Goals: Lecture 28 Chapter 20 Employ the wae model Visualize wae motion Analyze functions of two ariables Know the properties of sinusoidal waes, including waelength, wae number, phase, and frequency. Work with a few important characteristics of sound waes. (e.g., Doppler effect) Assignment HW12, Due Tuesday, May 4 th HW13, Due Friday, May 7 th For Tuesday, Read through all of Chapter 21 Physics 207: Lecture 28, Pg 1 Waes A traeling wae is an organized disturbance propagating at a well-defined wae speed. In transerse waes the particles of the medium moe perpendicular to the direction of wae propagation. In longitudinal waes the particles of the medium moe parallel to the direction of wae propagation. A wae transfers energy, but no material or substance is transferred outward from the source. Physics 207: Lecture 28, Pg 2 Page 1
Types of Waes Mechanical waes trael through a material medium such as water or air. Electromagnetic waes require no material medium and can trael through acuum. Matter waes describe the wae-like characteristics of atomicleel particles. For mechanical waes, the speed of the wae is a property of the medium. Speed does not depend on the size or shape of the wae. Examples: Sound waes (air moes locally back & forth) Stadium waes (people moe up & down no energy transfer) Water waes (water moes up & down) Light waes (an oscillating electromagnetic field) Physics 207: Lecture 28, Pg 3 Wae Graphs The displacement D of a wae is a function of both position (where) and time (when). A snapshot graph shows the wae s displacement as a function of position at a single instant of time. A history graph shows the wae s displacement as a function of time at a single point in space. The displacement, D, is a function of two ariables, x and t, or D(x,t) Physics 207: Lecture 28, Pg 4 Page 2
Wae Speed Speed of a transerse, mechanical wae on a string: = elastic property inertial property = T s µ µ = m L where T s is the string tension and µ is linear string density Speed of sound (longitudinal mechanical wae) in air at 20 C = 343 m / s Speed of light (transerse, EM wae) in acuum: c = 3x10 8 m/s Speed of light (transerse, EM wae) in a medium: = c / n where n = index of refraction of the medium (typically 1 to 4) Physics 207: Lecture 28, Pg 5 Wae Forms We will examine continuous waes that extend foreer in each direction! We can also hae pulses caused by a brief disturbance of the medium: And pulse trains which are somewhere in between. Physics 207: Lecture 28, Pg 6 Page 3
Continuous Sinusoidal Wae Waelength: The distance λ between identical points on the wae. Amplitude: The maximum displacement A of a point on the wae. Waelength λ A Animation Physics 207: Lecture 28, Pg 7 Wae Properties... Period: The time T for a point on the wae to undergo one complete oscillation. Speed: The wae displaces one waelength λ in one period T so its speed is = λ / T. = λ T Animation Physics 207: Lecture 28, Pg 8 Page 4
Exercise Wae Motion The speed of sound in air is a bit oer 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 wae and a light wae that both hae a waelength of 3 meters. What is the ratio of the frequency of the light wae to that of the sound wae? (Recall = λ / T = λ f ) (A) About 1,000,000 (B) About 0.000,001 (C) About 1000 Physics 207: Lecture 28, Pg 9 D Wae Properties ( 0 ( 0 x, t) = A cos[( 2π ( x / λ t / T ) + φ ] D x, t) = A cos[ kx ω t + φ ] A = amplitude k 2π/λ = wae number ω = 2πf = angular frequency φ 0 = phase constant Look at the spatial part (Let t =0). Waelength = λ D( x,0) Acos[( 2π / λ ) x)] D x = 0 D = A x = λ/4 D = A cos(π/2) = 0 x = λ/2 D = A cos(π) = -A A Physics 207: Lecture 28, Pg 10 x Page 5
Look at the temporal (time-dependent) part D ( x, t) = A cos[( 2π / λ ) x ωt )] Let x = 0 D( 0, t) = A cos( ωt) = Acos[ (2π / T ) t] t = 0 D = A Period Τ D t =T / 4 D = A cos(-π/2) = 0 t =T / 2 D = A cos(-π) = -A A t Animation Physics 207: Lecture 28, Pg 11 Look at the temporal (time-dependent) part D ( x, t) = A cos[( 2π / λ ) x ωt )] Let x = 0 D( 0, t) = A cos( ωt) = Acos[ (2π / T ) t] t = 0 D = A t =T / 4 D = A cos(-π/2) = 0 t =T / 2 D = A cos(-π) = -A D λ A Period Τ D t x Animation Physics 207: Lecture 28, Pg 12 Page 6
Exercise Wae Motion A harmonic wae moing in the positie x direction can be described by the equation D(x,t) = A cos ( (2π / λ) x - ωt ) = A cos (k x ω t ) = 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 wae traeling in the negatie 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 13 Exercise Wae Motion A boat is moored in a fixed location, and waes make it moe up and down. If the spacing between wae crests is 20 meters and the speed of the waes 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 = λ / T = λ f ) (A) 2 sec (B) 4 sec (C) 8 sec t t + t Physics 207: Lecture 28, Pg 14 Page 7
Exercise Wae Motion A boat is moored in a fixed location, and waes make it moe up and down. If the spacing between wae crests is 20 meters and the speed of the waes 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 waelength (A) 2 sec (B) 4 sec (C) 8 sec t t + t Physics 207: Lecture 28, Pg 15 Speed of Waes The speed of sound waes 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 waes follows a general form: = elastic inertial property property Waes on a string = T µ Physics 207: Lecture 28, Pg 16 Page 8
