© 2007 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. Lecture Outlines Chapter 14 Physics, 3 rd Edition James S. Walker

Chapter 14 Waves and Sound

Units of Chapter 14 Types of Waves Waves on a String Harmonic Wave Functions* Sound Waves Sound Intensity The Doppler Effect

Units of Chapter 14 Superposition and Interference Standing Waves Beats

Photo 14-1 A wave (Vilken typ?)

14-1 Types of Waves A wave is a disturbance that propagates from one place to another. The easiest type of wave to visualize is a transverse wave, where the displacement of the medium is perpendicular to the direction of motion of the wave.

14-1 Types of Waves In a longitudinal wave, the displacement is along the direction of wave motion. (förtätningar/förtunningar)

14-1 Types of Waves Water waves are a combination of transverse and longitudinal waves.

14-1 Types of Waves Wavelength λ: distance over which wave repeats Period T: time for one wavelength to pass a given point Frequency f : Speed of a wave:

Exercise 14-1 Ljudhastigheten i luft är 343 m/s. Hörbara frekvenser för örat ligger i intervallet 20 Hz till 20 kHz. Bestäm våglängderna för dessa frekvenser. λ = v/f λ low = 343 m/s / 20 Hz = 17,2 m λ high = 343 m/s / 20 kHz = 0,0172 m

14-2 Waves on a String The speed of a wave is determined by the properties of the material through which it propagates. For a string, the wave speed is determined by: 1. the tension (spänningen) in the string, and 2. the mass (egentligen massa per längdenhet) of the string. As the tension in the string increases, the speed of waves on the string increases as well.

14-2 Waves on a String The total mass of the string depends on how long it is; what makes a difference in the speed is the mass per unit length. We expect that a larger mass per unit length results in a slower wave speed.

14-2 Waves on a String As we can see, the speed increases when the force increases, and decreases when the mass increases.

Exercise 14-2 Ett 5,0 m långt rep väger 0,52 kg, spänns med kraften 46 N. Vad blir våghastigheten i repet? μ = m/L = 0,52 kg / 5,0 m v = (F/μ) ½ = (46 N5,0 m/0,52 kg) ½ = 21 m/s

Example 14-1 A Wave on a Rope

Exemple 14-1 A Wave on a Rope Ett 12 m långt rep spänns med kraften 92 N. När ena änden ges en “thunk” så tar det 0,45 s för störningen att propagera till den andra sidan. Vad väger repet? v = L/t μ = F/v 2 = 92 N(0,45 s) 2 / (12 m) 2 = 0,13 kg/m m = μ L = 1,6 kg

Conceptual Checkpoint 14–1 (p.436) Speed of a Wave När störningen propagerar uppåt, a) ökar b) minskar c) förblir densamma, hastigheten? (Tänk på figur 6-6! “Tension in a heavy rope”)

14-2 Waves on a String When a wave reaches the end of a string, it will be reflected. If the end is fixed (fast inspänd), the reflected wave will be inverted:

14-2 Waves on a String If the end of the string is free to move (fritt upphängd) transversely, the wave will be reflected without inversion.

14-3 Harmonic Wave Functions Since the wave has the same pattern at x + λ as it does at x, the wave must be of the form Also, as the wave propagates in time, the peak moves as

*14-3 Harmonic Wave Functions Combining yields the full wave equation:

14-4 Sound Waves Sound waves are longitudinal waves, similar to the waves on a Slinky: Here, the wave is a series of compressions and stretches.

14-4 Sound Waves In a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.

14-4 Sound Waves The speed of sound is different in different materials; in general, the denser the material, the faster sound travels through it.

Conceptual Checkpoint 14–2 How Far to the Lightning?

Conceptual Checkpoint 14-2 How far is the lightning? Fem sekunder efter att en magnifik blixt synts på himlen, skakar fönsterrutorna. Slog blixten ner a) 1,5 km b) närmare än 1,5 km c) längre bort än 1,5 km från huset? L = vt = 343 m/s 5 s = 1715 m (T = 20° C) Totaltiden (jmf; wishing well)? Kommentar?

