DOCSLIB.ORG

Physics I Notes: Chapter 13 – Sound

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Physics I Notes: Chapter 13 – Sound I. Properties of Sound A. Sound is the only thing that one can hear! Where do sounds come from?? Sounds are produced by VIBRATING or OSCILLATING OBJECTS! Sound is a longitudinal wave produced by a vibrating source that causes regular variations in air pressure ( P in diagram above). B. Audible range of sound for most young people is 20 Hz to 20,000 Hz 1. Infrasonic waves are below 20 Hz and ultrasonic waves are above 20,000 Hz. Infra- and ultra- have to do with frequencies below and above the normal range of human hearing. This is NOT to be confused with subsonic and supersonic…these terms have to do with the speeds of moving objects. For example: a subsonic car travels slower than the speed of sound in air, while a supersonic jet airplane travels faster than the speed of sound in air. C. Frequency determines the pitch – how high or low we perceive the sound to be. The higher the frequency, the higher the pitch you will hear. D. Loudness depends upon relative amplitude of the wave E. Intensity of a sound wave is the rate of energy flow (or Power) through a given area. Sound waves propagate spherically outward from the source. Since the original amount of energy is spread out over a larger amount of surface area, the intensity of sound decreases by the inverse square law as it moves away from the source. Power Intensity = 4πr 2 The value of the intensity of sound determines its loudness or volume, but the relationship is NOT directly proportional. This is because the sensation of loudness is approximately logarithmic in the human ear. Relative intensity, which is found by relating the intensity of a given sound to the threshold of hearing, corresponds more closely to human perceptions of loudness. Relative intensity is measured in decibels. The Decibel Scale: Relative I (in dB) = log (I/I o); I o = threshold of hearing ). A 10 dB increase (10 X the intensity) in sound level is heard as being as about twice as loud. Page 1 of 6 • Threshold of hearing (I o)– approximate lowest intensity of sound that can be heard by the average human ear (occurs at about 1000 Hz with an intensity of 1.0 X 10 –12 W/m 2) • Threshold of pain – approximate loudest sound that the human ear can tolerate (1.0 W/m 2) Example #1 : If the intensity of a person’s voice is 4.6 X 10 -7 W/m 2 at a distance of 2.0 m, how much power does that person’s voice generate? Example #1a: If you were 3 times farther away (at 6.0 meters from the source), how would the intensity change? How much intensity would you receive at that new location? F. Speed of sound waves depends upon the properties of the medium. The speed of sound in air @ 1.0 atm and 20 o C is 343 m/s. 1. The speed of sound in air generally increases by 0.6 m/s for each increase of 1 oC. Speed of sound is generally greater in liquids than gases and typically fastest in solids. Two factors that determine speed of sound: the elasticity and the density of the medium. Elasticity is the more important of the two factors. Increased elasticity increases the speed and increased density tends to slow it down . The interaction of these two factors determines the speed in a given medium. Elasticity is a measure of how quickly and easily a medium regains its original state or shape. Metal pipes are very dense BUT so elastic that the wave speed in metals is very fast. 2. Mach 1 is NOT Warp speed! Mach 1 is the speed of sound in air. When a plane exceeds this speed (~340 m/s or about 750 miles per hour) the plane is said to break through the sound barrier and go supersonic! The waves produced by the plane create a shock wave that travels along in a cone shape behind the plane. As the cone passes by observers they hear a sonic boom created by the overlap of all the waves produced by the plane. The angle of the cone is determined by how much above the speed of sound the plane is traveling. See the diagrams below . In the first picture below on the left, the source of the waves is sitting still. It moves in the second picture but slowly. In the third picture, the object is moving at exactly the speed of the sound waves it produces, and then in the fourth picture it’s moving faster than sound. http://www.lon- capa.org/~mmp/applist/doppler/d.htm Sboom.mpg When a plane is moving faster than the speed of sound, a sonic boom can be heard by people