Waves. 1. Types of waves. Mechanical Waves

Transcription

1 Waves 1. Types of waves 1. Mechanical Waves 2. Waves, a mathematical formulation 1. A basic wave 3. Speed of a traveling wave 1. Wave speed on a real string. 4. Energy 1. The wave equation 5. Sound waves 1. Speed of Sound 2. Bulk Modulus 3. Graphical Representation of sound waves 4. Wave fronts 5. Intensity 6. Doppler 1. General Doppler Shift 1. Types of waves Fig. 1 The Great Wave off Kanagawa, Hokusai, 1829 Fig. 2 An EM wave Fig. 3 The Quantum Corral Wave phenomena are ubiquitous in the natural world. Nearly every discipline in science must be able to deal with waves or wave-like behavior. Consider a geologist studying earthquakes. The motions of the earth following tectonic activity are waves. Understanding how they move through the earth's surface is paramount to dealing with these disasters. If you're an electrical engineer, you'll need to understand electromagnetic waves thoroughly. Even economists can analyze certain properties of the global economy in terms of wave physics. Perhaps the most important feature of all waves, is that they do not move matter in the direction of travel, only energy is transmitted. We'll see how this makes sense as we consider the various types of waves. Mechanical Waves These waves require a medium. A disturbance in the medium propagates through the medium and this is called the wave. updated on J. Hedberg 2018 Page 1

2 Water waves Sound waves Stadium waves (excited people) Electromagnetic Waves These are oscillations of electromagnetic fields. These waves don't require a medium. Light Radio waves wi-fi Matter Waves Very small particles have wave-like properties. (We'll have to save that for later.) Mechanical Waves These are the most familiar. They require a medium. A mechanical wave involves the motion of matter. However, as mentioned above, the material object that is experiencing the wave motion doesn't travel in the direction of the wave motion. Some examples of mechanical waves are sounds waves, ocean waves, and waves on a string. All of these require a medium. The medium must be an elastic material, which means that if part of it is displaced, it will experience a restoring force that seeks to return the displaced region back to equilibrium. A wave pulse Fig. 4 The left end of the rope is being displaced in the vertical direction. This displacement is propagated throughout the rest of the rope. The wave form, a pulse in this case is seen to travel down the length of the rope. In this example, the medium is the rope. We apply a disturbance to the medium by displacing part of the rope. Since the rope is somewhat elastic, the tension in the rope seeks to restore the displaced section back to equilibrium. A sinusoidal wave Fig. 5 updated on J. Hedberg 2018 Page 2

3 The left end of the rope is being displaced in the vertical direction. This displacement is propagated throughout the rest of the rope. If the displacement on the left end is continuously repeated, we observe the familiar sinusoidal shape. Now, instead of a single displacement, we keep oscillating one side of the rope in a sinusoidal fashion. This disturbance is propagated down the length of the rope. We can see also that the original displacement is perpendicular to the direction of travel of the wave. This is an important characteristic. Fig. 6 Here's another familiar example of a mechanical wave. A spring (aka Slinky). A compressed region propagates along the length of the spring. This is another example of a mechanical wave. This time however, our original displacement is parallel to the direction of travel for the wave. Still, the spring is an elastic medium, and we have a displacement of that medium which is propagated. Two types of mechanical waves Fig. 7 Transverse and Longitudinal Waves direction in which the wave travels. There are 2 general classes of waves: Longitudinal: The Particles move parallel to the Transverse: Particles in the medium move perpendicular to the direction that wave is traveling Fig. 8 Imagine one small segment of the rope. Let's look at its motion while the wave is passing through that point. We'll need to be able to quantify the motion of this little blue region. 2. Waves, a mathematical formulation Imagine a small segment of the rope, or spring, or water or whatever medium. To fully describe its motion, we'll need variables which tell us where it is and when -- essentially space and time. If we imagine a rope, which we can say only has two spatial dimensions, then we just need three total variables: x, y, and t. We can therefore completely describe the motion of a segment using: updated on J. Hedberg 2018 Page 3