Example 14-2 Wishing Well

Exemple 14-2 Wishing Well En sten släpps ned i en 7,35 m (= d) djup brunn. Efter hur lång tid hörs “plasket”? (T = 20° C) t ned = (2d/g) 1/2 = 1,22 s t upp = d/v = 7,35 m/343 m/s = 0,0214 s t tot = 1,22 s + 0,0214 s = 1,24 s (Jämför med ”blixttalet”)

14-4 Sound Waves Sound waves can have any frequency; the human ear can hear sounds between about 20 Hz and 20,000 Hz. Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are called infrasonic. Ultrasonic waves are familiar from medical applications; elephants and whales communicate, in part, by infrasonic waves.

14-5 Sound Intensity The intensity of a sound is the amount of energy that passes through a given area in a given time.

14-5 Sound Intensity Expressed in terms of power,

Exercise 14-3 En högtalare levererar ljudeffekten 0,15 W genom en kvadratisk ytan med sidan 2,0 m. Vad är ljudets intensitet? I = P/A = 0,15 W/(2,0 m) 2 = 0,038 W/m 2

14-5 Sound Intensity Sound intensity from a point source will decrease as the square of the distance.

14-5 Sound Intensity Bats can use this decrease in sound intensity to locate small objects in the dark.

Example 14-3 The Power of Song

Exemple 14-3 The Power of Song Två personer lyssnar på en sångfågel. Den ena personen står 1,00 m ifrån fågeln och hör ljudet med intensiteten 2,80 μW/m 2. a) Vilken intensitet hör den andra personen (på avståndet 4,25 m)? I i = P/4πr i 2 ; I j = P/4πr j 2 I j = 2,80 μW(1,0/4,25) 2 = 0,156 μW/m 2 b) Vilken effekt levererar fågeln? P = I i 4πr i 2 = 35,2 μW/m 2

Active Example 14-1 (p.443) The Big Hit: Find the Intensity En supporter 140 m bort, hör I = 0,380 μW/m 2. Vilken intensitet hör domaren på “first base”? (bleachers = los blanqueos, “lugar sol” på tjurfäktningen)

Active Exemple 14-1 The Big Hit Två personer lyssnar på en slagträträff. Den ena personen står 140 m ifrån slagmannen och hör ljudet med intensiteten 0,380 μW/m 2 (lågmält konversation). Vilken intensitet hör den andra personen (på avståndet 27,4 m)? I i = P/4πr i 2 ; I j = P/4πr j 2 I j = 0,380 μW(140/27,4) 2 = 9,9 μW/m 2 (livlig gatutrafik)

14-5 Sound Intensity When you listen to a variety of sounds, a sound that seems twice as loud as another is ten times more intense. Therefore, we use a logarithmic scale to define intensity values. Here, I 0 is the faintest sound that can be heard:

14-5 Sound Intensity The quantity β is called a bel; a more common unit is the decibel, dB, which is a tenth of a bel. The intensity of a sound doubles with each increase in intensity level of 10 dB.

Example 14-4 Pass the Pacifier

Exemple 14-4 Pass the Pacifier (nappen) Ett barn skriker med en intensitet av 8,0 μW/m 2 Bestäm intensiteten i dB. Vad blir intensiteten för två skrikande barn? (I 0 = W/m 2 ) β = (10 dB)log(I/I 0 ) = (10 dB)log( / ) = 69 dB [eller (10 dB)log(8)+(10 dB)log(10 6 ) = 69 dB] β = (10 dB)log(2I/I 0 ) = (10 dB)log(2) + (10 dB)log(I/I 0 ) = (10 dB)0, dB = 72 dB Av detta kan man lära sig I → 2I så ökar β med 3 dB och då 2I → I så minskar β med 3 dB

14-6 The Doppler Effect The Doppler effect is the change in pitch of a sound (tonhöjd) when the source and observer are moving with respect to each other. When an observer moves toward a source, the wave speed appears to be higher, and the frequency appears to be higher as well.