after the plane goes by…BUT A sonic boom is not the crash made as a plane “breaks the sound barrier”! The sonic boom that an observer on the ground would hear is created by the overlap and constructive interference of the compression waves that form a 3-d cone-shaped shock wave. As long as the plane is traveling faster than the speed of sound, it will drag this shock wave along behind it. The faster the plane, the longer Page 2 of 6 and narrower the cone becomes. You hear the sonic boom once the shock wave passes you by (this is usually not until the plane is very far past you)! III. Doppler Effect or Doppler Shift – the change in frequency (and wavelength) due to relative motion of source and/or detector. A. When the source is moving toward the detector, the observed frequency is higher and, since velocity does not change in a given medium , the wavelength is shortened. Higher frequency sounds have higher pitch, and higher frequency light is called “blue-shifted”. B. Observed frequency is lower (and wavelength longer) when source and observer are moving away from each other. This results in lower pitched sounds and “red-shifted” light. http://www.wfu.edu/physics/demolabs/demos/3/3 b/3B40xx.html Shortcut to caltraindopplermono22.wav.lnk IV. Resonance and Harmonics – in order for a musical instrument to sound good and proJect that sound outward to an audience - a standing wave must be formed in the instrument and its resonance chamber! A. Standing waves on a vibrating string 1. Fundamental frequency (or first harmonic) (f1); lowest possible frequency (lowest pitched sound) from a standing wave on a string that is fixed at both ends. The fundamental frequency is sounded when the length of the string is exactly ½ the wavelength . 2. Upper Harmonics – integral multiples of the fundamental frequency (f 2 = 2f1, f3 =3f1, f4 =4f1, etc.) L = length of string λ is wavelength Notice the # of nodes & anti-nodes. Example #2: The speed of waves in a particular guitar string is found to be 425 m/s. Determine the fundamental frequency (1st harmonic) of the string if its length is 76.5 cm. Page 3 of 6 Example #3: A guitar string with a length of 80.0 cm is plucked. The speed of a wave in the string is 400 m/s. Calculate the frequency of the first, second, and third harmonics. B. Standing waves in an air column – two basic examples: Closed and open 1. Closed-pipe resonator (closed at ONE end) Resonance occurs when the frequency of a force applied to an obJect matches the natural frequency of vibration of that obJect. When a sound wave has a wavelength that matches the resonance length of the tube, a standing wave is produced and the sound heard. *** The shortest column of air that can resonate in a closed-pipe resonator is ¼ wavelength. NOTICE: the wave must be open where the pipe is open and wave is closed where the pipe is closed!! Each additional resonance length is spaced by an increase of exactly ½ of a wavelength from that point since an antinode (wave is opened up) must be located at the opening of the tube in order for the sound to be heard. ***** Because of this restriction, there is no 2 nd harmonic nor any even # harmonics in a closed end pipe. Only odd harmonics are present in a closed pipe resonator, BUT you can be asked to give the first three harmonics that will be heard for this type of pipe… Answer: f 1, f3 =3f1, and f5 =5f1 For f1, L = 1/4 λ. For f3, L = 3/4 λ. And for f5, L = 5/4 λ L = length of pipe λ is wavelength Example #4: Titan Tommy and the Test Tubes are playing at Shades in Lincolnshire this weekend. The lead instrumentalist uses a test tube (closed end air column) with a 17.2 cm long air column. The speed of sound in the test tube is 340 m/s. Find the frequency of the first harmonic played by this instrument. Page 4 of 6 C. Open-pipe resonator (open at BOTH ends) The harmonic pattern for resonances in an open-pipe resonator is identical to the harmonic pattern for a string fixed at both ends. Minimum length of an open-pipe resonator is ½ wavelength and all harmonics are present. For f 1 find the wavelength and use v= λf to solve for f. Then the upper harmonics are (f 2 = 2f1, f3 =3f1, f4 =4f1 , etc.) λ is wavelength, L = length of pipe. L = ½ λ L = λ L = 3/2 λ Example #5: Determine the length of an open-pipe resonator required to produce a fundamental frequency of 480 Hz when the speed of sound in air is 340 m/s.
Recommended publications
  • No. 2990907 ACOUSTIC FILTER