Waes on a string... So we find: = F µ Animation tension F mass per unit length µ Making the tension bigger increases the speed. Making the string heaier decreases the speed. The speed depends only on the nature of the medium, not on amplitude, frequency etc of the wae. Physics 207: Lecture 28, Pg 17 Exercise Wae Motion A heay rope hangs from the ceiling, and a small amplitude transerse wae is started by jiggling the rope at the bottom. As the wae traels up the rope, its speed will: (a) increase (b) decrease (c) stay the same Physics 207: Lecture 28, Pg 18 Page 9
Sound, A special kind of longitudinal wae Consider a ibrating guitar string String Vibrates Piece of string undergoes harmonic motion Animation Air molecules alternatiely compressed and rarefied Physics 207: Lecture 28, Pg 19 Sound Consider the actual air molecules and their motion ersus time, time 0 time 1 time 2 Indiidual molecules undergo harmonic motion with displacement in same direction as wae motion. Physics 207: Lecture 28, Pg 20 Page 10
Speed of Sound in a Solid Rod The Young s modulus of the material is Y The density of the material is ρ The speed of sound in the rod is = Y ρ Speed of Sound in Liquid or Gas The bulk modulus of the material is B The density of the material is ρ The speed of sound in that medium is = B ρ Medium Air Helium Water Steel (solid) Speed (m/s) 343 972 1500 5600 Physics 207: Lecture 28, Pg 21 Speed of Sound in Air The speed of sound also depends on the temperature of the medium This is particularly important with gases For air, the relationship between the speed and temperature (if pressure is constant) is = (331 m/s) Tc 1+ o 273 C (331 m/s is the speed at 0 o C) T C is the air temperature in Centigrade Physics 207: Lecture 28, Pg 22 Page 11
Home Exercise Comparing Waes, He s. Air A sound wae haing frequency f 0, speed 0 and waelength λ 0, is traeling through air when in encounters a large helium-filled balloon. Inside the balloon the frequency of the wae is f 1, its speed is 1, and its waelength is λ 1 Compare the speed of the sound wae inside and outside the balloon (A) 1 < 0 (B) 1 = 0 (C) 1 > 0 Compare the frequency of the sound wae inside and outside the balloon (A) f 1 < f 0 (B) f 1 = f 0 (C) f 1 > f 0 Compare the waelength of the sound wae inside and outside the balloon (A) λ 1 < λ 0 (B) λ 1 = λ 0 (C) λ 1 > λ 0 Physics 207: Lecture 28, Pg 23 Waes, Wae fronts, and Rays Note that a small portion of a spherical wae front is well represented as a plane wae. Physics 207: Lecture 28, Pg 24 Page 12
Waes, Wae fronts, and Rays If the power output of a source is constant, the total power of any wae front is constant. I = Pa A = P 4πR a 2 Pa P I = = a A const Physics 207: Lecture 28, Pg 25 Exercise Spherical Waes You are standing 10 m away from a ery loud, small speaker. The noise hurts your ears. In order to reduce the intensity to 1/4 its original alue, how far away do you need to stand? (A) 14 m (B) 20 m (C) 30 m (D) 40 m Physics 207: Lecture 28, Pg 26 Page 13
Intensity of sounds Intensity of a sound wae is 2 Pmax 2ρ Proportional to (amplitude) 2 This is a general result (not only for sound) Threshold of human hearing: I 0 = 10-12 W/m 2 The range of intensities detectible by the human ear is ery large It is conenient to use a logarithmic scale to determine the intensity leel, β β = 10 I = log 10 I I 0 Physics 207: Lecture 28, Pg 27 Intensity of sounds I 0 is called the reference intensity It is taken to be the threshold of hearing I 0 = 1.00 x 10-12 W/ m 2 I is the intensity of the sound whose leel is to be determined β is in decibels (db) Threshold of pain: I = 1.00 W/m 2 ; β = 120 db Threshold of hearing: I 0 = 1.00 x 10-12 W/ m 2 ; β = 0 db Physics 207: Lecture 28, Pg 28 Page 14
Intensity of sounds Some examples (1 pascal 10-5 atm) : Sound Intensity Pressure Intensity (W/m 2 ) Leel (db) Hearing threshold 3 10-5 10-12 0 Classroom 0.01 10-7 50 Indoor concert 30 1 120 Jet engine at 30 m 100 10 130 Physics 207: Lecture 28, Pg 29 Sound Leel, Example What is the sound leel that corresponds to an intensity of 2.0 x 10-7 W/m 2? β = 10 log 10 (2.0 x 10-7 W/m 2 / 1.0 x 10-12 W/m 2 ) = 10 log 10 2.0 x 10 5 = 53 db Rule of thumb: An apparent doubling in the loudness is approximately equialent to an increase of 10 db. dbs are not linear with intensity Physics 207: Lecture 28, Pg 30 Page 15
Loudness and Intensity Sound leel in decibels relates to a physical measurement of the strength of a sound We can also describe a psychological measurement of the strength of a sound Our bodies calibrate a sound by comparing it to a reference sound This would be the threshold of hearing Actually, the threshold of hearing is this alue for 1000 Hz Physics 207: Lecture 28, Pg 31 Lecture 28 Assignment HW12, Due Tuesday, May 4 th HW13, Due Friday, May 7 th For Tuesday, Read through all of Chapter 21 Physics 207: Lecture 28, Pg 32 Page 16