he largest string of a guitar has a length of 9.9 cm and a mass of 5.58 g. Once on the guitar, there is 65.5 cm between the two points of attachment of the string. What should be the tension of the string so that the frequency of the first harmonic is 8.4 Hz (which is the frequency that the largest string of a guitar should have)? mathforum.org/mathimages/index.php/special:humb Discover how to solve this problem in this chapter.
What is a Mechanical Wave? Suppose that initially, there is some matter at rest. his matter will be called the medium. Suddenly, something moves part of this medium. his motion is called a disturbance. hen, this disturbance will propagate through the medium. his disturbance propagation is called a wave. Let s take an example to illustrate this: a stretched spring (shown in the figure). Initially, the spring (which is the medium) is stationary. Suddenly, a person slightly moves the end of the spring, which creates a disturbance. his disturbance then travels along the spring until it arrives at the other end. wave was then created in the spring. his type of wave in matter bears the name of progressive mechanical wave. mechanical wave is a disturbance that travels through matter. Some waves are not mechanical waves. Light, for example, is a travelling disturbance of electrical and magnetic fields. s these fields are not made of matter, light is not a mechanical wave. Progressive means that the wave travels through the medium. We will see later that stationary waves also exist. progressive mechanical wave is a disturbance that propagates through a medium. In this video, a progressive mechanical wave in a spring can be seen. http://www.youtube.com/watch?vubrlaccqfdk Haber-Schaim, Cross, Dodge, Walter, ougas, Physique PSSC, éditions CEC, 1974 here are several types of waves. For example, when a stone is thrown into a lake, there is a disturbance that appears at the place where the stone hits the water. his disturbance then starts propagating on the surface of the water as waves. he jump of this crazy dude generates waves on the surface of the water. http://www.youtube.com/watch?viy861uga1fo 018b Version -Mechanical Waves
Matter is not ransported by a Wave Before the passage of the wave, the particles composing the medium are at an equilibrium position. When the wave passes, the particles composing the medium moves. However, the wave only temporarily displaces the particles. hey return to their equilibrium positions after the passage of the wave. his phenomenon can be seen in the image to the right. In this image, a wave moves along a stretched spring. point of the medium has been identified by a black dot. When the wave passes, this point moves away from its equilibrium position and returns to its equilibrium position once the wave has passed. herefore, the matter composing the medium is not transported by the wave, it only temporarily moves away from the equilibrium position as the wave passes. fter the passage of the wave, each particle composing the spring is back to the same position it had before the passage of the wave. he following animation illustrates this phenomenon. http://gilbert.gastebois.pagespersoorange.fr/java/son/melde/melde.htm (nd click on Signal on the bottom.) In this video, a little duck moves around its equilibrium position during the passage of waves. It does not travel with the wave. http://www.youtube.com/watch?v-o-vgeabkji Haber-Schaim, Cross, Dodge, Walter, ougas, Physique PSSC, éditions CEC, 1974 However, it is obvious that waves on a beach can push objects, such as algae or surfers. hese waves are different because they are breaking waves. his is what happens when the height of the wave becomes too large relative to the depth. his is a very different type of wave that will not be treated here. Here, only non-breaking waves will be explored. ransverse and Longitudinal Waves ransverse Waves In a transverse wave, the motion of the matter forming the medium is in a direction perpendicular to the direction of propagation of the wave. his is the case, for example, for waves in a rope. 018b Version -Mechanical Waves 3
hyperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html Longitudinal Waves In a longitudinal wave, the motion of the matter forming the medium is in the same direction as the direction of propagation of the wave. his type of wave can be generated in a spring. he following video shows an animation of the two types of waves. http://www.youtube.com/watch?vrbuhdo0zdu yperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html he following video shows that both types of waves can travel in a spring. http://www.youtube.com/watch?vilzj8juvy8 he following applet allows you to explore the motion of matter during the passage of a transverse wave http://gilbert.gastebois.pagesperso-orange.fr/java/ondes/transversales/onde.htm or a longitudinal wave http://gilbert.gastebois.pagesperso-orange.fr/java/ondes/longitudinales/son.htm Waves are only Possible if there is a Force Opposed to the Deformation of the Medium If there is a wave in a medium, there must be a force opposed to the deformation of the medium (called a restoring force). If a stretched rope is moved from its equilibrium position, the tension of the rope will seek to bring the rope back to its equilibrium position. In this case, there is a restoring force, which means that there is a force that is trying to bring to rope back to its equilibrium position. 018b Version -Mechanical Waves 4
Longitudinal waves in matter are compression waves. When you compress an object, the elasticity of the body opposes this compression. s this is a restoring force, it is possible to have compression waves in any substance. (ctually, these are sound waves.) In transverse waves, matter is moved from one side to the other in a direction perpendicular to the direction of propagation of the wave. In a solid, this displacement of matter generates compression and elastic forces seeking to restore the equilibrium and transverse waves are then possible. In fluids (liquids and gases), there is no restoring force opposed to the matter displacement. If a little water is displaced in a pool, there are no restoring forces seeking to bring the water back to the starting location. ransverse waves are, therefore, impossible in fluids. his property allows us to know that the interior of the Earth is liquid. Earthquakes send both transverse and longitudinal waves in every direction inside the Earth. Since only longitudinal waves are received on the other side of the Earth, it is possible to conclude that transverse waves cannot pass through the interior of the Earth and that, therefore, the interior of the Earth must be liquid. In locating where transverse waves can be received on the surface of the Earth, the size of the liquid region can even be determined. principles.ou.edu/eq_seismo/seismo_interior_simple.html Why Sine Wave? We will now focus on waves whose shape is described by a sine function. his seems to be a little restrictive because a wave can have any shape but it is not since it can be shown that any waveform can be written as a sum of sine waves. his is Fourier theorem, a theorem that generated a 30-year debate at the beginning of the 19 th century. he figure to the right shows how a sum of sine waves can give a different wave shape. he sum of the 6 sine functions shown on top of the figure gives the signal at the bottom. his sum is a Fourier series, and it is an important topic of university mathematics. acousticslab.org/psychoacoustics/pmfiles/module0.htm 018b Version -Mechanical Waves 5
