PH 201-4A spring 2007 Waves and Sound Lectures 26-27 Chapter 16 (Cutnell & Johnson, Physics 7 th edition) 1
Waves A wave is a vibrational, trembling motion in an elastic, deformable body. The wave is initiated by some external force that acts on some parts of the body and deforms it. The elastic restoring force communicates this initial disturbance from one part of the body to the next, adjacent part. The disturbance gradually spreads outward through the entire elastic body. The elastic body in which the wave propagate is called the medium. When a wave propagates through a medium, the particles in the medium vibrate back and forth, but the medium as a whole does not perform translational ti l motion. 2
Transverse Wave Motion 1 up If we shake one end of the tightly stretched elastic string up and down with a flick of the wrist a disturbance travels along the string. String is regarded as a row of particles connected by small, massless springs. 1 particle up => it will pull the 2 nd particle up. 2 nd up 1 down 2 down Transverse wave motion the direction of the disturbance propagation is perpendicular to the back and forth vibration of the particles. 3
Longitudinal Wave Motion Disturbance generated by pushing the 1 st particle towards the 2 nd Longitudinal wave - direction of disturbance motion is parallel l to the vibration of the particles. 4
The wave is a propagation of the disturbance through the medium without any net displacement of the medium. Consider a horizontal string connected to the mass executing simple harmonic motion in the vertical direction. Each particle transmits the motion to the next particle along the entire length of the string. The resulting wave propagates in the horizontal direction with a velocity v, while any one particle of the string executes simple harmonic motion in the vertical direction. The particle of the string is moving perpendicular to the direction of wave propagation and is not moving in the direction of the wave. A transverse wave is a wave in which the particles of the medium execute simple harmonic motion in a direction perpendicular to its direction of propagation. 5
Characteristics of a simple wave (Transverse wave moving in a horizontal direction) The displacement of any particle of the wave is the displacement of that particle from its equilibrium position and is measured by the vertical distance y. The amplitude of the wave is the maximum value of the displacement is denoted d by A. The wavelength of a wave is the distance in the direction of the propagation in which the wave repeats itself ( λ ) The period ( T ) of a wave is the time it takes for one complete wave to pass a particular point. The frequency ( f ) of a wave is the number of waves passing a particular point per second. f = 1/T Wave equation: v = distance trav./time = λ / T = λf The wave causes a transfer of energy from one point in the medium to another point in 6 the medium without the actual transfer of matter between these points.
Mathematical Presentation of a Wave The wave is periodic in both space and time. The space period is represented by the wavelength λ, and the time by the period T. Consider a snapshot of the wave at t = 0. The displacement of every particle of the string at time t = 0. y = Asinx wave repeats itself for x = 360 = 2π rad y = AsinKx K is a wave number Repeats itself twice in the interval 2π Repeats itself 3 times in the interval 2π The wave repeats itself whenever Kλ = 2π => λ = 2π/K K is the number of waves contained in the interval of 2π 7
Consider the particle located at the point x = 0 and its up and down motion with time. y = Asinωt ωt = 2π sine function repeats itself T = 2π/ω ; f = 1/T ; ω = 2πf The general equation for a wave must represent every point x of the wave at every time t y = Asin(Kx ωt) {kλ = 2π ; ωf = 2π} => ωt = Kλ => ω = Kλ/T ω = Kv 8
y 1 = AsinK(x-vt) wave traveling to the right with a speed v at any time t A little later in time Δt, the wave has moved a distance Δx to the right such that the same point of the wave now has the coordinates x + Δx and t + Δt Then we represent the wave as y 2 = AsinK[(x + Δx) v(t + Δt)] or y 2 = AsinK[(x vt) + (Δx vδt)] In this equation for y 2 is to represent the same wave as y 1, then y 2 = y 1 y 2 = AsinK[(x vt) + (Δx vδt)] = AsinK[(x vt) + (Δx Δx/Δt Δt)] = AsinK[(x vt)] = y 1 => y 2 is the same wave as y 1 only displaced a distance Δx to the right in the time Δt => equation y = AsinK(x vt) or y = Asin(Kx ωt) represents a wave traveling to the right with a velocity of propagation v 9
y 3 is a wave traveling to the left. we will begin by representing it as y 3 = AsinK(x vt) in a time Δt, the wave y 3 moves a distance Δx to the left. The coordinates (x,t) of a point on y 3 now has coordinates x Δx and t + Δt for the same point on y 4. we can now write the new wave as: y 4 = AsinK[(x Δx) v(t + Δt)] = = AsinK[(x vt) + (-Δx vδt)] v = - Δx/Δt -Δx vδt = -Δx (- Δx/Δt)Δt = 0 => wave y 4 represents the same wave as y 3 only it is displaced a distance Δx to the left in the time Δt instead of writing y 3 = AsinK(x vt) as a wave to the left with v as a negative number, it is easier to write the equation for the wave where v is positive. y = AsinK(x + vt) = Asin(Kx + ωt) wave traveling to the left 10