14-6 The Doppler Effect The new frequency is: If the observer were moving away from the source, only the sign of the observer’s speed would change:

14-6 The Doppler Effect To summarize:

Example 14-5 A Moving Observer

Exemple 14-5 A moving Observer Violinisten spelar ettstrukna a på sin fiol. Cyklisten kör med hastigheten 11,0 m/s. Vad uppfattar han för frekvens då han kör a) mot b) ifrån violinisten? f´ mot = (1 + u/v)f = (1+11,0/343)440 Hz = 454 Hz f´ ifrån = (1 - u/v)f = (1-11,0/343)440 Hz = 426 Hz

14-6 The Doppler Effect The Doppler effect from a moving source can be analyzed similarly; now it is the wavelength that appears to change:

14-6 The Doppler Effect We find:

Example 14-6 Whistle Stop

Exemple 14-6 Whistle stop Ett tåg tutar på sin väg mot en tunnel. Ljudet har frekvensen 650,0 Hz och tåget har hastigheten 21,2 m/s. a) Vilken frekvens hör en person som står vid tunneln? b) Vad uppfattar tågföraren för frekvens (på det reflekterade, dopplerförskjutna ljudet)? f´ mot = [1/(1 - u/v)]f = [1/(1-21,2/343)]650 Hz = 693 Hz f´´ mot = [(1 + u/v)f = (1+21,2/343)693 Hz = 736 Hz

14-6 The Doppler Effect Here is a comparison of the Doppler shifts for a moving source and a moving observer. The two are similar for low speeds but then diverge. If the source moves faster then the speed of sound, a sonic boom is created.

14-6 The Doppler Effect Combining results gives us the case where both observer and source are moving: The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements.

14-6 The Doppler Effect At left, a Doppler radar shows the hook echo characteristic of tornado formation. At right, a medical technician is using a Doppler blood flow meter. Galaxernas RÖDFÖRSKJUTNING.

14-7 Superposition and Interference Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition.

14-7 Superposition and Interference If two pulses combine to give a larger pulse, this is constructive interference (left). If they combine to give a smaller pulse, this is destructive interference (right).

Conceptual Checkpoint 14-3 Amplitude of Resultant Wave Två vågor interfererar. Har den resulterande vågen (som är en superposition av vågorna y 1 och y 2 ) en amplitud som är > < = de individuella vågorna y 1 och y 2 :s amplituder?

14-7 Superposition and Interference Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.

14-7 Superposition and Interference Here is another example of an interference pattern, this one from two sources. If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.

Example 14–7 Sound Off

Example 14-2 Sound off Två högtalare, 4,30 m från varandra, sänder ut ljud av frekvensen 221 Hz. Högtalarna är i fas. En person står 2,80 m rakt framför den ena högtalaren. Har man i detta läge konstruktiv eller destruktiv interferens? Eftersom utbredningshastigheten v är produkten av våglängden och frekvensen är v = λ f så λ = 1,55 m d 2 = (D 2 + d 1 2 ) 1/2 = 5,13 m d 2 – d 1 = 5,13 m – 2,80 m = 2,33 m (d 2 –d 1 )/ λ = 1,50 som innebär att...?

Active Example 14-2 Opposite Phase Interference

Två högtalare separerade 5,20 m, sänder ut ljud av frekvensen 104 Hz. Högtalarna är i motfas. En person står 3,00 m framför högtalarna och 1,30 m från centrumlinjen. Har man i detta läge konstruktiv eller destruktiv interferens? v = λ f så λ = 3,30 m d 1 = (3, ,90 2 ) 1/2 = 4,92 m d1 = (3, ,30 2 ) 1/2 = 3,27 m d 2 – d 1 = 4,92 m – 3,27 m = 1,65 m (d 2 – d 1 )/ λ = 0,50 innebär att då högtalarna är i motfas att vi har konstruktiv interferens här!

14-8 Standing Waves A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air.

14-8 Standing Waves The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics.

Example 14-8 It’s Fundamental

En av egenfrekvenserna till en sträng (som är 1,30 m lång) är 15,60 Hz. Nästföljande egenfrevens är 23,40 Hz. Vilken är grundfrekvensen? Vad är våghastigheten längs strängen? f n = nv/2L = 15,60 Hz f n+1 = (n+1)v/2L = 23,40 Hz f n+1 – f n = 23,40 Hz – 15,60 Hz = 7,80 Hz (grundfrek.) v = 2L(f n+1 – f n ) = 20,28 m/s f 1 = v/2L = 7,80 Hz

14-8 Standing Waves There must be an integral number of half- wavelengths on the string; this means that only certain frequencies are possible. Points on the string which never move are called nodes; those which have the maximum movement are called antinodes (= bukar).