his next clip shows you how special waveforms can be obtained by adding sine functions (called harmonics here). http://www.youtube.com/watch?vlunnvyorec Bit of erminology For a sine wave in a string, we have the following elements. he amplitude () is the value of the maximum displacement of the medium. he crests are the places where the displacement of the medium reaches its largest positive value. he troughs are the places where the displacement of the medium reaches its largest negative value. he nodes are the places where the displacement of the medium is zero. Points in phase are points that are in the same position on the cycle of the sine function. here may be one or more complete cycles between these points in phase. hus, all the peaks are in phase and all the troughs are in phase. he wavelength (λ) is the distance between two adjacent crests. (Properly speaking, the wavelength is the distance between two points in phase that are closest to each other.) herefore, it is the length of one cycle of the sine function. he wave travels at a certain speed (v) determined by the characteristics of the medium. Whatever the amplitude and the wavelength, this speed is always the same. Let us now examine the motion of a specific part of the medium to see how it moves. transverse wave is considered to illustrate. 018b Version -Mechanical Waves 6
s the wave passes, the particle of the medium shown in the figure goes up and down. t the instant shown in the figure, the particle is going up since a crest is approaching. he particle must, therefore, moves upwards since it will form the crest of the wave in a few moments. It will be proven a little later (in an example) that the motion of this particle is actually a simple harmonic motion. ccording to what we have seen in Chapter 1, this motion of the particle of the medium is then characterized by a frequency and a period. In the following animation, a sine wave travels through in a medium. http://gilbert.gastebois.pagesperso-orange.fr/java/son/melde/melde.htm (and click on Progressive at the bottom) It can easily be noted that each piece of the medium seems to make a harmonic oscillation when the wave passes. he period of the wave () is the time taken by the particles of the medium to make a full oscillation. he frequency of the wave (f) is the number of oscillations made by the particles of the medium in one second. he relationship between these two quantities found in Chapter 1 is still valid. f 1 Relationship between the Frequency and the Wavelength o find the relationship between the wavelength and the frequency, let s examine the following situation on a rope. 018b Version -Mechanical Waves 7
he particle at the end of the medium, identified by *, is, at this time, at the maximum position of its motion. s the wave travels, it will move downwards and then upwards to reach its maximum position again when the next crest arrives at the end of the rope. he particle will then have made a complete cycle. his means that the time it takes to do one oscillation (period ) is equal to the time it takes for the next crest to arrive at the end of the rope. s this crest is initially at a distance λ and is travelling towards the end of the rope at the speed v, the time it takes to arrive is λ/v. herefore, his equation can then be written as λ v Link between λ and f or Link between λ and λ v λ f Function Describing the Wave Imagine there s a sine wave in a rope. In this case, the displacement of the medium is perpendicular to the direction of propagation of the wave. hyperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html he formula giving the displacement of the rope from the equilibrium position (dotted line at y 0) will now be sought. his displacement will be noted y. Obviously, the shape of a sine wave is given by a sine function. o have a wave that rises up to height, the sine function must be multiplied by because the maximum value of a sine function is 1. he length of a cycle of a sine function of π. he length of the cycle must, therefore, be adjusted to have any wavelength. his can be achieved with the function π y sin x λ 018b Version -Mechanical Waves 8
If a cycle starts at x 0, the cycle must end at x λ. If x λ is put in this equation, then sin (π) is obtained. his is indeed a full cycle because the period of a sine function is π. In this form, this function cannot describe a wave, because it is motionless. It must be changed so that this function moves as time passes. o find this moving function, two axes systems will be used. he first is a stationary axes system (x and y) and the second is a moving axes system which follows the sine function in its motion (x' and y'). s the axes x' and y' follow the sine function while it is moving, the sine function is stationary with respect to these axes. herefore, the equation of the sine with these axes is π y ' sin x ' λ In order to have this function with the stationary axes, the link between these two axes systems must be found. he x-value is the distance between a point in the plane and the y-axis, and x'-value is the distance between the point and the y'-axis. ccording to what is shown in the figure, we have x' x vt 018b Version -Mechanical Waves 9
he values of y and y' are the distances between the point and the x and x' axes. ccording to the figure, this means that y ' y By effecting these changes in the sine function, the following equation is obtained. π y ' sin x ' λ π y sin ( x vt) λ π π v y sin x t λ λ his wave travels towards the positive x-axis. o have a wave moving towards the negative x-axis, the sign of the speed must be inverted. he equation then becomes π π v y sin x ± t λ λ where is used if the wave moves towards the positive x-axis, and + is used if the wave moves towards the negative x-axis. Since v λ f the equation can be written as π y sin x ± π ft λ Certain combinations of variables are so common in the study of waves that symbols are used to represent them. Definitions of k and ω π k λ π ω π f 018b Version -Mechanical Waves 10
k (not to be confused with the spring constant seen in Chapter 1) is the wave number. It is in /m and represents the number of cycles made by the wave over a distance of π metres. ω, which is identical to the ω used in Chapter 1, is the angular frequency. It is in /s and represents the number of oscillation cycles made during π seconds. here is a link between these variables and the speed. his link is herefore, Link between ω and k v λ f λ v π f π 1 v ω k v ω k With these new variables, the function that describes the wave becomes y kx ± t sin ( ω ) It s a sine wave moving with a velocity v towards the positive x-axis or the negative x-axis according to the sign chosen in the sine function. Yet, this is not the most general form. For now, the amplitude, the wavelength, and the speed can be modified but the shape of the wave at t 0 cannot. With the current formula, the wave must have the following shape at t 0. he wave must have y 0 at x 0. But it is possible to have a wave that has a different shape at t 0. For example, the wave could have the following shape. 018b Version -Mechanical Waves 11
It is still described by a sine function but shifted towards the left. Fortunately, we know how to shift a sine function: simply add a phase constant φ. Depending on the value of the phase constant, the sine function at t 0 will shift to one side or the other along the x-axis. he following final result is thus obtained. Function Describing a Progressive Sine Wave y kx ± t + sin ( ω φ ) he negative sign is chosen if the wave moves towards the positive x-axis. he positive sign is chosen if the wave moves towards the negative x-axis. Let s check if a progressive wave travelling at the right speed is obtained. o do this, a wave with a 1 cm amplitude, a cm wavelength and a speed of 1 cm/s towards the positive x-axis is used. his implies that the frequency is 0.5 Hz. vanishing phase constant is used for this wave. he equation of this wave is thus ( π π ) y 1cm sin 100 x t t t 0, the wave equation is y 1cm sin ( 100π x) m m s. he graph of this function is t t 0.5 s, the equation is 1 sin ( 100 π y cm π x ). he graph of this function is m 018b Version -Mechanical Waves 1