A particular wave is given by y = (0.200m)sin[(0.500m -1 )x (8.20rad/s)t] Find: a) A e) f h) y at x = 10.0m and t = 0.500s b) K f) t c) λ g) v 11
The Speed of a Transverse Wave on a String The wave is moving to the right with a velocity v Observe the wave by moving with the wave at the same velocity v In this reference frame the wave appears stationary. The particles composing the string appear to be moving through the wave to the left at velocity v A small portion of the wave can by approximated by an arc of a circle of radius R Consider a small portion of the string l => θ is small => it is approximated by a mass m moving in uniform circular motion 2R 12
The speed of a transverse wave in a string is given by v = T/(m/l), where T is the tension in the string and m/l is the mass per unit length of the string. 13
Problem 22: The drawing shows a 15.0 kg ball being whirled in a circular path on the end of a string. The motion occurs on a frictionless horizontal table. The angular speed of the ball is ω = 12.0 rad/s. The string has a mass of 0.0230 kg. How much time does it take for a wave on the string to travel from the center of the circle to the ball? 14
The Speed of a Wave on a String Problem 19: To measure the acceleration due to gravity on a distant planet, an astronaut hangs a 0.086 kg ball from the end of a wire. The wire has length of 1.5 m and a linear density of 3.1 x 10-4 kg/m. Using electronic equipment, the astronaut t measures the time for a transverse pulse to travel the length of the wire and obtains a value of 0.083 s. The mass of the wire is negligible compared to the mass of the ball. Determine the acceleration due to gravity. 15
The Speed of a Wave on a String Problem 20: A horizontal wire is under a tension of 315 N and has a mass per unit length of 6.50 x 10-3 kg/m. A transverse wave with an amplitude of 2.50 mm and a frequency of 585 Hz is traveling on this wire. As the wave passes, a particle of the wire moves up and down in simple harmonic motion. Obtain: a) The speed of the wave b) The maximum speed with which the particle moves up and down 16
The Mathematical Description of a Wave Problem 27: The displacement (in meters) of a wave is y = (0.26)sin(πt 3.7πx), where t is in seconds and x is in meters. a) Is the wave traveling in the +x or x direction? b) What is the displacement y when t = 38 s and x = 13 m 17
Damped Oscillations Simple harmonic oscillator: F res = -Kx Pendulum: restoring torque = -mglsinθ Real oscillator, real pendulum: extra force, friction => result reducing amplitude of oscillation from one cycle to the next A gradually decreasing oscillation is called damped harmonic motion Energy decreases due to the work done by the oscillator against friction ΔE Energy loss per cycle ΔE = -(2π/Q)E; Q = quality factor of the oscillator The value of Q roughly coincides with number of cycles the oscillator completes before the oscillations damp away significantly Q of piano string = -1000 18
Forced Oscillations To maintain the oscillations of a damped harmonic oscillator at a constant level it is necessary to exert some extra force on the oscillator. - The energy fed into the oscillator by this new force compensates for the energy lost to friction. - The initial energy for the motion is needed to start the oscillations of any oscillator (extra force) Any extra force exerted on an oscillator is called driving force If the period of the driving force coincides with the period of the natural oscillations of the oscillator, then even a small driving force can gradually build up large amplitudes. Driving force feeds energy into the oscillations. Amplitude increases until friction becomes so large that it inhibits further growth. The buildup of a large amplitude by the action of a driving force in tune with the natural frequency of an oscillator is called resonance or sympathetic oscillation 19
Sound Wave propagating on a string one dimensional wave. Can be represented graphically by a plot of the displacement of the string vs. the distance. Sound waves emerging from a source (a loudspeaker, the horn of a car, a human mouth) spreads outward in all directions, filling the three-dimensional volume of air surrounding the source. Wave fronts Locations of the wave crests at a given instant of time simplest graphical representation of three-dimensional waves. The wave fronts of a sound wave emerging from the earpiece of a telephone headset. A spherical wave. The wave fronts of this wave are concentric spherical surfaces. A 1 >A 2 >A 3 >A 4 At a very large distance from the source the graphical wave fronts of a sound wave can be regarded as nearly flat. 20