In order for different strings to have different fundamental frequencies, they must differ in length and/or linear density. A guitar has strings that are all the same length, but the density varies (från sträng till sträng) Standing Waves

In a piano, the strings vary in both length and density. This gives the sound box of a grand piano (= flygel) its characteristic shape. Once the length and material of the string is decided, individual strings may be tuned to the exact desired frequencies by changing the tension. Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping (= snedvridningar) and other damage.

14-8 Standing Waves Standing waves can also be excited in columns of air, such as soda bottles, woodwind (träblåsinstrument) instruments, or organ pipes. As indicated in the drawing, one end is a node (N), and the other is an antinode (A).

14-8 Standing Waves In this case, the fundamental wavelength is four times the length of the pipe, and only odd- numbered harmonics appear.

14-8 Standing Waves If the tube is open at both ends (som tex i en pan- flöjt), both ends are antinodes, and the sequence of harmonics is the same as that on a string.

Figure Human response to sound

Example 14-9 Pop Music

Example 14-9 Pop music En tom flaska skall användas för att producera ettstrukna a (440 Hz) som grundfrekvens. Flaskan är 26,0 cm hög. Hur högt skall man fylla vatten för att få ett a? Behandla flaskan som en pipa med en sluten (vid vattenytan) och en öppen ände. f 1 = v/4L = 440 Hz L = 343 m/s /(4 440Hz) = 0,195 m Flaskan skall fyllas till 6,5 cm höjd

Conceptual Checkpoint 14-4 Om man fyller lungorna med helium och talar så låter man som Kalle Anka. Kan man ur denna observation sluta sig till att ljudhastigheten i He är > = < ljudhastigheten i luft? Oavsett om man betraktar munnen och näsan som en pipa med två öppna ändar* eller en pipa med en sluten och en öppen ände** så gäller f 1 ~ v/(2L)*(4L)** Ljudet har en högre frekvens → en högre hastighet ** Håll för näsan och checka om frekvensen ändras!

14-9 Beats ( = rytm, takt) Beats are an interference pattern in time, rather than in space. If two sounds are very close in frequency, their sum also has a periodic time dependence, although with a much lower frequency.

Summary of Chapter 14 A wave is a propagating disturbance. Transverse wave: particles move at right angles to propagation direction Longitudinal wave: particles move along propagation direction Wave speed: Speed of a wave on a string:

Summary of Chapter 14 If the end of a string is fixed, the wave is inverted upon reflection. If the end is free to move transversely, the wave is not inverted upon reflection. Wave function for a harmonic wave*: A sound wave is a longitudinal wave of compressions (förtätningar) and rarefactions (förtunningar) in a material.

Summary of Chapter 14 High-pitched (tonhöjd) sounds have high frequencies; low-pitched sounds have low frequencies. Human hearing ranges from 20 Hz to Hz. Intensity of sound: Intensity a distance r from a point source of sound:

Summary of Chapter 14 When the intensity of a sound increases by a factor of 10, it sounds twice as loud to us. Intensity level, measured in decibels: Doppler effect: change in frequency due to relative motion of sound source and receiver General case (both source and receiver moving):

Summary of Chapter 14 When two or more waves occupy the same location at the same time, their displacements add at each point. If they add to give a larger amplitude, interference is constructive. If they add to give a smaller amplitude, interference is destructive. An interference pattern consists of constructive and destructive interference areas. Two sources are in phase if their crests are emitted at the same time.

Summary of Chapter 14 Two sources are out of phase if the crest of one is emitted at the same time as the trough (vågdal, sänka) of the other. Standing waves on a string: Standing waves in a half-closed column of air:

Summary of Chapter 14 Standing waves in a fully open column of air: Beats occur when waves of slightly different frequencies interfere. Beat frequency: (Svävningsfrekvensen)