Longitudinal waves. Sound waves in fluid. Sound waves in fluid

Transcription

1 Lngitudinal waes Physics Enhancement Prgramme fr Gifted tudents The Hng Kng Academy fr Gifted Educatin and, HKBU Waes. Mechanical waes e.g. water waes, sund waes, seismic waes, strings in musical instruments. Electrmagnetic waes light (ultrailet, isible, infrared), mcrwaes, radi waes, teleisin waes, X rays 3. Matter (=quantum) waes electrns, prtns, ther fundamental particles, atms and mlecules 4. Graity waes neer bsered! und waes in fluid und waes in fluid und in a fluid (gas r liquid) is a lngitudinal mechanical ibratin with frequencies frm abut 0 Hz t abut 0000 Hz which is the typical range f human hearing. Lngitudinal waes f higher frequency, which are called ultrasnic waes ( ultrasund ), are used in lcating underwater bjects and in medical imaging. Lngitudinal mechanical waes f lwer frequency, called infrasnic waes ( infrasund ), ccur as seismic waes in earthquakes. The branch f physics and engineering deals with the study f mechanical waes f all frequencies, with bth transerse and lngitudinal ibratins in the case f slids. n this lecture, we cnsider mainly sund waes in air, which are strictly lngitudinal. Water waes (bth transerse and lngitudinal ibratins) 3 4

2 Traelling sund waes Traelling sund waes As the pistn scillates, it causes ariatins in the density f the air in the tube. The regins f high density are called cmpressins, the regins f lw density are called rarefactins. As the sund wae traels, the cmpressins and rarefactins trael alng the tube. f the pistn is drien s that its psitin can be described by a sine functin, then the density and pressure f air in the tube will als ary sinusidally. pxt (, ) p sinkxt m The air density in the tube can be expressed as ( x, t) ( x, t), where ( xt, ) is the small fluctuatin in the density caused by the sund wae 5 6 Relatinship between p m and m The relatinship between the pressure amplitude p m and the density amplitude m depends n the mechanical prperties f the medium. We intrduce the bulk mdulus B f the medium B p V / V which describes the relatie change in lume f an () element f fluid in respnse t a change in pressure. With density mv, then d m V dv V dv V V () V p Cmbine () and (), we hae p V B B n terms f the density and pressure amplitudes, p m m B The speed (elcity) f sund imilar t the case f the transerse mechanical wae, the speed f a sund wae depends n the rati f an elastic prperty f the medium and an inertial prperty. The speed f sund in fluids (gases and liquids) is in terms f the bulk mdulus B and the density m. Nte that the speed f sund in a fluid depends nly n the prperties f the medium and NOT n the frequency r waelength f the wae. B 7 8

3 The speed f sund ectin summary Human audi spectrum: 0Hz 0kHz Ultrasund: > 0kHz nfrasund: < 0Hz mv Lngitudinal scillatins f air density and pressure () cmpressins: regins f high density () rarefractins: regins f lw density Bulk mdulus B f medium, relatinship btw. p m and m B p V / V m pm B peed f sund depends n the prperties f the medium tiffer medium = faster sund wae B enser medium = slwer sund wae peed f sund in air (0 C) = 343 ms 9 0 ntensity f sund waes und Leel and ecibel (db) When we cmpare different sunds, it is useful t use the intensity (aerage pwer per unit area) f the wae. The respnse f the ear t sund f increasing intensity is apprximately lgarithmic, thus it is cnenient t intrduce a lgarithmic scale f intensity called the sund leel L: L 0lg The L is defined with respect t a reference intensity, which is chsen t be 0 W/m (a typical alue fr the threshld f human hearing). Unit: db r decibel und leel is measured in decibels (db). A sund f intensity (0 W/m ) has a L f 0 db, whereas sund at the upper range f human hearing, called the threshld f pain, has an intensity f W/m and a L f 0 db ( rders f magnitude strnger!). Multiplicatin f the intensity by a factr 0 crrespnds t adding 0 db t the L.

4 und Leel and ecibel (db) We can use db as a relatie measure t cmpare different sunds with ne anther. und Leel and ecibel (db) uppse we wish t cmpare tw sunds f intensities and : LL 0lg 0lg 0lg use lg (a) lg (b) = lg (a/b) Fr example, tw sunds whse intensity rati is differ in L by 0 lg = 3 db. 3 4 und Leel Example pherical sund waes are emitted unifrmly in all directins frm a pint surce, the radiated pwer P being 5 W. What are the intensity and the sund leel f the sund wae a distance r =.5 m frm the surce? P = 5 W and r =.5 m are gien. L 0lg is unknwn. Calculate using gien P and r. is knwn and = 0 - W/m 5 6

