Page 1Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 1Lecture 28Goals:Goals:••Chapter 20Chapter 20 Employ the wave model Visualize wave motion Analyze functions of two variables Know the properties of sinusoidal waves, including wavelength, wave number, phase, and frequency. Work with a few important characteristics of sound waves. (e.g., Doppler effect)••AssignmentAssignment HW12, Due Tuesday, May 4th HW13, Due Friday, May 7th For Tuesday, Read through all of Chapter 21Physics 207: Lecture 28, Pg 2Waves A traveling wave is an organized disturbance propagating at a well-defined wave speed v. In transverse waves the particles of the medium move perpendicular to the direction of wave propagation. In longitudinal waves the particles of the medium move parallelto the direction of wave propagation. A wave transfers energy, but no material or substance is transferred outward from the source.Page 2Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 3Types of Waves Mechanical waves travel through a material medium such as water or air. Electromagnetic waves require no material medium and can travel through vacuum. Matter waves describe the wave-like characteristics of atomic-level particles.For mechanical waves, the speed of the wave is a property of the medium. Speed does not depend on the size or shape of the wave. Examples: Sound waves (air moves locally back & forth) Stadium waves (people move up & down…no energy transfer) Water waves (water moves up & down) Light waves (an oscillating electromagnetic field)Physics 207: Lecture 28, Pg 4Wave Graphs The displacement D of a wave is a function of both position (where) and time (when). A snapshot graph shows the wave’s displacement as a function of position at a single instant of time. A history graph shows the wave’s displacement as a function of time at a single point in space. The displacement, D, is a function of twovariables, x and t, or D(x,t)Page 3Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 5Wave Speed Speed of a transverse, mechanical wave on a string:where Tsis the string tension and µ is linear string density Speed of sound (longitudinal mechanical wave) in air at 20°Cv = 343 m / s Speed of light (transverse, EM wave) in vacuum: c = 3x108 m/s Speed of light (transverse, EM wave) in a medium: v = c / nwhere n = index of refraction of the medium (typically 1 to 4)property inertialproperty elasticv =µsT=vLm=µPhysics 207: Lecture 28, Pg 6Wave Forms We will examine “continuous waves” that extend forever in each direction !vv We can also have “pulses”caused by a brief disturbanceof the medium:v And “pulse trains” which aresomewhere in between.Page 4Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 7Continuous Sinusoidal Wave Wavelength: The distance λ between identical points on the wave. Amplitude: The maximum displacement A of a point on the wave.AnimationλWavelengthAPhysics 207: Lecture 28, Pg 8Wave Properties... Period: The time T for a point on the wave to undergo one complete oscillation. Speed: The wave displaces one wavelength λin one period T so its speed is v = λ / T.Tλ=vAnimationPage 5Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 9Exercise Wave Motion The speed of sound in air is a bit over 300 m/s (i.e., 343 m/s), and the speed of light in air is about 300,000,000 m/s. Suppose we make a sound wave and a light wave that both have a wavelength of 3 meters. What is the ratio of the frequency of the light wave to that of the sound wave ? (Recall v = λ / T = λ f )(A) About 1,000,000(B) About 0.000,001(C) About 1000Physics 207: Lecture 28, Pg 10Wave PropertiesLook at the spatial part (Let t =0).])//(2cos[(),(0φλπ+−=TtxAtxD)] )/2cos[()0,(xAxDλπ=λWavelengthADx• x = 0 D = A• x = λ/4 D = A cos(π/2) = 0• x = λ/2 D = A cos(π) = -A]cos[),(0φω+−=tkxAtxDA = amplitude k 2π/λ = wave numberω = 2πf = angular frequency φ0= phase constantPage 6Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 11Look at the temporal (time-dependent) part Let x = 0)] )/2cos[(),(txAtxDωλπ−=ΤPeriodADt] )/2(cos[)cos(),0(tTAtAtDπω−=−=• t = 0 D = A• t =T / 4 D = A cos(-π/2) = 0• t =T / 2 D = A cos(-π) = -AAnimationPhysics 207: Lecture 28, Pg 12Look at the temporal (time-dependent) part Let x = 0)] )/2cos[(),(txAtxDωλπ−=] )/2(cos[)cos(),0(tTAtAtDπω−=−=ΤPeriodADt• t = 0 D = A• t =T / 4 D = A cos(-π/2) = 0• t =T / 2 D = A cos(-π) = -AλDxAnimationvPage 7Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 13Exercise Wave Motion A harmonic wave moving in the positive x direction can be described by the equation D(x,t) = A cos ( (2π/λ) x - ωt ) = A cos (k x – ω t )v = λ / T = λ f = (λ/2π ) (2π f) = ω / k and, by definition, ω > 0 Which of the following equation do you expect describes a harmonic wave traveling in the negative x direction ? Hint: cos α = cos – α so cos (k x – ω t ) = cos (- k x + ω t ) (A) D(x,t) = A sin ( k x − ωt )(B) D(x,t) = A cos ( k x + ωt )(C) D(x,t) = A cos (−k x + ωt )Physics 207: Lecture 28, Pg 14Exercise Wave Motion A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 meters and the speed of the waves is 5 m/s, how long ∆t does it take the boat to go from the top of a crest to the bottom of a trough ? (Recall v = λ / T = λ f )(A) 2 sec (B) 4 sec (C) 8 sectt + ∆tPage 8Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 15Exercise Wave Motion A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 meters and the speed of the waves is 5 m/s, how long ∆t does it take the boat to go from the top of a crest to the bottom of a trough ? T = 4 sec but crest to trough is half a wavelength(A) 2 sec (B) 4 sec (C) 8 sectt + ∆tPhysics 207: Lecture 28, Pg 16Speed of Waves The speed of sound waves in a medium depends on the compressibility and the density of the medium The compressibility can sometimes be expressed in terms of the elastic modulus of the material The speed of all mechanical waves follows a general form:Waves on a string property inertialproperty elasticv =µT=vPage 9Physics 207 – Lecture 28Physics 207: Lecture 28, Pg 17Waves on a string... So we find:µ=Fv Making the
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