4 y = h(x, t), where h is a function of space and time. y is simply the transverse displacement of a small element of the string. y x So, let's use a sine function to describe the position of a small section of the medium. y(x, t) = Asin(kx ωt) Notice how this is an equation for the y position as a function of x and t. This equation describes the motion of the material that the medium consists of. In this case of the wave on a string, it's a little bit of string. It moves up and down (in the y direction) based on this equation. The two independent variables on the R.H.S are x and t. This tells us that the motion is dependent on where the segment is, and what time value t has. y snapshot Now, we have 3 variables in one function. This means that in order to plot this on a 2D graphs, we'll need to keep one variable constant x Depending on which one, we'll end up with two different representations of the same wave. 1.0 history 0.5 y t Fig. 9 This is probably our first example of an equation that has three updated on J. Hedberg 2018 Page 4

5 variables, 2 of which are considered independent, meaning they can take on any value. Amplitude Here we refer to the maximum displacement of the elements away from their equilibrium positions. This plot shows two waves with amplitudes which differ by a factor of two. Wavelength y x Let's take a sine wave at time t = 0. Our previous formula will then be: y(x, 0) = Asin(kx) Now, the displacement is the same at both ends of the wavelength y Asin(k x 1 ) = = Asink( x 1 + λ) (1) Asin(k x 1 + kλ) (2) x This can only be true if k = 2π λ We'll call k the wavenumber, which we can see is inverse to the wavelength. The wavenumber k can be thought of as a spatial frequency. If k is large, then λ will be small. If the wavelength is small, then there will be many repetitions of it in a given length. The temporal frequency is big if there many repetitions in a given time, so this make sense by analogy. Think of walking north-south in Manhattan as opposed to eastwest. If you go north-south, you will encounter about 20 blocks in a mile. Whereas if you go east-west, there are on average only 7 blocks to a mile. Thus the spatial frequency of blocks is greater in the north-south direction compared to the east-west. The distance between blocks would be something like the wavelength in this analogy. Period and Frequency updated on J. Hedberg 2018 Page 5

6 T 1 f = = T ω 2π y t(s) The wavelength and wavenumber gave us spatial information about the wave. What about temporal? We can perform a similar analysis of a history graph, and obtain the period of the wave. That is, how long does it take for an element of the string to make one full oscillation. We'll also have what's called the angular frequency. It's the ω from our displacement equation. This is related to the frequency, f, by the following:. Quick Question 1 What should the horizontal axis be labeled as? 1. Position 2. Time 3. Amplitude 4. Something else ω = 2π T 1 f = = T ω 2π updated on J. Hedberg 2018 Page 6

7 Quick Question 2 Vectors are attached to several particles on this wave. What vectors are shown? 1. Displacement 2. Velocity 3. Acceleration 4. Force The velocity of a segment. Here is our position equation for the material of the wave. If we wanted to find the velocity of a given point, we would just need to take the time derivative of this equation. Quick Question 3 y(x, t) = Asin(kx ωt) Here is a snapshot of a traveling wave on a rope. It's moving in the x direction. Which of the labeled segments has the largest negative y-velocity value? E. All have zero y-velocity values. Quick Question 4 Here is a snapshot of a traveling wave on a rope. It's moving in the x direction. Which of the labeled segments has the largest positive y-acceleration value? E. All have zero y-acceleration values. updated on J. Hedberg 2018 Page 7

8 3. Speed of a traveling wave If we think about the crest of the wave as it moves, its displacement in the y axis is constant. 1.5 v 1.0 y(x, t) = Asin( kx ωt ) or, kx ωt = constant constant y Let's take the time derivative of that equation: dx k ω = 0 dt x For that to be the case, the argument of the sinusoidal term must be constant as well. dx dt = v = ω k -orrewriting in more familiar terms: v = λf Recall that the frequency, f, is just the inverse of the period, T. Using this, we can rearrange this expression: v = λf = λ T distance time We should be somewhat comforted by the fact that this is our familiar distance over time expression. (wavelegth is a distance, period is a time. ) Note that this is just the velocity of the wave, not the velocity of the material in the wave. That velocity is given by the derivative of the position equation: This is a critical distinction to make. The wave speed, sometimes also called c, does not change in time. dc/dt = 0, whereas the transverse velocity of a given segment most certainly does change.. dy dt = ẏ(x, t) = v(x, t) = Aωcos(kx ωt) dẏ dt = ÿ 0 updated on J. Hedberg 2018 Page 8