5 Answer t Example All the radiated pwer P must pass thrugh a sphere f radius r centered n the surce. Thus P 4 r We see that the intensity f the sund drps ff as the inerse square f the distance frm the surce. Numerically, we hae 5 W 0.3 W/m 4.5 m 0.3 W/m L 0lg 0lg 5 db 0 W/m A cmparisn f this result with the list in Table 9- shws this sund leel t be dangerus t a persn s hearing. 7 und leel: ntensity = pwer / area ectin summary L 0lg Threshld f hearing: = 0 W/m (i.e. 0 db) Threshld f pain: = W/m (i.e. 0 db) Threshld f hearing r pain frequency dependent Cmparisn between tw sunds: r Unit: db r decibel L L 0lg 8 nterference f sund waes The principle f superpsitin als applies t sund waes. The figure belw shws tw ludspeakers drien frm a cmmn surce. At pint P, the pressure ariatin due nly t speaker is p, and that due t alne is p. The ttal pressure disturbance at pint P is p = p + p. nterference f sund waes The type f interference that ccurs at pint P depends n the phase difference between the waes. The phase difference between the tw waes arriing at P depends n the path difference L rr and L are related by L (since ΔL = λ gies Δφ = π ) 9 0

6 nterference f sund waes Fr sme lcatins f pint P, the pressure ariatins arrie in phase ( = 0, π, 4π, ) and interfere cnstructiely. That is Path difference fr cnstructie interference m m 0,,,... L m ( m 0,,,...) Fr ther lcatins f P, the waes arries ut f phase ( = π, 3π, 5π, ) and interfere destructiely. That is m m 0,,,... Path difference fr Lm ( m 0,,,...) destructie interference L nterference f sund waes At the pints with destructie interference, that is, at lcatins where rr,3,5,..., the intensity has a minimum alue (nt necessarily zer, because in general the tw waes arrie at pint P with difference amplitudes). Lcatins f destructie interference crrespnd t dead spts in the listening enirnment f the speakers. Example n the gemetry f Fig. 9-6, a listener is seated at a pint a distance f. m directly in frnt f ne speaker. Tw speakers, which are separated by a distance f.3 m, emit pure tnes f waelength λ. The waes are in phase when they leae the speakers. Fr what waelengths will the listener hear a minimum in the sund intensity? Minimum intensity destructie interference r r and r are gien. L rr,3,5,... Find λ. Answer t Example The minimum sund intensity ccurs when the waes frm the tw speakers interference destructiely. f the listener is seated in frnt f speaker, then r =. m, and r can be fund frm the Pythagrean frmula,. m.3 m.6 m r r Then r r =.6 m. m =.4 m. Fr destructie interference, L rr,3,5,... crrespnding t.8 m, 0.93 m, 0.56 m,... Nte: Cmplete destructie interference will nt ccur at this lcatin, because the tw waes arriing at the bseratin pint hae different amplitudes, if they leaes the speakers with equal amplitudes. 3 4

7 ectin summary Phase difference Δϕ and path difference ΔL: L rr L urce f Musical und Clsed end nde Open end antinde tanding wae patterns Cnstructie interference in phase ( = 0, π, 4π, ) estructie interference ut f phase ( = π, 3π, 5π, ) Lm minimum intensity, i.e. dead spt (nt zer, because f different amplitudes) ( m 0,,,...) Lm ( m 0,,,...) Cnsider pipes with bth ends pen: Fundamental r first harmnic ( antindes at the ends, nde in the middle) L, L Resnant frequency: f L 5 6 urce f Musical und Case : pipes with bth pen ends: n general fr harmnic number n, Resnant frequency: L Ln, fr n,,3,... n f n fr n,,3,... L Case : pipes with nly ne pen end (anther end clsed): urce f Musical und n general, when a musical instrument prduces a tne, the fundamental as well as higher harmnics are generated simultaneusly. This gies rise t the different waefrms generated by different instruments. Hence different instruments hae different sunds: different timbres L,,,, n general, 4L n Resnant frequency: fr n,3,5,... 4L 4L 4L 4 L,,,, f n fr n,3,5,... 4L 7 8

8 Example 3 Weak backgrund nises frm a rm set up the fundamental standing wae in a cardbard tube f length L = 67.0 cm with tw pen ends (due t resnance). Assume that the speed f sund in the air within the tube is 343 ms -. (a) What frequency d yu hear frm the tube? (b) f yu jam yur ear against ne end f the tube, what fundamental frequency d yu hear frm the tube? *Fundamental standing wae *Gien L = 67.0 cm and ν = 343 ms - (a) Case: tw pen ends (b) Case: ne pen end, ne clsed end Answers t Example 3 (a) Tw pen ends L L 343 f 56 Hz L (0.67) L = / (b) Ear clses ne end: One fixed end and ne pen end, L 4 4L 343 f 8 Hz 4L 4(0.67) L = / Beats Cnsider tw sund waes with slightly different frequencies: s m s cs t and Resultant displacement: s m s cs t s s s sm (cst cst). Using trignmetric identity cs cs =cs ( )cs ( ) we btain s s cs ( ) t cs ( ) t m lw beat frequency s. high frequency ' beat f f f beat Beats The slwly arying amplitude s m cs t (where = ½( - )) is maximum when cs t =. i.e. times in each repetitin f the csine functin. (There are beats in each perid T, s we appear t hear duble the frequency cmpared t!) Hence the beat frequency is: ee emnstratin f Beats f 3 3