t t 1 s, the equation is y 1cm sin ( 100π x π ). he graph of this function is m 3 t t 1.5 s, the equation is 1 sin ( 100 π y cm π x ). he graph of this function is m It is easy to see in this series of graphs that sine function moves towards the right with a speed of 1 cm/s, as it is supposed to do. Longitudinal Waves Remember that in a longitudinal wave, the oscillation of matter is in the same direction as the direction of propagation of the wave. hyperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html 018b Version -Mechanical Waves 13
Since the motion of the particles composing the medium is not in the y-direction, the displacement of those particles cannot be given by y. he variable x cannot be used either, because it is already used to indicate the position of the particles in the medium. So, the letter s is used to indicate the displacement of the particles of the medium from their equilibrium positions. he displacement of the particles composing the medium for a longitudinal sine wave is, therefore, given by s kx ± t + sin ( ω φ ) For a wave in a spring, the function looks like this. hyperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html In the calculations that follow, the formulas for transverse waves are used, but the formulas for longitudinal waves can be easily obtained by replacing y by s. Speed and cceleration of the Particles of the Medium From the formula for the position of the particles of the medium as a function of time, the formulas for the velocity and the acceleration of the particles of the medium can be obtained. Since the velocity is the derivative of the position the velocity is v y y t Velocity of the Particles of the Medium v ± ω cos( kx ± ωt + φ ) y he negative sign is chosen if the wave moves towards the positive x-axis. he positive sign is chosen if the wave moves towards the negative x-axis. vy max ω 018b Version -Mechanical Waves 14
Common Mistake: o Confuse v and v y Do not confuse the speed of the wave and the speed of a particle composing the medium. For a rope, the speed of the wave (v) is the speed of the crest in the x-direction whereas the speed of the medium (v y ) (here the speed of the rope) is in the y-direction. here s no direct link between these two speeds. he acceleration is found by deriving once more. his derivative gives a t cceleration of the Particles of the Medium v y a ω kx ± ωt + φ sin ( ) he negative sign is chosen if the wave moves towards the positive x-axis. he positive sign is chosen if the wave moves towards the negative x-axis. amax ω Calculation of and φ from Initial Conditions he values of and φ can be calculated if the position and the velocity of the medium at a certain place and at a certain time are known. he formula for the position gives sin ( kx ωt φ ) ± + y and the formulas for the velocity gives 018b Version -Mechanical Waves 15
± v cos( ) y kx ± ωt + φ ω Since the following result can be obtained ( kx ωt φ ) ( kx ωt φ ) sin ± + + cos ± + 1 v y y + 1 ω By multiplying by ², this equation becomes Calculation of the mplitude of the Wave lso, since sin cos y ( kx ± ωt + φ ) ( kx ± ωt + φ ) v y + ω ( kx ωt φ ) tan ± + we have his gives tan ( kx t ) y ( ) ± ω + φ ± vy ( ω ) Calculation of the Phase Constant Warning: ω y ± + ± v tan ( kx ωt φ ) he negative sign is chosen if the wave moves towards the positive x-axis. he positive sign is chosen if the wave moves towards the negative x-axis. - he value of φ is in ians. Put your calculator in ian mode to obtain the correct value. 018b Version -Mechanical Waves 16 y
- If the value of ± v y is negative, π ians must be added to the answer given by the calculator. - If the speed is zero, the inverse tangent of infinity or -infinity is obtained according to the sign of y. Do not panic, the answers of these inverse tangents are π/ and -π/. Example..1 sine wave is travelling towards the positive x-axis on a taut rope. he wavelength is 40 cm, and the frequency is 8 Hz. t t 0 s and x 0 m, the displacement of the rope is y 15 cm, and the velocity of the rope is v y 1000 cm/s. a) What is the period? he period is 1 1 0.15s f 8Hz b) What is the speed of the wave? he speed is v λ f 0.4m 8Hz 3. m s c) What is the function describing this wave? he function is y kx t + sin ( ω φ ) he negative sign was chosen since the wave is travelling towards the positive x-axis. k, ω,, and φ must then be found. he wave number is he angular frequency is he amplitude is found with π π k 5π λ 0.4m 018b Version -Mechanical Waves 17 m ω π f π 8Hz 16π s
and the phase constant is found with vy y + ω + 4.9cm cm 1000 s ( 15cm) 16π s tan tan ( kx ωt φ) ( φ) ω y + v 16π 15cm 1000 ( φ) tan 0.754 φ.496 (π was added to the value given by the calculator because the divisor is negative.) herefore, the function describing the wave is cm s ( π π ) y 4.9cmsin 5 x 16 t +.496 m s y d) What is the position as a function of time of the piece of rope located at x 0 cm? Using x 0 cm in the equation previously found, we obtain ( π m π s ) ( π ) y 4.9cm sin 5 0.m 16 t +.496 4.9cm sin 16 t + 5.638 s Perhaps it is not obvious (because of the minus sign in front of ω) but this is the equation of a harmonic motion. If you don t like this minus sign, it can be removed using some properties of the sine function. ( π s ) ( π s ) ( π ) y 4.9cmsin 16 t + 5.638 4.9cmsin 16 t 5.638 4.9cmsin 16 t.496 For this last step, sin x sin (x + π) was used. s 018b Version -Mechanical Waves 18
formula describing a harmonic motion is thus obtained. similar sine function (but with a different φ) would be obtained for any x-position. his proves that each piece of rope performs a harmonic motion during the passage of a sine wave. e) What is the maximum speed of the rope? he maximum speed of the rope is v max ω 16π 0.49m 1.5 m y s s f) What is the velocity of the rope at x 1 m and t 1 s? he formula giving the velocity of the rope is 1 π ( π π ) ( π x π t ) v 0.49m 16 s cos 5 x 16 t +.496 y m s m 1.5 cos 5 16 +.496 s m s t t 1 s and x 1 s, the velocity is ( π π ) m v 1.5 cos 5 1m 16 1s +.496 y s m s 10.00 m s he Wave Speed is Determined only by the Medium (in a Non- Dispersive Medium) Most of the time in these notes, waves in non-dispersive media are considered. his means that the waves, regardless of their shape, are all travelling at the same speed in a medium. Speed in a Non-Dispersive Medium he waves are all travelling at the same speed in a non-dispersive medium. Sound waves propagating in air are travelling in a non-dispersive medium. So, low-pitched sounds and high-pitched sounds travel at the same speed in air. If it were not the case, a sound could be dramatically distorted if, for example, high-pitched sounds were to travel faster than low-pitched sounds. We would then hear high-pitched sounds before the lowpitched sounds. his would drastically alter the beauty of a musical piece if the sound of the bass was heard 1 second after the sound of the guitar. 018b Version -Mechanical Waves 19