    No. 2990907 ACOUSTIC FILTER

    July 4, 1961 W. S. EVERETT 2,990,907 ACOUSTIC FILTER Filed June 11, 1959 3 Sheets-Sheet 1 74 40 0.---.>2 .58 to 70 "1l!I.~a--34 22. /8 20 , 30 28 _~X 70 24 ~~~~~~~f;:;26 \:l Z 58 ~2B.6. 32 =- 76 y-r~.z. 77 J~ 00 7- rf ~8 /' oX. / 54 / 0 I ~' 40 7~- 58 70 8'1 July 4, 1961 W. S. EVERETT 2,990,907 ACOUSTIC FILTER Filed June 11, 1959 3 Sheets-Sheet 2 120 104~~--;18~ INVE IV TOR. 140 ILH£L~. ErR.£TT :By ~~ ATTCRN£'t, July 4, 1961 W. S. EVERETT 2,990,907 ACOUSTIC FILTER Filed June 11, 1959 3 Sheets-Sheet 3 o o No o (\,J o , .... 0 In° '\ 0;)2 I""'" I\. t- ID ....... ~ '\ \Cl ~ Q , I\. \ Z "- ", 0 I\. ~ tJ ~ ~ ~ I\. ru U) ~ " 1\ IL1 -r-J 1'\ ll. c:r 1\ r\ V'J '\ -OLiJ 1110,.J C()~v t- ~ lD 1..1 U) Z -t >- v II) Z UJ 1/ ::> (\j C!J IIJ \ J Q: I \ IJ.. I '" 1'0... I ........ II . I--.. I--~ ~ 1....- .... ~ 1/ IJ INVENTOR WILHELI'1 S. EVERETT B)/ ~t1~ ~ ATTORNEY, 2,990,907 United States Patent Office Patented July 4, 1961 1 2 acoustic spectrum without introducing any substantial 2,990,907 back pressure into the system. ACOUSTIC FILTER Wilhelm S. Everett, 1349, E. Main St., Santa P.aula, Calif. I have devised an acoustic mute which absorbs se­ Filed June 11, 1959, Ser. No. 819,779 lected frequencies of the acoustic speotrum, which may 6 Claims.
  • Ultrasonic Concentration of Microorganisms

    Ultrasonic Concentration of Microorganisms

    University of Kentucky UKnowledge Theses and Dissertations--Biosystems and Agricultural Engineering Biosystems and Agricultural Engineering 2012 Ultrasonic Concentration of Microorganisms Samuel J. Mullins University of Kentcuky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Mullins, Samuel J., "Ultrasonic Concentration of Microorganisms" (2012). Theses and Dissertations-- Biosystems and Agricultural Engineering. 7. https://uknowledge.uky.edu/bae_etds/7 This Master's Thesis is brought to you for free and open access by the Biosystems and Agricultural Engineering at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Biosystems and Agricultural Engineering by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained and attached hereto needed written permission statements(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine). I hereby grant to The University of Kentucky and its agents the non-exclusive license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless a preapproved embargo applies. I retain all other ownership rights to the copyright of my work.
  • Generation of Sub-Wavelength Acoustic Stationary Waves in Microfluidic Platforms

    Generation of Sub-Wavelength Acoustic Stationary Waves in Microfluidic Platforms

    GENERATION OF SUB-WAVELENGTH ACOUSTIC STATIONARY WAVES IN MICROFLUIDIC PLATFORMS: THEORY AND APPLICATIONS TO THE CONTROL OF MICRO-NANOPARTICLES AND BIOLOGICAL ENTITIES. A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Muhammet Kursad Araz February 2010 c 2010 Muhammet Kursad Araz ALL RIGHTS RESERVED GENERATION OF SUB-WAVELENGTH ACOUSTIC STATIONARY WAVES IN MICROFLUIDIC PLATFORMS: THEORY AND APPLICATIONS TO THE CONTROL OF MICRO-NANOPARTICLES AND BIOLOGICAL ENTITIES. Muhammet Kursad Araz, Ph.D. Cornell University 2010 Presented in this dissertation are the theoretical modeling and experimental re- sults of a novel method for manipulating micro and nanoparticles in an acoustically actuated microfluidic glass capillary. Here, the PZT (Lead Zirconate Titanate)- glass capillary actuator mechanism performs bioanalytical methods such as collec- tion, separation and mixing at microscale, at low voltage drives, enabling produc- tion of battery operated inexpensive portable microfluidic systems. Analytical and finite element modeling of the vibrational modes of the fluid filled thick walled cylindrical capillary has been also studied. Torsional, longitudi- nal and flexural modes and their dispersion relationships are presented. Through the excitation of the various vibrational modes of the silica capil- lary, sub-wavelength acoustic pressure modes in the microfluidic cavity are formed. More than 20 such sub-harmonic modes are generated harmonically in the 20kHz- 2MHz regime whereas naturally occurring radial modes have a cut off frequency around 9 MHz. The amplitude of these stationary acoustic pressure fields are high enough to generate nonlinear acoustic forces and streaming effects for micro and nanoparticle manipulation.

