Presentation on theme: "Types of mechanical waves"— Presentation transcript:
1 Types of mechanical waves Chapter 14: Wave MotionTypes of mechanical wavesMechanical wavesare disturbances that travel through some material or substancecalled medium for the waves.travel through the medium by displacing particles in the mediumtravel in the perpendicular to or along the movement of theparticles or in a combination of bothtransverse waves:waves in a string etc.longitudinal waves:sound waves etc.waves in water etc.
2 Types of mechanical waves (cont’d) Longitudinal and transverse wavessound wave = longitudinal waveC = compressionR = rarefactionair compressedair rarefied
3 Types of mechanical waves (cont’d) Longitudinal-transverse waves
4 Types of mechanical waves (cont’d) Periodic wavesWhen particles of the medium in a wave undergo periodicmotion as the wave propagates, the wave is called periodic.lwavelengthAamplitudet=0x=0xt=T/4t=Tperiod
5 Mathematical description of a wave Wave functionThe wave function describes the displacement of particlesin a wave as a function of time and their positions:A sinusoidal wave is described by the wave function:sinusoidal wave moving in+x directionangular frequencyvelocity of wave, NOT ofparticles of the mediumperiodwavelengthsinusoidal wave moving in-x directionv->-vphase velocity
6 Mathematical description of a wave (cont’d) Wave function (cont’d)lwavelengtht=0xx=0t=T/4t=Tperiod
7 Mathematical description of a wave (cont’d) Wave number and phase velocitywave number:phaseThe speed of wave is the speed with which we have tomove along a point of a given phase. So for a fixed phase,phase velocity
8 Mathematical description of a wave (cont’d) Particle velocity and acceleration in a sinusoidal waveu in textbookvelocityaccelerationAlsowave equation
9 Mathematical description of a wave (cont’d) General solution to the wave equationwave equationSolutions:such asThe most general form of the solution:
10 Speed of a transverse wave Wave speed on a stringConsider a small segment of string whoselength in the equilibrium position isThe mass of the segment isThe x component of the force (tension) at bothends have equal in magnitude and opposite indirection because this is a transverse wave.The total y component of the forces is:Newton’s 2nd lawmassacceleration
11 Speed of a transverse wave (cont’d) Wave speed on a string (cont’d)The total y component of the forces is:wave eq.
12 Energy in wave motion At point a, the force does work on the Total energy of a short string segment of massAt point a, the forcedoes work on thestring segment right of point a.work donePower is the rate of work done :aPmax
13 Energy in wave motion (cont’d) Maximum power of a sinusoidal wave on a string:Average power of a sinusoidal wave on a stringThe average ofover a period:The average power:
14 Wave intensity source: power/unit area Wave intensity for a three dimensional wave from a pointsource:power/unit area
15 Wave interference, boundary condition, and superposition The principle of superpositionWhen two waves overlap, the actual displacement of anypoint at any time is obtained by adding the displacementthe point would have if only the first wave were present andthe displacement it would have if only the second wave werepresent:
16 Wave interference, boundary condition, and superposition (cont’d) Constructive interference (positive-positive or negative-negative)Destructive interference (positive-negative)
17 Wave interference, boundary condition, and superposition (cont’d) Reflectionincident wavereflected waveFree endFor x<xBAt x=xBVertical component of the forceat the boundary is zero.
18 Wave interference, boundary condition, and superposition (cont’d) Reflection (cont’d)Fixed endFor x<xBAt x=xBDisplacement at the boundary is zero.
19 Wave interference, boundary condition, and superposition (cont’d) Reflection (cont’d)At high/low density
20 Wave interference, boundary condition, and superposition (cont’d) Reflection (cont’d)At low/high density
21 Standing waves on a string Superposition of two waves moving in the same directionSuperposition of two waves moving in the opposite direction
22 Standing waves on a string (cont’d) Superposition of two waves moving in the opposite directioncreates a standing wave when two waves have the samespeed and wavelength.incidentreflectedN=node, AN=antinode
23 Normal modes of a string There are infinite numbers of modes of standing wavesfunda-mentalfirstovertonesecondovertonethirdovertoneLfixed endfixed end
24 Sound waves Sound is a longitudinal wave in a medium The simplest sound waves are sinusoidal waves whichhave definite frequency, amplitude and wavelength.The audible range of frequency is between 20 and 20,000 Hz.