o change the speed of a wave, the medium must be changed. For a wave travelling in a string, the tension of the rope, for example, has to be changed to change the speed of the wave. For a sound wave travelling in air, the air temperature must be changed to change the speed. non-dispersive medium also implies that a wave will always keep the same shape while propagating. In a dispersive medium, the wave shape changes as the wave travels. Common Mistake: hinking that the Speed Changes if λ or f Changes Sometimes, students use the formula v λf to deduce that the speed of the wave changes if the wavelength or the frequency is changed. his is wrong. he speed of the wave is the same regardless of the wavelength. his formula rather indicates that the frequency must change if the wavelength is changed since the speed must remain the same. demonstration of this can be seen in the following video. In the top part of the clip, there is a wave with a smaller frequency than the wave in the bottom part. It is clear that a lower frequency generates a wave with a longer wavelength. It is also possible to note that the speed of the wave is the same even if the wave is different. http://www.youtube.com/watch?v7iv4gmyxscq ransverse and Longitudinal Waves Can Have Different Speeds in the Same Medium In a solid, longitudinal waves and transverse waves can coexist. s the restoring forces at the origin of the propagation of these waves are different for both types of waves, they can have different speeds. Even if the waves propagate in the same medium, the speeds of these two types of waves can be different because they are not of the same nature. ll longitudinal waves have the same speed and all transverse waves have the same speed, a speed that can be different from the speed of longitudinal waves. For example, earthquakes create longitudinal and transverse waves in the ground. In this case, longitudinal waves are travelling faster than transverse waves. pproximately, longitudinal waves have a speed of 6 km/s near the Earth s surface while transverse waves have a speed of 3 km/s. By measuring the difference in time of arrival between longitudinal waves (called primary waves, because they arrive first) and transverse waves (known as the secondary waves) the distance of the epicentre can be calculated. 018b Version -Mechanical Waves 0
Wave Speed in a Rope o find the speed of waves in a stretched rope, the forces exerted on a small piece of rope will be considered. Suppose that the wave has the following shape. www.wikipremed.com/image.php?img01001_68zzzz10700_1960_68.jpg&image_id10700 he forces acting on a small piece of rope on top of the wave are as follows. he tension forces and the angles being the same on each side of the piece, the sum of the x-component of the forces vanishes by symmetry. On the other hand, the y-components of the forces do not cancel out. he sum of these components is F y θ F sin If the piece of rope is very small, the angle is small and the approximation sinθ θ can be used. he sum then becomes θ Fy F Ftθ he angle is simply the length of the arc of a circle (the length of the small piece of rope) divided by the ius of curvature. herefore, l F F r y Now, the acceleration of the piece of rope must be found. here is a trick to figure out its value quite easily: using the point of view of an observer moving at the same speed as the 018b Version -Mechanical Waves 1
wave. For this observer, the wave is stationary and the rope travels towards the left at speed v. t the highest point of the wave, the rope moves along an arc and is, therefore, making a circular motion. www.wikipremed.com/image.php?img01001_68zzzz10700_1960_68.jpg&image_id10700 s the rope makes part of a circular motion at the top of the wave, its acceleration is herefore, v a r F y ma l v F m r r F l mv y he mass of the small piece of rope depends on its length. If the linear density of the rope (in kg/m) is then, the mass of the piece is mass µ length mass µ length m µ l he equation then becomes F l mv F l µ lv F µ v If this equation is solved for the speed, the following result is obtained. 018b Version -Mechanical Waves
Wave Speed in a Rope v F µ It can be seen that this speed does not depend on the shape of the wave. It depends only on the characteristics of the medium such as the tension force and the linear density of the rope. Example.3.1 kg mass is suspended at the end of a 6 m long rope as shown in the figure. he mass of the rope is 300 g. What is the speed of the waves travelling on the rope? he speed of waves in a rope is given by v F µ o find the speed, the tension and the linear density must be found first. www.chegg.com/homework-help/questions-andanswers/physics-archive-01-february-11 he tension of the rope is found with the sum of the forces acting on the mass. he linear density of the string is herefore, the speed is F mg + F 0 y F F mg 19.6N µ mass 0.3kg 0.05 length 6m F 19.6N v 19.8 µ 0.05 kg m kg m m s 018b Version -Mechanical Waves 3
wave does not transport matters, but it carries energy. he amount of energy in a wave will now be determined. he rate at which energy is received at the end of the rope (which is the power of the wave) will also be calculated. Energy in a Sine Wave Suppose that there is a sine wave with a certain length on a string, as shown in this figure. Each particle (of mass m) composing the string is undergoing a harmonic oscillation. he energy of a mass in a harmonic oscillation being the energy of all the oscillating masses is 1 E mω 1 E mω 1 ( m) ω 1 Mω where M is the mass of the rope in the area where there is the wave. his mass is M µ length µ D he energy is then Energy of a Sine Wave of Length D (in joules) 1 E µ Dω 018b Version -Mechanical Waves 4
Power of a Sine Wave he rate at which energy arrives at the end of the rope will now be calculated. his power is Power Energy time he total amount of energy in the wave is already known but not the time taken for this energy to arrive at the end of the rope. he time starts when the front of the wave arrives and it stops when the back of the wave arrives. herefore, the time is the time taken by the end of the wave to arrive when the front of the wave has just arrived at the end of the rope. t this instant, the end of the wave is at a distance D and is coming towards the receiver with a speed v. hus, the time is time D v and the power is Power 1 µ Dω D v Simplifying, the result is Power of a Sine Wave (in watts) 1 P µ vω 018b Version -Mechanical Waves 5
Example.4.1 sine wave travelling in a string has a length of 3 m, a wavelength of 5 cm and an amplitude of 1 cm. he tension of the rope is 500 N, and its linear density is 50 g/m. a) What is the total energy in the wave? he energy is 1 E µ Dω he speed of the wave and frequency are needed to solve this problem. he speed is he frequency is F 500N v 100 µ 0.05 kg m m s m 100 0.5m s v λ f f 400Hz f herefore, the energy is 1 E µ Dω 1 0.05 kg m 3 m 400 Hz 0.01 m 47.37J ( π ) ( ) b) What is the power of this wave? he power is 1 P µ vω 1 0.05 kg 100 m m s 400 0.01 1579W ( π Hz) ( m) 018b Version -Mechanical Waves 6
Interference is a superposition of waves. he result of the encounter of two waves is particularly simple: the displacements caused by each wave add up. his is called the superposition principle. Superposition Principle y y + y + y + tot 1 3 Let s illustrate this principle with an example. wo waves with a rectangular shape are heading towards each other. he distances indicated by a grid in the figure are in centimetres. Each wave has a speed of 1 cm/s and causes the rope to shift 1 cm from the equilibrium position. t t 1 s, the fronts of each wave come into contact and the interference begins. t t s, the 1 cm long wave is found in the middle of the 3 cm long wave. here, the displacements caused by each wave (which is 1 cm) add up to obtain a displacement of cm. t t 3 s, the interference ends and at t 4 s, the two original waves are found again and each is continuing on its way. he passage of the two waves one through the other does not affect at all the shape of the waves. hey are back to their original shape once they have interfered. In this case, the displacement of the rope is greater at the places where the waves overlap than the displacement caused by a single wave. When this happens, constructive interference is obtained. his beautiful clip illustrates the result of two waves overlapping on a string. he top part of the clip shows the wave going towards the right. he middle part shows the wave going towards the left and at the bottom part of the clip shows the addition of these two waves. http://www.youtube.com/watch?v8irzyoc7deu You can also watch this demonstration. http://www.youtube.com/watch?vyvir5th8jw 018b Version -Mechanical Waves 7