  • Structural Acoustic Coupling Resonance of Hindu Temples

    STRUCTURAL ACOUSTIC COUPLING RESONANCE OF HINDU TEMPLES 1R WIJESIRIWARDANA, 2M. VIGNARAJAH, 3P.KATHIRGAMANATHAN, 4K. WALGAMA 1,2,3Faculty of Engineering University of Jaffna Sri Lanka, 4Faculty of Engineering University of Peradeniya Sri Lanka E-mail: [email protected], [email protected], [email protected], [email protected] Abstract- Some ancient constructions over the world exhibit acoustic resonance chambers [1, 7]. Acoustic resonance of Pyramids [1], Stupas [2,6,7], Cathedrals [3,4,12] and Mosques [5] have been studied and they closely resembles the Alpha, Theta and Gamma waves of EEG spectrum [11]. Moreover, these EEG frequencies corresponds to the alert calmness that conducive for mental composure and general well being [6, 7]. Research has also showed that these frequencies are also closer to the Shumann resonance frequencies [9,10]. Even though, acoustic resonances are in the mechanical vibration domain the researchers have found these vibration energies may have transformed into equivalent electromagnetic frequencies (EEG and Shumann bandwidth) [1,6]. This conversion is mostly may have achieved in ancient resonance chambers by properly using special quartz crystals [2,6,7] and getting them to vibrate by acoustic energies. In addition, some of these acoustic resonance chambers also have vibrating frequencies which are octaves of 528 Hz and 432 Hz [13]. These frequencies are also known to be healing frequencies commonly used in sound therapies [14]. Limited research has been done to understand the acoustic resonance chambers of Hindu Temples. Two types of acoustic resonance chambers can be found in Hind Temples. The first one is the “Moolastana” chamber and the other is the inner core (path).
  • W O 2019/118921 Al 20 June 2019 (20.06.2019) W IPO I PCT

    W O 2019/118921 Al 20 June 2019 (20.06.2019) W IPO I PCT

    (12) INTERNATIONAL APPLICATION PUBLISHED UNDER THE PATENT COOPERATION TREATY (PCT) (19) World Intellectual Property (1) Organization11111111111111111111111I1111111111111i1111liiiii International Bureau (10) International Publication Number (43) International Publication Date W O 2019/118921 Al 20 June 2019 (20.06.2019) W IPO I PCT (51) International Patent Classification: (72) Inventors: LIPKENS, Bart; 380 Main Street, Wilbraham, H04R 3/04 (2006.01) H04R 17/00 (2006.01) MA 01095 (US). MUSIAK, Ronald; 380 Main Street, H04R 1/24 (2006.01) Wilbraham, MA 01095 (US). ARTIS, John; 380 Main (21) International Application Number: Street, Wilbraham, MA01095 (US). PCT/US2018/065839 (74) Agent: KENNEDY, Brendan J.; FloDesign Sonics, Inc., (22) International Filing Date: 380 Main Street, Wilbraham, MA 01095 (US). 14 December 2018 (14.12.2018) (81) Designated States (unless otherwise indicated, for every (25) Filing Language• English kind ofnational protection available): AE, AG, AL, AM, AO, AT, AU, AZ, BA, BB, BG, BH, BN, BR, BW, BY, BZ, (26) Publication Language: English CA, CH, CL, CN, CO, CR, CU, CZ, DE, DJ, DK, DM, DO, (30)PriorityData: DZ, EC, EE, EG, ES, FI, GB, GD, GE, GH, GM, GT, HN, (30)/Priorit : 14 December 2017 (14.12.2017) US HR, HU, ID, IL, IN, IR, IS, JO, JP, KE, KG, KH, KN, KP, LY, MA, MD, ME, 62/614,354 05 January 2018 (05.01.2018) US KR, KW, KZ, LA, LC, LK, LR, LS, LU, MG, MK, MN, MW, MX, MY, MZ, NA, NG, NI, NO, NZ, (71) Applicant: FLODESIGN SONICS, INC. [US/US]; 380 OM, PA, PE, PG, PH, PL, PT, QA, RO, RS, RU, RW, SA, Main Street, Wilbraham, MA 01095 (US).
  • Estimation of the Acoustic Waste Energy Harvested from Diesel Single Cylinder Engine Exhaust System