9 Wave speed on a real string. What determines the speed of a traveling wave on a stretched string? We could show by looking at the tensions in the string that the velocity must be determined by the tension, τ, and the linear density, μ. v = τ μ Here is a derivation of the wave speed based on the 2nd law. Quick Question 5 Which of the following actions would make a pulse travel faster down a stretched string? 1. Use a heavier string of the same length, under the same tension. 2. Use a lighter string of the same length, under the same tension. 3. Move your hand up and down more quickly as you generate the pulse. 4. Move your hand up and down a larger distance as you generate the pulse. 5. Use a longer string of the same thickness, density, and tension. updated on J. Hedberg 2018 Page 9

10 Quick Question 6 A thick heavy (i.e. not massless)rope is hanging from a very tall ceiling. A person grabs the end of the rope and begins moving it back and forth with a constant amplitude and frequency. A transverse wave moves up the rope. Which of the following statements describing the speed of the wave is true? 1. The speed of the wave decreases as it moves upward. 2. The speed of the wave increases as it moves upward. 3. The speed of the wave is constant as it moves upward. 4. The speed of the wave does not depend on the mass of the rope. 5. The speed of the wave depends on its amplitude. Example Problem #1: A wave travels along a string at speed v 0. What will the speed be if the string is replaced by one made of the exact same material, but having twice the radius. (The tension is the same) Example Problem #2: A sinusoidal wave with an amplitude of 1.00 cm and a frequency of 100 Hz travels at 200 m/s in the positive x-direction. At t=0s, the point x = 1.00 m is on a crest of the wave. 1. Determine A, v, λ, k, f, ω, T, and ϕ for this wave. 2. Write the equation for the wave's displacement as it travels. 4. Energy An interesting aspect of waves is that they are a means of transferring energy, but not matter. Here is a frame from the earlier animation. The vectors show the velocity of each element of mass, dm. At the crest, 2, the velocity is zero. While, at the y = 0 point, 1, the velocity will be a maximum. Kinetic energy is given by the square of the updated on J. Hedberg 2018 Page 10

11 velocity, thus we can see that the K.E. of the dm element will oscillate between a minimum and maximum during the wave travel. Energy, cont'd We should also consider the potential energy of the element dm. At point 2, the string is not stretched at all, while at point 1, the string length is elongated as it passes through the origin. This change in length will change the elastic potential energy. Quantify energy The kinetic energy of a particle in motion is given by KE = section of a rope, dm, we write: 1 2 v 2 m. In the case of a little dk = 1 dm 2 y 2 dy ( y = which is the time derivative of y, or speed in the transverse direction) dt Going back to our original definition for the y displacement: y(x, t) = Asin(kx ωt), we can see that y is just: y = Aωcos(kx ωt) Therefore, our dk can be written: 1 dk = dm( Aω ) 2 cos 2 1 (kx ωt) = μdx( Aω ) 2 cos 2 (kx ωt) 2 2 Quantify energy If we continue, and take the derivative of dk with respect to time: dk 1 = μv A 2 ω 2 cos 2 (kx ωt) dt 2 updated on J. Hedberg 2018 Page 11