In this other example, one of the waves generates a negative displacement on the rope. t t s, the short wave is located in the middle of the long wave. t the places where the waves are superimposed, the displacement made by the long wave (1 cm) is added to the displacement made by the short wave (-1 cm). he result is 0 cm and the rope is at its equilibrium position everywhere where the short wave interferes with the longer wave. In this case, the displacement of the rope is zero where the waves overlap. When the resulting displacement on the two waves is smaller than the displacement made by only one wave (like here), destructive interference is obtained. It can still be seen that, after the passage of the waves one through the other, each wave retakes exactly the same shape it had before the encounter of the waves. gain, the interference of two waves can be seen in this clip but, this time, a wave generates a positive displacement (the one moving towards the right) while the other generates a negative displacement (the one moving towards the left). http://www.youtube.com/watch?v95macpu6xgm Here s a demonstration of destructive interference. http://www.youtube.com/watch?vurre-hokums hese elements are only true for waves travelling in a linear medium. In a non-linear medium, the wave resulting from the superposition of waves is not simply the sum of each wave. lso, in a non-linear medium, the shape of a wave after its passage through another wave is different from the shape it had before the encounter. Non-linear media will not be considered in these notes. Passing from One Medium to nother Sometimes, a wave travels from one medium to another. his could be a wave passing from one rope to another rope when the ropes are connected end to end. 018b Version -Mechanical Waves 8
Speed, Frequency, and Wavelength When a wave passes from one medium to another, its frequency cannot change. If 50 oscillations per second arrive at the place where the medium changes, 50 oscillations per second must also leave. Oscillations cannot accumulate or vanishes at the place where the media meet. Frequency in a Change of Medium he frequency does not change when a wave passes from one medium to another f f 1 On the other hand, the speed may be different in the media because the speed depends on the characteristics of the medium. s f v λ and f f 1 We obtain, by substituting v/λ for f, Wavelength in a Change of Medium v1 v λ λ 1 hus, if the speed is greater in medium, the wavelength will be greater in this medium. his change can be seen in the following video. http://www.youtube.com/watch?viejdckubcc Reflected and ransmitted Waves We ll examine waves on a rope to illustrate what happens when a wave passes from one medium to another. For example, the wave could pass from a rope having a certain linear density to another rope with a different linear density. 018b Version -Mechanical Waves 9
www.kshitij-iitjee.com/reflection-and-transmission-of-waves Part of the wave will pass in the other medium while part of the wave will be reflected back in the same medium as the incident wave. he goal here is to know the amplitude of these transmitted and reflected waves. However, the amplitude of these waves depends on a characteristic the medium: the impedance. Impedance he impedance is a characteristic of the medium. For a wave propagating on a rope, the impedance of the medium is defined by Z µv ctually, different versions of the impedance formula can be obtained by using the formula for the wave speed. hese results are then obtained. Impedance of the Medium for a Wave on a String Z µv F µ F v o understand a bit more what this impedance is, it must be observed that the power of a wave on a string Can be written as 1 P µvω 018b Version -Mechanical Waves 30 1 P Zω
(In fact, this last equation is the general equation for the power of a wave, valid for all types of wave. With the formula for the impedance of the medium for the specific type of wave studied, the power can always be calculated with this equation.) Now, imagine that two waves with the same power and the same frequency are send in ropes with different impedances. he power equation indicates that the amplitude must be smaller in the rope with the biggest impedance. ( must decrease if Z increases if all other variables remain the same.) herefore, the impedance can be interpreted as the response of the medium to a disturbance. he more impedance there is, the smaller the response of the medium will be for a similar disturbance. mplitudes of the Reflected and ransmitted Waves If the initial wave has an amplitude, then the amplitude of the reflected and transmitted waves are as follows. mplitudes of the Reflected and ransmitted Waves R Z Z Z + Z 1 1 Z1 Z + Z 1 If you wish, you can see proof of these formulas in this document. http://physique.merici.ca/waves/proofrandt.pdf (It is a proof for waves on a rope, but the result is valid for every type of wave.) 018b Version -Mechanical Waves 31
Example.6.1 sine wave with a 1 cm amplitude, and a 50 Hz frequency arrives at a junction between two strings. he following figure gives the properties of the two strings. What are the powers of the incident, reflected and transmitted waves? he power of a wave is 1 1 P µ vω Zω o calculate the powers, the impedances of the media are needed. hese impedances are Z F µ 400N 0.065 5 kg kg 1 1 m s Z F µ 400N 0.01 kg kg m s hus, the power of the incident wave is 1 1 1 5 kg ω s ( π 50 ) ( 0.01 ) 4.67 Pincident Z Hz m W o find the powers of reflected and transmitted waves, the amplitudes of these waves are needed. hese amplitudes are found with Z Z Z R Z Z Z Z 1 1 1 + 1 + he amplitude of the reflected wave is R Z Z 1 1 kg 5 s kg 5 + s Z + Z kg s kg s 0.49cm 1cm 018b Version -Mechanical Waves 3
he amplitude of the transmitted wave is Z1 Z + Z 1 kg 5 s kg 5 + s kg s 1.49cm 1cm It may seem odd that the transmitted wave has a greater amplitude than the incident wave, but it is possible. s the second medium has a smaller impedance, the wave can easily create a large disturbance in this medium. We will see that the power of this wave is less even if it has a greater amplitude. he power of the reflected and transmitted wave are 1 1 1 5 kg ω s ( π 50 ) ( 0.0049 ) 4.53 PR Z Hz m W 1 1 kg ω s ( π 50 ) ( 0.0149 ) 0.14 P Z Hz m W It can be noted that the sum of the powers of the reflected and transmitted wave is equal to the power of the incident wave. his must always be true. In this example, 81.6% of the wave energy is transmitted and 18.4% of the wave energy is reflected. Reflection and ransmission and Impedance Difference Very Different Impedances (Impedance Mismatch) Let s consider what happens if the impedance Z 1 is much larger than the impedance Z. In this case, the amplitude of the reflected wave is Z Z Z 1 1 R Z1 + Z Z1 he reflected wave has virtually the same amplitude as the initial wave. his means that almost all the energy of the initial wave is reflected back and that there is very little energy in the transmitted wave. he wave is, therefore, almost entirely reflected. If the impedance Z 1 is much smaller than the impedance Z, then Z Z Z 1 R Z1 + Z Z 018b Version -Mechanical Waves 33
he reflected wave has virtually the same amplitude as the incident wave. (We will deal with the meaning of the minus sign later.) gain, this means that there is a lot a reflection and that there is practically no energy in the transmitted wave. Similar Impedances (Impedance Matching) If the impedance Z 1 is very close to the impedance Z, then R Z Z Z + Z 1 1 0 In this case, there s no reflected wave and the wave is fully transmitted in the other medium. his is called impedance matching. For example, this situation can be achieved with ropes having the same tension and made of different materials, but with identical linear densities. his is a very important concept for waves of currents (alternating currents) in electrical circuits. If the impedances are identical, there is no signal reflection. his is needed, for example when speakers are connected to a stereo system. hen, the amplifier, which has a certain impedance, sends signals to the speakers, who also have a certain impedance. If the impedance is not the same, there will be reflections of the signal between the receiver and speakers, which will result in an echo in sound. herefore, it is better to use speakers with the right impedance. he impedance values are often written on the back of the speakers and of the amplifier. he following conclusion can then be drawn. Rules for Reflection and ransmission. s the difference of impedance of the media gets larger, the reflection is larger and the transmission is smaller. If the impedances of the media are identical, the wave is completely transmitted and there is no reflection. Inversion of the Reflected Wave What is the meaning of the minus sign in front of the amplitude of the reflected when Z is greater than Z 1? his sign means that the reflected wave is inverted compared to the incident wave. his is not so obvious to see with a sine wave, but it s pretty easy with a wave that consists of a single bump. If the impedance Z 1 is larger than Z, the amplitude of the reflected wave is positive and we have the following situation. 018b Version -Mechanical Waves 34
