Page 1Physics 207 – Lecture 27Physics 207: Lecture 27, Pg 1Lecture 27, Dec. 3Goals: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 HW11, Due Friday, Dec. 5th HW12, Due Friday, Dec. 12th For Monday, Read through all of Chapter 21Physics 207: Lecture 27, 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 27Physics 207: Lecture 27, Pg 3Energy is transported in wave but the motion of matter is localPhysics 207: Lecture 27, Pg 4Types 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) Water waves (water moves up & down) Light waves (an oscillating electromagnetic field)Page 3Physics 207 – Lecture 27Physics 207: Lecture 27, Pg 5Wave 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)Physics 207: Lecture 27, Pg 6Wave 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=µPage 4Physics 207 – Lecture 27Physics 207: Lecture 27, Pg 7Wave Forms So far we have examined “continuous wavescontinuous waves” that go on 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.Physics 207: Lecture 27, Pg 8Continuous Sinusoidal WaveλWavelength Wavelength: The distance λ between identical points on the wave. Amplitude: The maximum displacement A of a point on thewave.AAnimationPage 5Physics 207 – Lecture 27Physics 207: Lecture 27, Pg 9Wave Properties... Period: The time T for a point on the wave to undergo one complete oscillation. Speed: The wave moves one wavelength λ in one period T so its speed is v = λ / T.Tλ=vAnimationPhysics 207: Lecture 27, Pg 10Exercise Wave Motion The speed of sound in air is a bit over 300 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 1000Page 6Physics 207 – Lecture 27Physics 207: Lecture 27, Pg 11Wave PropertiesLook at the spatial part (Let t =0).Animation])//(2cos[(),(0φλπ+−=TtxAtxD)] )/2cos[()0,(xAxDλπ=λWavelengthAyx• x = 0 y = A• x = λ/4 y = A cos(π/2) = 0• x = λ/2 y = A cos(π) = -A]cos[),(0φω+−=tkxAtxDA = amplitude k = 2π/λ = wave numberω = 2πf = angular frequency φ0= phase constantPhysics 207: Lecture 27, Pg 12Look at the temporal (time-dependent) part Let x = 0)] )/2cos[(),(txAtxDωλπ−=ΤPeriodAyt] )/2(cos[)cos(),0(tTAtAtDπω−=−=• t = 0 y = A• t =T / 4 y = A cos(-π/2) = 0• t =T / 2 y = A cos(-π) = -APage 7Physics 207 – Lecture 27Physics 207: Lecture 27, Pg 13Exercise Wave Motion A harmonic wave moving in the positive x directioncan be described by the equation (The wave varies in space and time.) v = λ / T = λ f = (λ/2π ) (2π f) = ω / k and, by definition, ω > 0D(x,t) = A cos ( (2π/λ) x - ωt ) = A cos (k x – ω t ) Which of the following equation describes a harmonic wave moving in the negative x direction ?(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 27, 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 27Physics 207: Lecture 27, 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 27, Pg 16Sound, A special kind of longitudinal waveConsider a vibrating guitar stringString VibratesPiece of string undergoes harmonic motionAir molecules alternatively compressed and rarefied AnimationPage 9Physics 207 – Lecture 27Physics 207: Lecture 27, Pg 17Soundtime 0time 1time 2Individual molecules undergo harmonic motion with displacement in same direction as wave motion.Consider the actual air molecules and their motion versus time,Physics 207: Lecture 27, Pg 18Speed of Sound Waves, General 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
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