12 Now, the kinetic energy of one little element is clearly changing all the time, but we can consider the average change in kinetic energy: The last term: change: dk ( ) dt avg [ cos 2 (kx ωt)] avg 1 = μva 2 ω 2 [ cos 2 (kx ωt)] 2 1 is equal to, so in the end, for the average rate of kinetic 2 dk ( ) dt avg 1 = μva 2 ω 2 4 The wave also transmits elastic potential energy (since the rope is kinda springy). This should be equal to the average kinetic energy since they are conserved quantities. Thus, the total rate of energy transmission (aka Power) will be twice what we figured for the kinetic energy: avg The wave equation This: y(x, t) = Asin(kx ωt) was just a specific case of a wave. (A sinusoidal traveling wave). We'll need a more general wave equation which can be used to describe any travelling wave. **Derive** 1 P avg = μva 2 ω Sound waves 2 y 1 2 y = x 2 v 2 t 2 Quick Question 7 A bell is ringing inside of a sealed glass jar that is connected to a vacuum pump. Initially, the jar is filled with air at atmospheric pressure. What does one hear as the air is slowly removed from the jar by the pump? 1. The sound intensity gradually increases. 2. The sound intensity gradually decreases. 3. The sound intensity of the bell does not change. 4. The frequency of the sound gradually increases. 5. The frequency of the sound gradually decreases. updated on J. Hedberg 2018 Page 12

13 Speed of Sound We saw before that the speed of a wave (transverse) was equal to: In this case, τ was a type of 'elastic property' while μ would be classified as an 'inertial property'. i.e. These two quantities only made sense in reference to a string, but the speed of other types of waves can be determined by analogous considerations. Bulk Modulus v = τ μ elastic proptery = v inertial property Δp = B ΔV V B is the Bulk modulus of a material. It tells us how the volume of given material will change if pressure is applied to it. Steel for example, has a B = N/m 2, while water, which is a bit more compressible, has a B = N/m 2 We'll use this parameter to calculate how compression waves travel in a medium (i.e. sound waves) updated on J. Hedberg 2018 Page 13

14 Material v (m/s) Gases Hydrogen (0 C) 1286 Helium (0 C) 972 Air (20 C) 343 Air (0 C) 331 Liquids Sea water 1533 Water 1493 Solids Diamond Pyrex glass 5640 Iron 5130 Aluminum 5100 Copper 3560 Gold 3240 Rubber 1600 Example Problem #3: Here is a table that lists the velocity of sound in various materials. The speed of sound is dictated by the material properties of the medium. Just like it was for the string. v sound B = ρ A hammer taps on the end of a 4.00 m long metal bar at room temperature. A microphone at the other end of the bar picks up two pulses of sound, one that travels through the metal and one that travels through the air. The pulses are separated in time by 9.00 ms. What is the speed of sound in this metal? updated on J. Hedberg 2018 Page 14

15 Quick Question 8 You are observing a thunderstorm. In the distance, you see a flash of lightning. Five seconds later, you hear thunder. How far away was the lightning flash? 1. 1 mile (1.6 km) 2..5 mile (.8 km) 3. 2 miles (3.2 km) miles (.4 km) 5. 5 miles (8.0 km) Traveling Sound Waves To describe the longitudinal motion of an element of the medium, we can use a sinusoidal function: updated on J. Hedberg 2018 Page 15

16 s(x, t) = s m cos(kx ωt) All of our wave parameters are still present: f,λ,ω, k, T However, it's easier to work with pressure, p. The volume of our element will just be the length times its cross-sectional area: V = AΔx. While the change in volume of this element will be given by: ΔV = AΔs. We can substitute these back into our formula for the bulk modulus: Δs Δp = B Δx s = B x s x = k sin(kx ωt) s m or Δp = Bk Thus, we can see the pressure at a given location oscillates with time. Graphical Representation of sound waves s m sin(kx ωt) s longitudinal displacement x We'll see plots like this a lot. Here we have a speaker producing a sound wave. We'll plot the displacement of the elements of air as a function of position. Understanding this plot is paramount. It looks upon first glance that an particle is oscillating up and down as the wave propagates. This is not true. Sound is a longitudinal wave, so the displacements of the air molecules will be in the direction of the wave motion. updated on J. Hedberg 2018 Page 16

17 Quick Question 9 A particle of dust is floating in the air approximately one half meter in front of a speaker. The speaker is then turned on produces a constant pure tone of 267 Hz, as shown. The sound waves produced by the speaker travel horizontally. Which one of the following statements correctly describes the subsequent motion of the dust particle? dust particle y 267 Hz x 1. The particle of dust will oscillate in the ±x direction with a frequency of 267 Hz. 2. The particle of dust will oscillate in the ±y with a frequency of 267 Hz. 3. The particle of dust will be accelerated toward the right ( +x) and continue moving in that direction. 4. The particle of dust will move toward the right (+ x) at constant velocity. 5. The dust particle will remain motionless as it cannot be affected by sound waves. Radially Propagating Waves Here is our standard, circular wave pattern. updated on J. Hedberg 2018 Page 17