www.transtutors.com/physics-homework-help/waves/reflection-and-refraction-of-waves.aspx he reflected wave is in the same direction as the incident wave. If the impedance Z 1 is smaller than Z, the amplitude of the reflected wave is negative and we have the following situation. www.transtutors.com/physics-homework-help/waves/reflection-and-refraction-of-waves.aspx he reflected wave is inverted compared to the incident wave. he amplitude of the transmitted wave can never be negative and so it is never inverted compared to the incident wave. Reflected and ransmitted Waves Inversion If Z > Z 1, the reflected wave is inverted. Si Z < Z 1, the reflected wave is not inverted. he transmitted wave is never inverted. Let s see what this means for waves travelling on ropes. In this case, the impedance is Z F v F µ Generally, the tensions of the strings tied end to end are the same and only the linear density is different. Note that the impedance is greater if the linear density is greater or if the speed of the wave is smaller. he rules for the inversion of the reflected wave then become 018b Version -Mechanical Waves 35
If µ > µ 1 or if v < v 1 the reflected wave is inverted. If µ < µ 1 or if v > v 1, the reflected wave is not inverted. In the images on the following page, a wave is travelling along two stretched springs (which act as ropes) tied end-to-end. he largest spring has a smaller linear density than the smallest spring. (Yes, you read correctly. he smallest spring is perhaps made of metal while the largest is perhaps made of plastic.) he impedance of the big spring is, therefore, smaller than the impedance of the small spring. Haber-Schaim, Cross, Dodge, Walter, ougas, Physique PSSC, éditions CEC, 1974 In the series of images to the left, the wave arrives in the medium having the smaller impedance and attempts to go into the medium with a higher impedance. here s a 018b Version -Mechanical Waves 36
transmitted wave, which is in the same direction as the initial wave. here is also a reflected wave that is inverted because the impedance of medium is greater than the impedance of medium 1. It can be seen that the speed of the wave is smaller in medium, which is an indication that the impedance of medium is greater. In the series of images to the right, the wave arrives in the medium having the greater impedance and attempts to go into the medium with a smaller impedance. he transmitted wave is once again in the same direction as the initial wave. he reflected wave is also in the same direction since the impedance medium is smaller than the impedance of medium 1. It can be seen here that the speed in medium is greater than in medium 1, confirming that the impedance of medium is smaller than that the impedance of medium 1. Wave rriving at the End of a Rope he formulas for the amplitude of the reflected wave will help to determine what happens when a wave arrives at the end of a rope. Rope ttached at the End wave is travelling along a rope whose end is securely attached to something. When the wave arrives at the end of the rope, the same thing happens as when the wave passes from one medium to another. Here the two media are the rope and the object on which the rope is attached. s this object is fixed, it will not be disturbed at all by the motion of the rope, which means that this object behaves as a medium with a huge impedance. his situation is, therefore, similar to the situation where Z is much greater than Z 1. It was then found that the amplitude of the reflected wave is -, which means that the wave is completely reflected, but inverted. www.schoolphysics.co.uk/age16-19/wave%0properties/wave%0properties/text/phase_change/index.html Here is a demonstration. http://www.youtube.com/watch?vlwhxz6jvjs his wave is inverted since the rope pulls on the point of attachment when the wave arrives at the end of the rope. ccording to Newton s third law, the attachment point then pulls down on the rope, which creates a downwards wave that moves towards the left. 018b Version -Mechanical Waves 37
Rope with a Free End Now imagine that the rope is not attached at its end or that it is fixed to a rod by a massless ring that can slide without any friction along the rod. his is similar to a situation where the media are a rope and a massless rope. s the linear density of the second medium is zero, its impedance is zero. his corresponds to the situation where Z is much smaller than Z 1, and this means that the amplitude of the reflected wave is. his means that the wave is completely reflected and is not inverted. www.schoolphysics.co.uk/age16-19/wave%0properties/wave%0properties/text/phase_change/index.html Here is a demonstration. http://www.youtube.com/watch?vavcqq5kepi In this case, the end of the rope rises much higher than expected when the wave reaches the end because there is less tension force acting on the last piece of rope. Usually, each piece of rope is forced back to its equilibrium position by tension forces acting on each side of the piece. However, the tension force of only one side is exerted on the last piece of rope. his lack of tension force makes this last piece rise higher than expected. his upwards motion will exert an upwards force on the rest of the rope, which then creates a new upwards wave moving towards the left. It is also possible to obtain standing waves. In this type of wave in a string, each piece of rope oscillates, but the wave crests and the nodes are not moving. he video following shows several possible standing waves. It can easily be seen that these waves are standing waves as the nodes remain motionless. http://www.youtube.com/watch?vjovixzvfoxo http://gilbert.gastebois.pagesperso-orange.fr/java/son/melde/melde.htm (and click on Stationnaire at the bottom of the page.) Standing Wave is the Result of the Superposition of wo Identical Progressive Waves Going in Opposite Directions Standing waves are the result of the superposition of two waves: a wave moving towards the right and an identical wave moving towards the left. In the previous video, the standing 018b Version -Mechanical Waves 38
wave is the result of the wave made by a person, which is travelling towards the left, and the wave reflected at the other end of the medium, which is travelling towards the right. It will now be demonstrated that this sum is a standing wave. he wave moving towards the positive x-axis is described by y kx t 1 sin ( ω ) and the wave travelling towards the negative x-axis (same amplitude and wavelength) is described by y kx + ωt sin ( ) (he amplitude of each wave is, and the resulting amplitude of the two superimposed waves will be noted tot.) Vanishing phase constants were used to simplify. he same results would be obtained if non-vanishing constant were used. he sum of the two waves is ( ω ) sin ( ω ) ytot sin kx t + kx + t o write this equation into another form, the following trigonometric identity is used he sum then becomes ( ) ( ) sinθ + sinθ sin θ + θ cos θ θ 1 1 1 1 1 (( ω ) ( ω )) ( ω ) ( ω ) 1 1 ( ) y sin kx t + kx + t cos kx t kx + t tot Simplifying, the following result is obtained. Standing Wave Equation Here is the interpretation of this equation. y sin kx cosωt tot ytot sin kx cosωt mplitude that depends on the position oscillation as a function of time Each piece of the medium, therefore, makes an oscillation with an amplitude that depends on the position. his was seen previously in the video: matter oscillates with some 018b Version -Mechanical Waves 39