    Estimation of the Acoustic Waste Energy Harvested from Diesel Single Cylinder Engine Exhaust System

    Estimation of the Acoustic Waste Energy Harvested from Diesel Single Cylinder Engine Exhaust System Claudiu Golgot a, Nicolae Filip b and Lucian Candale Department of Automotive Engineering and Transport, Technical University of Cluj-Napoca, Romania Keywords: Noise, Diesel, Acoustic, Energy Harvesting, FFT, Resonant Frequency. Abstract: Noise generated in the operation of an internal combustion engine is an energy waste that produces noise pollution. Recovering some of this energy and transforming it into another form of usable energy brings significant benefits. We proposed to develop a device to recover this energy waste produced by the internal combustion engines, in the gases changing process. The developed energy recovery system is based on the Helmholtz resonator principle. For the conversion of acoustic waves into electricity, we used an audio speaker as a low-cost electromagnetic transducer located at the end of the resonant chamber. By audio playback of the acoustic signal recorded at the engine exhaust, we measured the electricity generated with the proposed recovery system. We found that the noise level measured at the exhaust depending on the engine speed range, follows a linear distribution law, instead, the harvested electric power varies nonlinearly. To find out the cause of the electric power variation, we performed a detailed FFT analysis. We found that at most engine speeds, the dominant amplitudes in the frequency spectrum are close to the resonant frequency of the system. With the proposed conversion system, we obtained a maximum value of the harvested electric power of 165 µW. 1 INTRODUCTION The conversion of acoustic energy is also found in research on thermo-acoustic generators, they Road vehicles are generally recognized as a major transform thermal energy into acoustic energy and source of urban noise pollution, mainly due to the then into electricity using an acoustic-electric noise produced by the exhaust system.
  • Principles of Sound Resonance in Human Whistling Using Physical Models of Human Vocal Tract

    Principles of Sound Resonance in Human Whistling Using Physical Models of Human Vocal Tract

    Acoust. Sci. & Tech. 37, 2 (2016) #2016 The Acoustical Society of Japan Principles of sound resonance in human whistling using physical models of human vocal tract Toshiro Shigetomi and Mikio Morià Graduate School of Engineering, University of Fukui, 3–9–1 Bunkyo, Fukui, 910–8507 Japan (Received 16 June 2015, Accepted for publication 28 October 2015) Keywords: Whistling, Human vocal tract modeling, Helmholtz resonance, Air-column resonance PACS number: 43.64.Yp [doi:10.1250/ast.37.83] 1. Introduction produce the vocal tract forms corresponding to the five Whistling competitions have been held regularly in Japan Japanese vowels through changes in the plate combination. in recent years, and some prize-winning whistlers have Because the shape of the mouth at the time of whistling is established music schools to teach musical whistling. How- considered to be close to the mouth structure required for the ever, no theoretical textbook on whistling has been written pronunciation of the vowel /u/, Mori et al. reported adjust- thus far, and the principles of human whistling sound ments made to physical models to achieve this shape [5]. production are relatively unknown. A good understanding of Following this approach, the vocal cord parts in the model these principles would be very beneficial to both the trainer used in this study were detached and various components and the trainee. (described below) were added. By employing simplified Previously, Rayleigh reported that the whistling frequency models, Mori et al. also demonstrated that the resonant is determined by the Helmholtz resonator frequency in the frequency and mode in human whistling involve not only the mouth cavity.
  • D. Guicking Patents on Active Control of Sound and Vibration – An