18 We can approximate the circular waves as parallel wave fronts if we are far enough away from the source. (far would mean d λ). Fig. 10 Wave fronts Fig. 11 A wave front plane waves with power = P We'll consider an intensity, I, to describe and quantify the loudness of the waves. v I = P A area = a Here, P, is the rate of energy transfer (power) and A is the area over which we are considering. Imagine a sound source: If it s in the center of the room, the waves will propagate outwards in a circle. These are two different ways of representing the pressure waves: a) on top, a model of the particles b) underneath: a color gradient where blue is high pressure and white is low pressure. updated on J. Hedberg 2018 Page 18

19 To find the power at some distance, r, from the source, we need to use the surface area of a sphere at that distance: Example Problem #4: P I = = A P 4πr 2 A helium-neon laser emits 1 mw of light power into a 1.0 mm diameter laser beam. What is the intensity of the laser beam? Quick Question 10 Nancy Reagan is a distance d in front of a speaker emitting sound waves. She then moves to a position that is a distance 2d in front of the speaker. By what percentage does the sound intensity decrease for Nancy between the two positions? 1. 10% 2. 25% 3. 50% 4. 75% 5. The sound intensity remains constant because it is not dependent on the distance. Intensity To quantify how loud a given sound is, we'll define a term β: the sound intensity level. It is calculated by comparing the intensity of the sound in question, I, to a base level, the threshold of hearing:. I 0 β = (10dB) log I I 0 The threshold of hearing I 0 defined as I 0 = W/ m 2 The units of β are given in decibels. If a sound has an intensity of W/ m 2, then we can see that it will have an intensity level of 0 db. I β = (10dB) log 0 10 = (10dB) log (1) = 0 db I 10 0 updated on J. Hedberg 2018 Page 19

20 A very loud sound, one that might damage your ears, can have an intensity of 10 W/m 2. How many decibels is that? Sound β (db) I W/m2 Threshold of Hearing A Whisper at 1m Conversation at 1m Vacuum Cleaner Home Stereo Threshold of PAIN Example Problem #5: We can see that there is wiiiide range of intensities heard during a normal day out and about. That is why it makes more sense to use a log scale when describing this phenomenon. If the sound intensity level at distance d of one trombone is β = 70 db, what is the sound intensity level of 99 identical trombones, all at distance d? Example Problem #6: 6. Doppler General Doppler Shift The Grateful Dead and their crew built a speaker system that was able to generate 26,400 Watts of audio power. It was called the Wall of Sound. How many decibels would this make at ¼ mile away from the stage? (Assume an isotropic sound distributioin) f v ± v = f D v ± v S 1. f = emitted frequency 2. f = detected frequency 3. v = speed of sound in air 4. v D = speed of detector 5. = speed of source v S When the motion of detector or source is toward the other, the sign on its speed must give an upward shift in frequency. When the motion of detector or source is away from the other, the sign on its speed must give a downward shift in frequency. updated on J. Hedberg 2018 Page 20

21 Quick Question 11 A child is swinging back and forth with a constant period and amplitude. Somewhere in front of the child, a stationary horn is emitting a constant tone of frequency f. Five points are labeled in the drawing to indicate positions along the arc as the child swings. At which position(s) will the child hear the lowest frequency for the sound from the horn? stationary horn at 2 when moving toward 1 2. at 2 when moving toward 3 3. at 3 when moving toward 2 4. at 3 when moving toward 4 5. at both 1 and 4 Example Problem #7: A 2kHz sine wave generator is swung around in a circle with a rope of length 1m at a speed of 100 rotations per minute. Find the highest and lowest frequencies heard by stationary listeners out side the circle but in the plane of rotation. updated on J. Hedberg 2018 Page 21