PHYS 1443 – Section 501 Lecture #26AnnouncementsWavesSpeed of Transverse Waves on StringsSpeed of Waves on Strings cont’dExample for Traveling WaveSinusoidal WavesSinusoidal Waves cont’dExample for WavesSinusoidal Waves on StringsRate of Energy Transfer by Sinusoidal Waves on StringsRate of Energy Transfer by Sinusoidal Waves cont’dExample for Wave Energy TransferReflection and TransmissionTransmission Through Different MediaSuperposition Principle of WavesWave InterferencesMonday, May 3, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt1PHYS 1443 – Section 501Lecture #26Monday, May 3, 2004Dr. Andrew Brandt1. Waves2. Sinusoidal Waves3. Rate of Wave Energy Transfer4. Reflection and Transmission5. Superposition and InterferenceMonday, May 3, 2004 2 PHYS 1443-501, Spring 2004Dr. Andrew BrandtAnnouncements•On May 5, I will answer any questions that you may have on any of the material. If you have none it will be a short class—if you can use the time better with independent study that’s fine•Final exam will be Monday May 10 @5:30. It will be multiple choice only, so bring a Scantron form. It will be comprehensive, perhaps with a little more emphasis on ch 10-14.Monday, May 3, 2004 3 PHYS 1443-501, Spring 2004Dr. Andrew BrandtWaves•Waves are closely related to vibrations. • Mechanical waves involve oscillations of a medium and carry energy from one place to another•Two forms of waves–Pulse–Continuous or periodic wave•Wave can be characterized by–Amplitude–Wave length–Period•Two types of waves–Transverse Wave–Longitudinal wave•Sound waveMonday, May 3, 2004 4 PHYS 1443-501, Spring 2004Dr. Andrew BrandtSpeed of Transverse Waves on StringsHow do we determine the speed of a transverse pulse traveling on a string?If a string under tension is pulled sideways and released, the tension is responsible for accelerating a particular segment of the string back to the equilibrium position.The speed of the wave increases.So what happens when the tension increases? Which law is this hypothesis based on?Based on the hypothesis we have laid out above, we can construct a hypothetical formula for the speed of wave For a given tension, acceleration decreases, so the wave speed decreases.Newton’s second law of motionThe acceleration of a particular segment increases Tv Which means? Now what happens when the mass per unit length of the string increases? T: Tension on the string: Unit mass per lengthIs the above expression dimensionally sound?T=[MLT-2], =[ML-1](T/)1/2=[L2T-2]1/2=[LT-1]Monday, May 3, 2004 5 PHYS 1443-501, Spring 2004Dr. Andrew BrandtSpeed of Waves on Strings cont’dLet’s consider a pulse moving right and look at it in the frame that moves along with the the pulse.Since in the reference frame moving with the pulse, the segment is moving to the left with the speed v, and the centripetal acceleration of the segment is What is the mass of the segment when the line density of the string is ?Using the radial force componentNow what are the force components when is small? TTFrO svRRvar2tFmrFTherefore the speed of the pulse isTv coscos TT 0rFsin2T2s 2RR2maRvm2RvR22T2Monday, May 3, 2004 6 PHYS 1443-501, Spring 2004Dr. Andrew BrandtExample for Traveling WaveA uniform cord has a mass of 0.300kg and a length of 6.00m. The cord passes over a pulley and supports a 2.00kg object. Find the speed of a pulse traveling along this cord.Thus the speed of the wave isSince the speed of wave on a string with line density and under the tension T is 2/6.1 98 0.90 0.2 smkgMgT M=2.00kg1.00m5.00mTv The line density ismkgmkg/1000.500.6300.02The tension on the string is provided by the weight of the object. ThereforesmTv /8.191000.56.192Monday, May 3, 2004 7 PHYS 1443-501, Spring 2004Dr. Andrew BrandtA-ASinusoidal WavesEquation of motion of a simple harmonic oscillation is a cosine/sine function.But it does not travel. What is the expression for a travelling wave?2( ) siny x A xpl� �=� �� �The wave form can be described by the y-position (displacement) of a particle of the medium at a location x. First for t=0: Wave LengthWe want the wave form of the wave traveling at the speed v in +x at any given time t( )2( , ) siny x t A x vtpl� �= -� �� �AmplitudexAfter time t the original wave form (crest in this case) has moved to the right by vt thus the wave form at x becomes the wave form which used to be at x-vtvtNote this gives same displacement at x=0, , 2,…Monday, May 3, 2004 8 PHYS 1443-501, Spring 2004Dr. Andrew BrandtSinusoidal Waves cont’dThus the wave form can be rewrittenBy definition, the speed of wave in terms of wave length and period T is Defining, angular wave number k and angular frequency 2kTv( , ) sin 2x ty x t ATpl� �� �= -� �� �� �� �( )( , ) siny x t A kx tw= -The wave form becomesFrequency, f, Tf1Wave speed (or phase velocity) vvGeneralized wave form( )( , ) siny x t A kx tw f= � +T2fTll= =kTravels to leftTravels to rightMonday, May 3, 2004 9 PHYS 1443-501, Spring 2004Dr. Andrew BrandtExample for WavesA sinusoidal wave traveling in the positive x direction has an amplitude of 15.0cm, a wavelength of 40.0cm, and a frequency of 8.00Hz. The vertical displacement of the medium at t=0 and x=0 is also 15.0cm. a) Find the angular wave number k, period T, angular frequency , and speed v of the wave.At x=0 and t=0, y=15.0cm, therefore the phase becomesUsing the definition, angular wave number k is b) Determine the phase constant , and write a general expression of the wave function.mradk /7.1500.540.022sradfT/3.5022Period is sec125.000.811fTAngular frequency is Using period and wave length, the wave speed is smfTv /2.300.8400.0 ( )0.150sin 0.150y f= =Thus the general wave function is 2txtkxAy 3.507.15sin150.0sinsin 1;f = pf =2Monday, May 3, 2004 10 PHYS 1443-501, Spring 2004Dr. Andrew BrandtSinusoidal Waves on StringsLet’s consider the case where a string is being oscillated in the y-direction. The trains of waves generated by the motion will travel through the string, causing the particles in the string
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