9 Cnclusin: s [ sm cs' t]cst, where Beats ' ( ) and ( ). ince and are nearly equal, >>.. Hence the resultant displacement cnsists f an scillatin with angular frequency and a slwly changing amplitude with angular frequency. Example: Beats A tuning frk with a frequency f 440Hz alng with the middle A string f a ilin, and beats per secnd are heard. What are the pssible frequencies f the A string? ifference between the string and tuning frk frequencies = Hz f beat = f f Pssible frequencies = 438 Hz, 44 Hz Pipes with bth pen ends ectin summary Any harmnic can be f n fr n,,3,... L set up in the pipe. Pipes with nly ne pen end (anther end clsed) f n fr n,3,5,... 4L Beat frequency: f f f beat s [s m cs ' t]cs t, where Only dd harmnic can be set up in the pipe. lw freq. term ppler Effect ee Yutube ppler effect, shck wae, and snic bm Ming etectr; tatinary urce ' ( ) and = ( ) high freq. term lw freq. term 35 36

10 ppler Effect Ming etectr; tatinary urce The frequency bsered by the detectr appraching the surce: ince =/f, ppler Effect Ming etectr; tatinary urce f ( t t)/ t ' f ' f / f n time t the waefrnts me a distance t, the detectr mes a distance t, the range f waes intercepted by the detectr = t+ t, the number f waefrnts intercepted by the detectr = (t+ t)/. imilarly, if the detectr mes away frm the surce, ummarizing, f ' f f ' f ppler Effect Ming urce ; tatinary etectr ppler Effect Ming urce ; tatinary etectr n a perid T, the distance med by the waefrnt W = T, the distance med by the surce = T, the distance between the waefrnts W and W = T T. The frequency bsered by the detectr (surce is appraching): f the surce mes away frm the detectr, f ' f ' T T / f / f f ' f ummarizing, f ' f 39 40

11 General ppler Effect Equatin When bth the surce and detectr are ming (alng the same line; therwise ectrial), f ' f = 0 reduces t the equatin fr statinary surce. = 0 reduces t the equatin fr statinary detectr. f ' f upper +e sign detectr mes twards surce lwer e sign detectr mes away frm surce upper e sign surce mes twards detectr lwer +e sign surce mes away frm detectr 4 Example 4 Bats naigate and search ut prey by emitting, and then detecting reflectins f, ultrasnic waes, which are sund waes with frequencies greater than what can be heard by a human. uppse a bat emits ultrasund at frequency f be = 8.5 khz while flying with elcity b = 9 ms. t chases a mth that flies with elcity m = 8 ms. (a) What frequency f md des the mth detect? (b) What frequency f bd des the bat detect in the returning ech frm the mth? f be f bd speed f sund in the air b m = 343 ms - f m f m 4 Answers t Example 4 (a) etectin by mth (mth is the detectr, bat is the surce): detectr away frm surce e f ' f upersnic speeds; hck waes f md m fbe khz b surce twards detectr e (b) etectin f ech by bat (bat is detectr, mth is surce f the reflected wae with ppler-shifted frequency): detectr twards surce +e f bd b fmd khz m surce away frm detectr +e ee Yutube nic Bm Physics

12 upersnic speeds; hck waes When appraches, the ppler frequency f becmes infinite since f ' f When the sund speed exceeds the surce speed, the ppler effect equatin des nt apply. All waefrnts bunch alng a V-shaped enelpe. This is called the Mach cne. A shck wae is prduced. Nte that the enelpe tuches the circular waefrnts. Therefre the radius ending at the tangent pint is nrmal t the Mach cne. t Mach cne angle: sin / is called the t Mach number. When is the bm heard? h x The snic bm detected by a grund bserer was generated by a supersnic jet befre it flew erhead. Hweer, when the snic bm arries at the grund bserer, the supersnic jet has flwn ahead (lcated at the tip f the Mach cne) Example 5 Answer t Example 5 The speed f sund is 340 ms -. A plane flies hrizntally at an altitude f 0,000 m and a speed f 400 ms -. When an bserer n the grund hears the snic bm, what is the hrizntal distance x frm the pint n its path directly abe the bserer t the plane? 340t 400t 0,000 m x Gien: () h = 0000 m () speed f sund = 340 ms - (3) speed f plane = 400 ms - h x * trignmetry prblem abut length (r distance) and angle * distance = speed x time Frm the figure, 340t sin t 58. x 0000ct 600 m 47 48

13 ectin summary ppler effect - the apparent change in f f a wae caused by relatie mtin between the surce f the wae and the bserer. General ppler effect equatin: f ' f nic bm, supersnic waes, shck waes () > ppler effect des nt apply () V-shaped enelpe Mach cne t (3) Mach cne angle: sin t (4) Mach number = 49