Page Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 1Lecture 29Goals:Goals:••Chapter 20, WavesChapter 20, WavesNo labs this week.HW 11 is due tomorrow night.HW 12 (a short one) is due Thursday night. Physics 207: Lecture 29, Pg 2Relationship between wavelength and periodD(x,t=0)xλvx0T=λ/vPage Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 3Mathematical formalismD(x=0,t)tTD(x=0,t) ~ A cos (ωt + φ) ω: angular frequency ω=2π/ΤD(x,t=0)xλD(x,t=0) ~ A cos (kx+ φ) k: wave number k=2π/λPhysics 207: Lecture 29, Pg 4Mathematical formalismD(x,t) = A cos (kx -ωt + φ)A: Amplitudek: wave numberω : angular frequencyϕ: phase constantlThe two dimensional displacement function for a sinusoidal wave traveling along +x direction:Page Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 5Mathematical formalismlNote that there are equivalent ways of describing a wave propagating in +x direction:D(x,t) = A cos (kx -ωt + φ)D(x,t) = A sin (kx -ωt + φ+π/2)D(x,t) = A cos [k(x – vt) + φ]Physics 207: Lecture 29, Pg 6Why the minus sign?lAs time progresses, we need the disturbance to move towards +x:at t=0, D(x,t=0) = A cos [k(x-0) + φ]at t=t0, D(x,t=t0) = A cos [k(x-vt0) + φ]xvt0vPage Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 7lWhich of the following equations describe a wave propagating towards -x:A) D(x,t) = A cos (kx –ωt )B) D(x,t) = A sin (kx –ωt )C) D(x,t) = A cos (-kx + ωt )D) D(x,t) = A cos (kx + ωt )Physics 207: Lecture 29, Pg 8Speed of waveslThe speed of mechanical waves depend on the elastic and inertial properties of the medium.lFor a string, the speed of the wave can be shown to be:v =TstringµTstring: tension in the stringμ=M/L : mass per unit lengthPage Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 9Waves on a stringv =TstringµlMaking the tension bigger increases the speed.lMaking the string heavier decreases the speed.lThe speed depends only on the nature of the medium, not on amplitude, frequency etc of the wave.Physics 207: Lecture 29, Pg 10Exercise Wave MotionlA heavy rope hangs from the ceiling, and a small amplitude transverse wave is started by jiggling the rope at the bottom. lAs the wave travels up the rope, its speed will:(a) increase(b) decrease(c) stay the samevPage Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 11Sound, A special kind of longitudinal waveIndividual molecules undergo harmonic motion with displacement in same direction as wave motion.λPhysics 207: Lecture 29, Pg 12Waves in two and three dimensionslWaves on the surface of water:circular waveswavefrontPage Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 13Plane waveslNote that a small portion of a spherical wave front is well represented as a “plane wave”. Physics 207: Lecture 29, Pg 14Intensity (power per unit area)I=P/A : J/(s m2) lA wave can be made more “intense” by focusing to a smaller area. RI =Psource4πR2Page Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 15lYou are standing 10 m away from a very loud, small speaker. The noise hurts your ears. In order to reduce the intensity to 1/4 its original value, how far away do you need to stand? Exercise Spherical Waves(A) 14 m (B) 20 m (C) 30 m (D) 40 mPhysics 207: Lecture 29, Pg 16Intensity of soundslThe range of intensities detectible by the human ear is very largelIt is convenient to use a logarithmic scale to determine the intensity level,ββ=10log10II0 I0: threshold of human hearingI0=10-12 W/m2Page Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 17Intensity of sounds lSome examples (1 pascal ≅ 10-5atm) :Sound IntensityPressure Intensity (W/m2)Level (dB) Hearing threshold3 × 10-510-120Classroom 0.01 10-750Indoor concert30 1 120Jet engine at 30 m100 10 130Physics 207: Lecture 29, Pg 18The Doppler effect lThe frequency of the wave that is observed depends on the relative speed between the observer and the source. observerPage Physics 207 – Lecture 29Physics 207: Lecture 29, Pg 19vsobserverPhysics 207: Lecture 29, Pg 20The Doppler effect f=f0/(1-vs/v)lApproaching source:lReceding
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