amplitude in some places while there are no oscillations at all (the amplitude is zero) at some other places. Oscillation mplitude for a Standing Wave sin kx tot tot is the amplitude of the resulting wave and is the amplitude of each interfering wave. he amplitude is the absolute value since a negative value for the amplitude is simply a hidden π ians phase shift. he oscillation has a maximum amplitude of at the centre of the antinodes (see figure). he amplitude is zero at the nodes of the standing wave. Remember that is the amplitude of each progressive wave forming the standing wave and not the amplitude of the standing wave itself. www.allaboutcircuits.com/vol_3/chpt_/.html his figure shows the rope when it is at its farthest position (curved black line) from its equilibrium position (dashed line) and at its other farthest position from its equilibrium position (grey line). he rope swings between these two positions. Oscillation Modes of the Standing Wave Consider a taut rope fixed to something at each end. hen, no motion whatsoever is possible at each end of the rope. his means that a node of the wave must be located at each end of the rope (x 0 and x L). t x 0, the amplitude of the wave is already zero because sin k 0 0 tot (his is nice but we arrive at this result because vanishing phase constants have been chosen previously. If non-vanishing phase constants had been used, a relation concerning these phase constants would have been obtained to have a node at x 0.) o have a node at x L, the following equation must hold. 018b Version -Mechanical Waves 40
sin kl 0 sin kl 0 kl 0, ± π, ± π, ± 3 π, ± 4 π, he last line is obtained by finding all possible solutions of the inverse sine function. Negative solutions do not make sense since the length of the rope cannot be negative. Forgetting these negative solutions, this result can be written in the following way. his means that kl nπ (n is a positive integer) kl nπ π L nπ λ L λ n herefore, the wavelength of a standing wave can have only certain values. Many values are still possible, though. o help identify these solutions, an index is added to the wavelength. hus, λ 3 is the value of the wavelength for n 3. he formula for the possible wavelength of a standing wave is then Possible Wavelengths of a Standing Wave Here are some of these possibilities. L λ n (n is a positive integer) n 018b Version -Mechanical Waves 41
universe-review.ca/r1-03-wave.htm In this figure, the solutions for n 1,, 3 and 4 are shown. hose different solutions are called harmonics or modes. he standing wave with n 5, for example, is called the 5 th harmonic or the 5 th mode. Note that the distance between the nodes is always equal to half the wavelength. Distance between the Nodes of a Standing Wave distance λ If only some values of the wavelength are possible, then only some values of the frequency are possible. he possible frequencies are found with f v λ Using the formula for the possible wavelength, the possible frequencies are given by Possible Frequencies of a Standing Wave nv fn L 018b Version -Mechanical Waves 4
he frequency of the fundamental mode (f ) is called the fundamental frequency. Since f 1 v L the possible frequencies formula can also be written as Possible Frequencies of a Standing Wave fn nf 1 It can be noted that the frequency increases for higher harmonics. Let s return to the video previously seen to observe the different harmonics of a standing wave. Notice how the frequency of oscillation of the hand increases to create higher-order modes. http://www.youtube.com/watch?vjovixzvfoxo If the person were to try to generate a wave with a frequency that does not correspond to a possible frequency of a standing wave, the medium would move, but this motion would not be a standing wave. more chaotic motion would then be obtained. For a standing wave on a string, the formula obtained for the speed can be used to get Possible Frequencies of a Standing Wave on a Rope f n n F L µ When a guitar string is plucked, waves are sent along the rope. s the rope is attached at each end, reflections at these ends create a standing wave on the string. Several harmonics can be present at the same time on the string, but the note played by the instrument always corresponds to the frequency of the first harmonic. hus, the note played by a rope will be f note 1 F f1 L µ his equation represents very well what is happening with the strings of a guitar. (Note that high frequencies correspond to high-pitch tones and that low frequencies correspond to low-pitch sounds as we shall see in the next chapter.) If you pluck a string, a sound is created. If the length of the string is reduced, by pressing a fret, the sound is more high-pitched. he formula does indeed tell us that f increases if L decreases. 018b Version -Mechanical Waves 43
If you pluck the largest string, the sound has a lower pitch than with the smallest string. he big string has a larger linear density than the little string (it is more massive for the same length). he formula also tells us that the frequency is lower if the linear density is larger. If the string is stretched by turning the tuning peg, the tension increases. hen a higherpitched sound is obtained. his time, the formula indicates that the frequency increases if the tension increases. he rope can also be stretched with a slight sideways motion while pressing the fret. vibrato effect can be obtained by changing the tension in this way. (he following paragraph is for those having a more advanced knowledge in music.) he formula also indicates that the distance between the frets will be smaller and smaller as the length of the string is decreased. Suppose that the length of the rope has to be reduced from 0.6 m to 0.3 m to change the pitch from 300 Hz to 600 Hz (one octave). hen there are 1 semitones on a distance of 30 cm. o go one octave higher (from 600 Hz to 100 Hz), the length of the rope must be reduced from 0.3 m to 0.15 m as the frequency is proportional to 1/L. here are now 1 semitones on a distance of 15 cm. hey are, therefore, closer to each other since the same number of semitones are located on a distance twice as short. Mr. Borowicz makes a beautiful demonstration of all these effects in the first 5 minutes of this video. http://www.youtube.com/watch?vtxwjnr56lfo Example.7.1 he largest string of a guitar has a length of 9.9 cm and a mass of 5.58 g. Once on the guitar, there is 65.5 cm between the two points of attachment of the string. a) What should be the tension of the string so that the frequency of the first harmonic is 8.4 Hz (which is the frequency that the largest string of a guitar should have)? he tension can be found with the formula for the frequency of the first harmonic f 1 1 F L µ if the linear density is known. s the linear density of the string is the tension is µ mass 0.00558kg 0.00601 length 0.99m kg m 018b Version -Mechanical Waves 44
f 1 1 L 1 F 8.4Hz 0.655m 0.00601 F F µ 70.03N kg m b) What are the frequencies of the nd, 3 rd, and 4 th harmonics? he frequency of the other harmonics is fn 3 4 nf 1 f 8.4Hz 164.8Hz f 3 8.4Hz 47.Hz f 4 8.4Hz 39.6Hz Each of these harmonics will be heard at the same time when the string is plucked. c) If the amplitude of the oscillation of the standing wave is 0.1 mm at a position located 3 cm from the end of the string for the third harmonic, what is the amplitude of the oscillation in the centre of the antinode? he amplitude at the centre of the antinode is as seen previously. www.allaboutcircuits.com/vol_3/chpt_/.html o find this amplitude from the amplitude at another position, we will use the following formula. sin kx tot Knowing that tot 0.1 mm at x 0.03 m, we will be able to find provided the value of k is known. k can be found with the wavelength. For the 3 rd harmonic, the wavelength is 018b Version -Mechanical Waves 45
