Physics 207: Lecture 29, Pg 1 Lecture 29Goals: Chapter 20, Waves Chapter 20, Waves Final test review on Wednesday. Final exam on Monday, Dec 20, at 5:00 pm. HW 11 due Wednesday.

Physics 207: Lecture 29, Pg 2 Relationship between wavelength and period D(x,t=0) x v x0x0 T= /v

Physics 207: Lecture 29, Pg 3 Exercise 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 T =  / v ) (A) 2 sec (B) 4 sec (C) 8 sec t t +  t

Physics 207: Lecture 29, Pg 4 Mathematical formalism D(x=0,t) t T D(0,t) ~ A cos (  t +  )  angular frequency   D(x,t=0) x λ D(x,0) ~ A cos ( kx +  )  k  wave number  k 

Physics 207: Lecture 29, Pg 5 Mathematical formalism D(x,t) = A cos (kx -  t +  ) A : Amplitude k : wave number  : angular frequency  phase constant l The two dimensional displacement function for a sinusoidal wave traveling along +x direction:

Physics 207: Lecture 29, Pg 6 Mathematical formalism l Note that there are equivalent ways of describing a wave propagating in +x direction: D(x,t) = A cos (kx -  t +  ) D(x,t) = A sin (kx -  t +  ) D(x,t) = A cos [k(x – v t) + 

Physics 207: Lecture 29, Pg 7 Why the minus sign? l As time progresses, we need the disturbance to move towards +x: at t=0, D(x,t=0) = A cos [k(x-0) +  at t=t 0, D(x,t=t 0 ) = A cos [k(x- v t 0 ) +  x vt 0 v

Physics 207: Lecture 29, Pg 8 l Which of the following equations describe a wave propagating towards -x: A)D(x,t) = A cos (kx –  t  B)D(x,t) = A sin (kx –  t  C)  D(x,t) = A cos (-kx +  t  D  D(x,t) = A cos (kx +  t 

Physics 207: Lecture 29, Pg 9 Speed of waves l The speed of mechanical waves depend on the elastic and inertial properties of the medium. l For a string, the speed of the wave can be shown to be: T string : tension in the string  L : mass per unit length

Physics 207: Lecture 29, Pg 10 Waves on a string l Making the tension bigger increases the speed. l Making the string heavier decreases the speed. l The speed depends only on the nature of the medium, not on amplitude, frequency etc of the wave.

Physics 207: Lecture 29, Pg 11 Exercise Wave Motion l A heavy rope hangs from the ceiling, and a small amplitude transverse wave is started by jiggling the rope at the bottom. l As the wave travels up the rope, its speed will: (a) increase (b) decrease (c) stay the same v

Physics 207: Lecture 29, Pg 12 Sound, A special kind of longitudinal wave Individual molecules undergo harmonic motion with displacement in same direction as wave motion. λ

Physics 207: Lecture 29, Pg 13 Waves in two and three dimensions l Waves on the surface of water: circular waves wavefront

Physics 207: Lecture 29, Pg 14 Plane waves l Note that a small portion of a spherical wave front is well represented as a “plane wave”.

Physics 207: Lecture 29, Pg 15 Intensity (power per unit area) I=P/A : J/(s m 2 ) l A wave can be made more “intense” by focusing to a smaller area. R

Physics 207: Lecture 29, Pg 16 l You 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 m

Physics 207: Lecture 29, Pg 17 Intensity of sounds l The range of intensities detectible by the human ear is very large It is convenient to use a logarithmic scale to determine the intensity level,  I 0 : threshold of human hearing I 0 = W/m 2

Physics 207: Lecture 29, Pg 18 Intensity of sounds l Some examples (1 pascal  atm) : Sound Intensity PressureIntensity (W/m 2 ) Level (dB) Hearing threshold 3  Classroom Indoor concert Jet engine at 30 m