25 Sound waves (cont’d) Sinusoidal sound wave function: Change of volume: Sound wave (sinusoidal wave)Sinusoidal sound wave function:Change of volume:undisturbedcyl. of airdisturbedcyl. of airPressure:pressurebulk modulusSDxxx+Dx
26 Pressure amplitude and ear Pressure amplitude for a sinusoidal sound wavePressure:Pressure amplitude:Ear
27 Perception of sound waves Fourier’s theorem and frequency spectrumFourier’s theorem:Any periodic function of period T can be written asfundamental freq.whereImplication of Fourier’s theorem:
28 Perception of sound waves Timbre or tone color or tone qualityFrequency spectrumnoisemusicpianopiano
29 Speed of sound waves (ref. only) velocity of waveThe speed of sound waves in a fluid in a pipemovable pistonlongitudinal momentum carriedby the fluid in motionfluid inequilibriumoriginal volume of the fluid inmotionvelocityof fluidchange in volume of the fluidin motionbulk modulus B:-pressure change/frac. vol. changechange in pressure in the fluidin motionfluid in motionfluid at restboundary moves at speed of wave
30 Speed of sound waves (ref. only) (cont’d) The speed of sound waves in a fluid in a pipe (cont’d)longitudinal impulse = change in momentumspeed of a longitudinalwave in a fluidThe speed of sound waves in a solid bar/rodYoung’s modulus
31 Speed of sound waves (cont’d) The speed of sound waves in gasesbulk modulus of a gasratio of heat capacitiesequilibrium pressure of gasIn textbook- P in textbook(backgroundpressure).- r densityspeed of a longitudinalwave in a fluidgas constant J/(mol K)temperature in Kelvinmolar mass
32 Sound level (Decibel scale) As the sensitivity of the ear covers a broad range of intensities,it is best to use logarithmic scale:Definition of sound intensity:( unit decibel or dB)Sound intensity in dBIntensity (W/m2)Military jet plane at 30 m140102Threshold of pain1201Whisper2010-10Hearing thres. (100Hz)10-12
33 Standing sound wavesSound wave in a pipe with two open ends
34 Standing sound wavesStanding sound wave in a pipe with two open ends
35 Standing sound wavesSound wave in a pipe with one closed and one open end
36 Standing sound waves Displacement Standing wave in a pipe with two closed endsDisplacement
37 Normal modesNormal modes in a pipe with two open ends2nd normal mode
38 Normal modes (stopped pipe) Normal modes in a pipe with an open and a closed end(stopped pipe)
39 ResonanceResonanceWhen we apply a periodically varying force to a system that canoscillate, the system is forced to oscillate with a frequency equalto the frequency of the applied force (driving frequency): forcedoscillation. When the applied frequency is close to a characteristicfrequency of the system, a phenomenon called resonance occurs.Resonance also occurs when aperiodically varying force is appliedto a system with normal modes.When the frequency of the appliedforce is close to one of normalmodes of the system, resonanceoccurs.
40 Interference of waves destructive constructive d2 d1 Two sound waves interfere each otherdestructiveconstructived2d1
41 Beats Two waves with different frequency create a beat Two interfering sound waves can make beatTwo waves with differentfrequency create a beatbecause of interferencebetween them. The beatfrequency is the differenceof the two frequencies.
42 Beats (cont’d) Suppose the two waves have frequencies and Two interfering sound waves can make beat (cont’d)Suppose the two waves have frequenciesandFor simplicity, consider two sinusoidal waves of equal intensity:Then the resulting combined wave will be:As human ears does not distinguish negative and positive amplitude,they hear two max. or min. intensity per cycle, so 2 x (1/2)|fa-fb|=|fa-fb| is the beat frequency fbeat.
43 Doppler effect Source at rest Source at rest Listener moving left Moving listenerSource at restListener moving leftSource at restListener moving right
44 Doppler effect (cont’d) Moving listener (cont’d)The wavelength of the sound wave does not change whetherthe listener is moving or not.The time that two subsequent wave crests pass the listenerchanges when the listener is moving, which effectively changesthe velocity of sound.freq. listener hearsfreq. source generatesvelocity of sound at source- for a listener moving away from+ for a listener moving towardsthe source.velocity of listener
45 Doppler effect (cont’d) Moving sourceWhen the source moves
46 Doppler effect (cont’d) Moving source (cont’d)The wave velocity relative to the wave medium does notchange even when the source is moving.The wavelength, however, changes when the source is moving.This is because, when the source generates the next crest, thethe distance between the previous and next crest i.e. the wave-length changed by the speed of the source.The source at rest When the source is moving+ for a receding source- for a approaching source
47 Doppler effect (cont’d) Moving source and listener- for a listener moving away from+ for a listener moving towardsthe source.+ for a receding source- for a approaching sourceThe signs of vL and vS are measuredin the direction from the listener L to thesource S.Effect of change of source speed
48 Doppler effect (cont’d) Example 1A police siren emits a sinusoidal wave with frequency fs=300 Hz.The speed of sound is 340 m/s. a) Find the wavelength of the wavesif the siren is at rest in the air, b) if the siren is moving at 30 m/s, findthe wavelengths of the waves ahead of and behind the source.a)b) In front of the siren:Behind the siren:
49 Doppler effect (cont’d) Example 2If a listener l is at rest and the siren in Example 1 is moving awayfrom L at 30 m/s, what frequency does the listener hear?Example 3If the siren is at rest and the listener is moving toward the left at 30m/s, what frequency does the listener hear?