    D. Guicking Patents on Active Control of Sound and Vibration – An

    D. Guicking Patents on Active Control of Sound and Vibration { an Overview { Second revised and enlarged edition May 2009 Contact: Dr. Dieter Guicking Schl¨ozerweg6 37085 G¨ottingen Germany Tel.: +49-551-4 61 06 E-Mail: Dieter.Guicking(at)phys.uni-goettingen.de Homepage: http://www.guicking.de/dieter Emeritus member of: Drittes Physikalisches Institut Universit¨atG¨ottingen Friedrich-Hund-Platz 1 37077 G¨ottingen Germany ⃝c Dr. Dieter Guicking 2009 PREFACE Patent titles and patent abstracts often are not really informative. During my work on a patent bibliography it appeared to me therefore reasonable to write this text. Based on about 2060 patent families out of which about 1740 have been selected (removing redundancies and less important ones), this text gives an overview of old and new patents on Active Noise and Vibration Control (ANVC), including related fields such as algorithms, sound design, active flow control, transducers, etc. The patents are grouped in 17 sections which are subdivided into subsections if appropriate. Some emphasis is laid upon patents on algorithms (Section 2). The References in Section 18 are given in short form; concerning the citation style, see [1]. This text is the second revised and enlarged edition of the first version, published in January 2001 [2, 3]. Although I have done my best to deliver a reliable and useful document, I cannot take any re- sponsibility for the correctness and completeness of the data given. Gottingen,¨ April 2009 Dieter Guicking 3 CONTENTS 1. Introduction . 5 2. Algorithms. 5 3. ANC in Ducts and Mufflers . 14 3.1. Ducts .
  • Structural Acoustic Coupling Resonance of Hindu Temples

    Structural Acoustic Coupling Resonance of Hindu Temples

    International Journal of Advanced Computational Engineering and Networking, ISSN: 2320-2106, Volume-5, Issue-7, Jul.-2017 http://iraj.in STRUCTURAL ACOUSTIC COUPLING RESONANCE OF HINDU TEMPLES 1R WIJESIRIWARDANA, 2M. VIGNARAJAH, 3P.KATHIRGAMANATHAN, 4K. WALGAMA 1,2,3Faculty of Engineering University of Jaffna Sri Lanka, 4Faculty of Engineering University of Peradeniya Sri Lanka E-mail: [email protected], [email protected], [email protected], [email protected] Abstract- Some ancient constructions over the world exhibit acoustic resonance chambers [1, 7]. Acoustic resonance of Pyramids [1], Stupas [2,6,7], Cathedrals [3,4,12] and Mosques [5] have been studied and they closely resembles the Alpha, Theta and Gamma waves of EEG spectrum [11]. Moreover, these EEG frequencies corresponds to the alert calmness that conducive for mental composure and general well being [6, 7]. Research has also showed that these frequencies are also closer to the Shumann resonance frequencies [9,10]. Even though, acoustic resonances are in the mechanical vibration domain the researchers have found these vibration energies may have transformed into equivalent electromagnetic frequencies (EEG and Shumann bandwidth) [1,6]. This conversion is mostly may have achieved in ancient resonance chambers by properly using special quartz crystals [2,6,7] and getting them to vibrate by acoustic energies. In addition, some of these acoustic resonance chambers also have vibrating frequencies which are octaves of 528 Hz and 432 Hz [13]. These frequencies are also known to be healing frequencies commonly used in sound therapies [14]. Limited research has been done to understand the acoustic resonance chambers of Hindu Temples.
  • Lecture 6: Waves in Strings and Air

    Lecture 6: Waves in Strings and Air

    Matthew Schwartz Lecture 6: Waves in strings and air 1 Introduction In Lecture 4, we derived the wave equation for two systems. First, by stringing together masses and springs and taking the continuum limit, we found ∂2 ∂2 v2 A(x,t)=0 (1) ⇒ ∂t2 − ∂x2 where A(x,t) is the displacement from equilibrium of the mass at position x. These are longitu- dinal waves. In this equation, for waves in a solid, the wave speed is E v = (2) µ r where E is the elastic modulus and µ is the density per unit length. Now we consider two more cases: transverse oscillations on a string and longitudinal motion of air molecules (sound waves). 2 Transverse oscillations Consider a string of tension T . We define the amplitude of the string at a point x at time t as A(x,t). In this section, we’ll sometimes write A(x, t) just as A(x) to avoid clutter. Let us treat the string as a bunch of massless test probes connected by a elastic strings. Then we can draw the picture as What is the force acting on the test mass at position x (in red)? First, consider the downward component of the force pulling on the test mass at x from the mass to the left (at x ∆x). We can draw a triangle: − The force is given by ∆A Fdownwards, from left mass = T sinθ = T (3) √∆A2 + ∆x2 If the system is close to equilibrium, then the slope will be small. That is, ∆A ∆x.