he wave number is thus L 0.655m λ 0.4367m 3 3 π k 14.39 m λ With an amplitude of 0.1 mm cm at x 3 cm, we obtain sin kx tot 0.0001m sin14.39 0.03m m 0.0001195m herefore, the amplitude at the centre of the antinode is 0.00039m 0.39mm Sometimes, you ll have the feeling that some information is missing in order to solve the problem, but it is not the case. By applying the same equation twice, the solution becomes simpler when one equation is divided by the other. Here is an example. Example.7. standing wave on a rope has a fundamental frequency of 100 Hz when the tension is 00 N. What should be the tension to have a fundamental frequency of 500 Hz? We know that f 1 1 F L µ 1 00N 100Hz L µ and we want to have 1 F f1 L µ 1 F 500Hz L µ 018b Version -Mechanical Waves 46
Dividing the second equation by the first, we arrive at 1 F 500Hz L µ 100Hz 1 00N L µ F 5 00N F 5000N Link between λ and f or Link between λ and λ v λ f Definition of k and ω Link between ω and k π k λ π ω π f v ω k Function Describing a Progressive Sine Wave y kx ± t + sin ( ω φ ) he negative sign is chosen if the wave moves towards the positive x-axis. he positive sign is chosen if the wave moves towards the negative x-axis. Velocity of the Particles of the Medium vy ± ω cos( kx ± ωt + φ ) he negative sign is chosen if the wave moves towards the positive x-axis. he positive sign is chosen if the wave moves towards the negative x-axis. v ω y max 018b Version -Mechanical Waves 47
cceleration of the Particles of the Medium a ω kx ± ωt + φ sin ( ) he negative sign is chosen if the wave moves towards the positive x-axis. he positive sign is chosen if the wave moves towards the negative x-axis. a ω max Calculation of the mplitude of the Wave Calculation of the Phase Constant y v y + ω ω y ± + ± v tan ( kx ωt φ ) he negative sign is chosen if the wave moves towards the positive x-axis. he positive sign is chosen if the wave moves towards the negative x-axis y Wave Speed in a Rope v F µ Energy of a Sine Wave of Length D (in joules) on a rope 1 E µ Dω Power of a Sine Wave (in watts) on a rope Superposition Principle 1 1 P µ vω Zω tot y y + y + y + 1 3 Frequency and Wavelength in a Change of Medium f f 1 v1 v λ λ 1 018b Version -Mechanical Waves 48
Impedance of the Medium for a Wave on a String F v Z F µ µv mplitudes of the Reflected and ransmitted Waves R Z Z Z + Z 1 1 Z1 Z + Z 1 Rules for Reflection and ransmission. s the difference of impedance of the media gets larger, the reflection is larger and the transmission is smaller. If the impedances of the media are identical, the wave is completely transmitted and there is no reflection. Reflected and ransmitted Waves Inversion If Z > Z 1, the reflected wave is inverted. Si Z < Z 1, the reflected wave is not inverted. he transmitted wave is never inverted. Standing Wave Equation y sin kx cosωt tot Oscillation mplitude for a Standing Wave sin kx tot Possible Wavelengths of a Standing Wave L λ n (n is a positive integer) n 018b Version -Mechanical Waves 49
Distance between the Nodes of a Standing Wave distance Possible Frequencies of a Standing Wave λ f n nv nf L 1 Possible Frequencies of a Standing Wave on a Rope n F fn L µ 018b Version -Mechanical Waves 50
. Progressive Sine Waves 1. sound wave has a speed of 350 m/s and a frequency of 400 Hz. a) What is the period of the wave? b) What is the wavelength of the wave?. In 4 seconds, a wave on a rope travelled 30 m while a piece of the rope has made 0 complete oscillations. What is the wavelength of the wave? 3. his wave moves at 40 m/s towards the right on a rope. (Distances are in centimetres on the figure.) a) What is the frequency of the wave (approximately)? b) What is the maximum speed of the rope (approximately)? 4. he equation of a wave on a string is π y 0.msin 10 m x + 00 s t + 4 a) In which direction is this wave travelling? b) What is the wavelength? c) What is the speed of the wave? d) What is the velocity of the rope at x 1 m and t 1 s? 5. transverse wave is moving towards the positive x-axis with a speed of 50 m/s. t t 0 s, the string at the position x 0 m is at y cm and has a speed of -100 cm/s. Knowing that the wave period is 0.1 s, write the equation of the wave. y kx ± t + sin ( ω φ ) 018b Version -Mechanical Waves 51
.3 Waves Speed 6. wave with a 50 cm wavelength is moving at 30 m/s on a rope. How fast will a wave with a wavelength of 0 cm travel if it moves on the same string with the same tension? 7. m long rope has a 00 N tension. wave passes from one end of the rope to the other in 0.05 s. What is the mass of the rope? 8. When the tension of a rope is 50 N, the speed of the wave on the rope is 40 m/s. What will the speed of the waves be if the tension is increased to 80 N? 9. On a rope with a tension of 50 N, there is a wave described by the equation π y 0, msin 10 m x + 00 s t + 4 a) What is the linear density of the string? b) What is the maximum speed of the rope?.4 Energy in Sine Waves 10. wave has the following characteristics: mplitude: mm Wavelength: 40 cm Wave speed: 50 m/s Length of the wave on the rope: 10 m a) How much energy is in this wave if the linear density of the string is 5 g/m? b) What is the power of the wave if the linear density of the string is 5 g/m? 11. source with a power of 0 W generates a wave with a wavelength of 1.5 cm and a frequency of 00 Hz. What is the amplitude of the wave if the tension of the rope is 80 N? 1. he wave described by this equation y cmsin 10π x 50π t ( ) is travelling along a rope having a linear density of 50 g/m. What is the power of this wave? 018b Version -Mechanical Waves 5 m s
.5 Interference 13. Draw the result of the superposition of these waves. www.kwantlen.ca/science/physics/faculty/mcoombes/p1101_solutions/waves/p1101_13_solutions.htm 14. Draw the resulting wave.5 seconds later. 15. Draw the resulting wave.5 seconds later. www.chegg.com/homework-help/questions-and-answers/wave-pulses-string-approach-timet-0-shown-figure-pulse-isinverted-downward-deflection-st-q14130 www.chegg.com/homework-help/questions-and-answers/wave-pulses-string-approach-timet-0-shown-figure-pulse-isinverted-downward-deflection-st-q14130 018b Version -Mechanical Waves 53
.6 Wave Reflection and ransmission 16. In the situation shown in the figure, a sine wave with a wavelength of 0 cm and an amplitude of 5 mm starts on the left side of the aluminum wire and moves towards the steel wire. www.chegg.com/homework-help/questions-and-answers/figure-aluminum-wire-length-l1-600-cm-cross-sectional-area - 100-10--cm-density -60-g-cm3--q11706 a) What is the wave speed in the aluminum wire? b) What is the frequency of the wave when it is travelling in the aluminum wire? c) What is the wave speed in the steel wire? d) What is the frequency of the wave when it is travelling in the steel wire? e) What is the wavelength of the wave when it is travelling in steel wire? f) What is the impedance of the aluminum wire? g) What is the impedance of the steel wire? h) What is the amplitude of the reflected wave? (Specify if the reflected wave is inverted.) i) What is the amplitude of the transmitted wave? j) What percentage of the power of the initial wave is transmitted in the steel cable? 17. wave on a string arrives at a place where the linear density of the string changes. www.acs.psu.edu/drussell/demos/reflect/reflect.html a) In what direction (upwards or downwards) will the transmitted wave be? b) In what direction (upwards or downwards) will the reflected wave be? c) What will the speed of the transmitted wave be? d) What will the speed of the reflected wave be? 018b Version -Mechanical Waves 54