50 Doppler effect (cont’d) Example 4If the siren is moving away from the listener with a speed of 45 m/srelative to the air and the listener is moving toward the siren with aspeed of 15 m/s relative to the air, what frequency does the listenerhear?Example 5The police car with its 300-MHz siren is moving toward a warehouseat 30 m/s, intending to crash through the door. What frequency doesthe driver of the police car hear reflected from the warehouse?Freq. reachingthe warehouseFreq. heard bythe driver
51 Exercises Problem 1 A transverse wave on a rope is given by: (a) Find the amplitude, period, frequency, wavelength, and speed ofpropagation. (b) Sketch the shape of the rope at the following valuesof t : s, and s. (c) Is the wave traveling in the +x or –xdirection? (d) The mass per unit length of the rope is kg/m.Find the tension. (e) Find the average power of this wave.SolutionA=0.75 cm, l=2/0.400 = 5.00 cm, f=125 Hz, T=1/f= s andv=lf=6.25 m/s.(b) Homework(c) To stay with a wave front as t increases, x decreases. Therefore thewave is moving in –x direction.(d) , the tension is(e)
52 ExercisesProblem 2A triangular wave pulse on a taut string travels in the positive +x directionwith speed v. The tension in the string is F and the linear mass density ofthe string is m. At t=0 the shape of the pulse I given byDraw the pulse at t=0. (b) Determine the wave function y(x,t) at alltimes t. (c) Find the instantaneous power in the wave. Show that thepower is zero except for –L < (x-vt) < L and that in this interval thepower is constant. Find the value of this constant.Solutiony(a)hL-Lx
53 Exercises Problem 2 (cont’d) Solution (b) The wave moves in the +x direction with speed v, so in the experessionfor y(x,0) replace x with –vt:(c)Thus the instantaneous power is zero except for –L < (x-vt) < L whereIt has the constant value Fv(h/L)2.
54 ExercisesProblem 3The sound from a trumpet radiates uniformly in all directions in air. At adistance of 5.00 m from the trumpet the sound intensity level is 52.0 dB.At what distance is the sound intensity level 30.0 dB?SolutionThe distance is proportional to the reciprocal of the square root of theintensity and hence to 10 raised to half of the sound intensity levelsdivided by 10:
55 ExercisesProblem 4An organ pipe has two successive harmonics with frequencies 1,372 and1,764 Hz. (a) Is this an open or stopped pipe? (b) What two harmonics arethese? (c) What is the length of the pipe?SolutionFor an open pipe, the difference between successive frequencies isthe fundamental, in this case 392 Hz, and all frequencies are integermultiples of this frequency. If this is not the case, the pipe cannot bean open pipe. For a stopped pipe, the difference between the successivefrequencies is twice the fundamental, and each frequency is an oddinteger multiple of the fundamental. In this case, f1 = 196 Hz, and1372 Hz = 7f1 , 1764 Hz = 9f1 . So this is a stopped pipe.(b) n=7 for 1,372 Hz, n=9 for 1,764 Hz.(c) so
56 Exercises Problem 5 Two identical loudspeakers are located at points A and B, 2.00 m apart. The loud-speakers are driven by the same amplifierand produce sound waves with a frequencyof 784 Hz. Take the speed of sound in air tobe 344 m/s. A small microphone is moved outfrom Point B along a line perpendicular to theline connecting A and B. (a) At what distancesfrom B will there be destructive interference?(b) At what distances from B will there beconstructive interference? (c) If the frequencyis made low enough, there will be no positionsalong the line BC at which destructiveinterference occurs. How low must thefrequency be for this to be the case?A2.00 mBCx
57 Exercises Problem 5 Solution (a) If the separation of the speakers is denoted by h, the condition fordestructive interference iswhere is an odd multiple of one-half. Adding x to both sides, squaring,canceling the x2 term from both sides and solving for x gives:Using and h from the given data yields:(b) Repeating the above argument for integral values for , constructiveinterference occurs at 4.34 m, 1.84 m, 0.86 m, 0.26 m.(c) If , there will be destructive interference at speaker B.If , the path difference can never be as large asThe minimum frequency is then v/(2h)=(344 m/s)/(4.0 m)=86 Hz.
58 ExercisesProblem 6A 2.00 MHz sound wave travels through a pregnant woman’s abdomenand is reflected from fetal heart wall of her unborn baby. The heart wall ismoving toward the sound receiver as the heart beats. The reflected soundis then mixed with the transmitted sound, and 5 beats per second aredetected. The speed of sound in body tissue is 1,500 m/s. Calculate thespeed of the fetal heart wall at the instance this measurement is made.SolutionLet f0=2.00 MHz be the frequency of the generated wave. The frequencywith which the heart wall receives this wave is fH=[(v+vH)/v]f0, and this isalso the frequency with which the heart wall re-emits the wave. The detectedfrequency of this reflected wave is f’=[v/(v-vH )]fH, with the minus sign indicatingthat the heart wall, acting now as a source of waves, is moving toward thereceiver. Now combining f’=[(v+vH)/(v-vH)]f0, and the beat frequency is:Solving for vH ,