Sound Waves In Air A sound wave in air consists of alternating zones of low density and high density (or zones of low pressure and of high pressure.) Such zones are generated by the vibrating diaphragm of a loudspeaker which exerts successive pushes on the air that is in contact with it. The alternating zones of low and high density travel away from the source. The air as a whole does not travel (the air molecules oscillate back and forth.) Travel only density disturbances. Sound wave longitudinal wave. Oscillation along the direction of propagation of the wave. The restoring force that drives these oscillations is the pressure of the air. Wherever the density of molecules is higher than normal, the pressure is higher than normal and pushed molecules apart. The frequency of the sound determines the pitch we hear (like color for light) White noise mixture of harmonic waves of all frequencies with equal strength. Musical tones mixture of just a few harmonic waves, fundamental and first few overtones. 21
Anatomy of a Human Ear The ear converts the mechanical oscillations of a sound wave into electric nerve impulses. The ear is unmatched in its ability to accommodate a wide range of intensities of sound (dynamic range is about 10 12 ) Three main parts of the human ear: The outer ear auricle and the ear canal Auricle focus sound waves into the ear. Ear canal 2.7 cm long tube closed off at the inner end by the eardrum. Ear canal guides sound waves toward the eardrum. The middle ear is an airfilled chamber in the temporal bone of the skull. Contains three small bones: the hammer, the anvil and the stirrup The inner ear complex system of fluid-filled cavities in the temporal bone. 20Hz 20,000Hz the range of frequencies audible for ear f>20,000hz ultrasound, propagate through liquids and solids sonography (10 6 Hz) 22
As a sound wave spreads out from its source, its intensity falls off. Concentric spherical wave fronts of a sound wave in air at successive instants of time. The intensity is inversely proportional to the square of the distance. 23
Example 1: Express the threshold of hearing (2.5 x 10-12 W/m 2 ) and the threshold of pain (1 W/m 2 ) in decibels. Example 2: At a distance of 60m from a jet airline the intensity is 1 W/m 2. I 180 =? 24
The intensity of sound near a loud rock band is 120dB. What is the intensity of sound near two such rock bands playing together? 25
Problem 10 (Decibels): Two identical rifles are shot at the same time and the sound intensity level is 800dB. What would be the sound intensity level if only one rifle were shot? (hint: the answer is not 400db) 26
Problem 60: Two sound sources each emit sound power uniformly in all directions. There are no reflections. Both sources are located on the x axis, one at the origin and the other at x = +123m. The source of the origin emits four times more power than the other source. Where on the x axis is the intensity of each sound equal? Note there are 2 answers. 27
The Speed of Sound 28
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Example: Thunder, lightning and a rule of thumb There is a rule of thumb for estimating how far away a thunderstorm is. You can estimate your distance from a bolt of lightning i by counting the seconds between seeing the flash and hearing the thunder and then dividing by 3 to obtain the distance in km. Why does this rule work? 30
The Speed of Sound Problem 47: Both krypton (Kr) and neon (Ne) can be approximated at monatomic ideal gases. The atomic mass of krypton is 83.8 u, while that of neon is 20.2 u. A loudspeaker produces a sound whose wavelength in krypton is 1.25m. If the loudspeaker were used to produce sound of the same frequency in neon at the same temperature, what would be the wavelength? 31
The Doppler Effect A train approaching a siren. The train encounters more wave fronts per unit time than when stationary. A train receding from the siren. The train encounters fewer wave fronts per unit time than when stationary. f = f + additional # of condensations additional # of condensations in a time t= (V r t)/λ in 1 sec additional # = V r /λ -> f = f + V r /λ = f(1 + V r /fλ) = f(1 + V r /V) A receiver on the train will detect a higher frequency when approaching a siren, and a lower frequency when receding. 32
The source (train) is in motion, the receiver is stationary A stationary receiver will detect a higher frequency when it is front of the train and a lower frequency when behind the train. The wavelength ahead of the train is shorter λ = λ V E /f and behind the train is longer λ = λ + V E /f when the train is stationary. ti 33
Problem 83: The trucks travel at the same speed. They are far apart on adjacent lanes and approach each other essentially head-on. One driver hears the horn of the other truck at a frequency that is 1.20 times the frequency he hears when the truck is stationary. ti The speed of sound is 343 m/s. At what speed is each truck moving? 34