorbed (but we won t talk about that now). Law of reflection: angle of incidence = angle of reflection DEMO: Laser and mirror Source: Physics for the IB Diploma, 5 th Ed, Tsokos 9
10 Source: Physics, 8 th Ed, Cutnell & Johnson B. SNELL S LAW c = 3.0 x 10 8 ms -1 in a vacuum. In all other media, v < c. v depends on the optical density of the media through which light is passing. DEMO Because it slows down, it also changes direction when transmitted through another medium. DEFINE: refraction = the change in direction of a wave when travelling from one media to another due to a change in speed of the wave. ALL WAVES REFRACT. f does not change, but v does. Therefore, λ changes. Water waves change direction from one depth to another DEMO If v 1 > v 2, wave bent towards the normal. If v 1 < v 2, wave bent away from the normal. THEREFORE, FOR LIGHT: The wave is always bent toward the normal if going from less dense to more dense media. Usually some of the wave is reflected and some is refracted. Snell s Law: Source: Physics for the IB Diploma, 5 th Ed, Tsokos For light: every optically transparent material has an index of refraction given by: where c = speed of light in vaccum, c m = speed of light in the media Snell s Law again: n1 sinθ1 = n2 sinθ2 derive? In your data booklet: NOTE: n must always be greater than 1. n depends on λ, so rays with same angle of incidence but different λs will be bent differently. Called dispersion. DEMO 10
11 Source: Physics for the IB Diploma, 5 th Ed, Tsokos Question: Why do ocean waves (almost always) break parallel to the shore? For these examples, draw diagrams to help visualize the problem. Use a straightedge and protractor. EXAMPLE 8 A ray of light is incident in water at an angle of i) 30.0 degrees and ii) 70.0 degrees on a water-glass plane surface. Calculate the angle of refraction in the glass in each case. [25.9, 55.3 ] EXAMPLE 9 Given the information in example 8, sketch the path of the ray in the diagram below showing all relevant angles, and calculate the angles of the ray within the glass and the water. Use a straightedge and make your diagram as neat as possible. 30 glass water 11
12 VII. WHEN WAVES INTERACT WITH EACH OTHER A. THE SUPERPOSITION PRINCIPLE 2 waves travelling along a slinky towards one another. Source: Physics, 8 th Ed, Cutnell & Johnson They go through one another. If there are 2 up pulses, they make a big wave, but if one is up and one is down, they cancel out for an instant. Waves interfere with one another Principle of Superposition: When 2 or more waves meet at the same time in the same place, the resulting disturbance is the sum of the disturbances from the individual waves. All waves interfere according to this principle (water, sound, light, etc). DEMO: Rope, slinky Add waves 1 and 2: WKSHT: The Linear Superposition Principle Source: Physics for the ib Diploma, Oxford Wave 1 = Wave 2 + Wave 3 DO HW #9,10 12
13 Source: Mathematics for the ib Diploma, Oxford B. CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE if resulting wave is bigger, waves reinforce each other : interact constructively if resulting wave is smaller, waves cancel part of each other : waves interact destructively STRICTLY SPEAKING: Constructive interference = 2 waves are in phase (the phase difference is 0) and the resultant wave has a large amplitude. Destructive interference = 2 waves are exactly out of phase (the phase difference is ½ of a cycle, or π) and the resultant wave has zero amplitude. Source: Physics for the ib Diploma, Oxford REMEMBER?? Discuss PHASE SHIFTS 13
14 Source: Physics for the ib Diploma, Oxford C. INTERFERENCE: PATH AND PHASE DIFFERENCE Consider: a string between two points S 1 and S 2, and point P between them: Send identical waves from S 1, S 2 at same time. Will they hit P at the same time? NO! and so difference in arrival times is What if: Wave from S 1 arrives at P after 2 s. Wave from S 2 arrives at P after 4 s. When waves arrive at P, crests/troughs match up. Destructive interference. In this case: T = 4 s, and difference in arrival times is 2 s. Would also happen if difference in arrival times was 2 s, 6 s, 10 s, etc These are half-integer multiples of T. THEREFORE: If difference in arrival times is a half-integer multiple of T = destructive interference. Since Tv = λ, condition for destructive interference is when path difference is: 14
15 Source: Physics for the ib Diploma, Oxford NOW: What if waves arrive as shown: Resultant wave at P? Between 4 and 6 s, only wave from S 1 reaches P. But at 6 s onwards, crests/crests match up between waves from S 1 and S 2. interference. Constructive In this case: T = 4 s, and difference in arrival times is 4 s. Would also happen if difference in arrival times was 4 s, 8 s, 12 s, etc These are integer multiples of T. THEREFORE: If difference in arrival times is an integer multiple of T = constructive interference. Since Tv = λ, condition for destructive interference is when path difference is: 15
16 VIII. DIFFRACTION A. WHEN WAVES GO THROUGH OPENINGS DEFINE: diffraction = spreading out of waves when they encounter an obstacle or go through an opening ALL WAVES DIFFRACT For an opening of width W: When W < λ : more diffraction When W > λ : little or no diffraction DEMO: Ripple tank Can you hear sound around open doors? DIFFRACTION! Can you see around open doors? WHY NOT? Source: Physics, 8 th Ed, Cutnell & Johnson Source: Source: B. WHEN WAVES GO AROUND OBSTACLES Around corners/obstacles, the same thing happens. Size of obstacle must be about same size as λ for diffraction to occur Source: Physics for the ib Diploma, Oxford 16
17 Diffraction affects the lives of people living in mountainous areas. For example, radio waves (longer wavelength) can diffract around a hill to be received by a radio in a house, but TV waves (shorter wavelength) do not bend as much. Therefore, the house will have poor TV reception. The same principle applies for devices such as mobile phones, whose EM communications depend upon transmission/reception towers. Source: Physics for the IB Diploma Study Guide, Kirk DEMO: Show overheads with moire patterns DO HW #11,15 Source: nalakagunawardene.com Lines are wavefronts (crests) in this diagram. Constructive and destructive interference patterns! If these were sound waves, where would it be louder/softer? Source: Physics for the ib Diploma, Oxford 17
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