Waves and the One-Dimensional Wave Equation

Transcription

1 Waves and the One-Dimensinal Wave Equatin Earlier we talked abut the waves n a pnd. Befre we start lking specifically at sund waves, let s review sme general infrmatin abut waves. Types There are tw general classificatins f waves, lngitudinal and transverse: Transverse Wave A traveling wave in which the particles f the disturbed medium mve perpendicularly t the wave velcity. An example is the wave pulse n a stretched rpe that ccurs when the rpe is mved quickly up and dwn. Lngitudinal Wave A traveling wave in which the particles f the medium underg displacement parallel t the directin f the wave mtin. Sund waves are lngitudinal waves. One thing t nte is that sme waves exhibit characteristics f bth types f waves. The waves n ur pnd are a cmbinatin f bth types. Transverse: Lngitudinal: -1

2 Characteristics Just like the peridic mtin f the simple harmnic scillatr, waves have certain characteristics. The nes we will cncentrate n are the frequency, perid, wave speed and the wavelength. Recall frm SP11, a picture f a transverse wave in a medium at sme time, maybe t=0 sec. Traveling Wave at t = 0 s (m) s x (m) λ We wrte an equatin t describe this picture: π s( x) = s0 sin x λ where: s = particle displacement Distance that the fluid particle is mved frm its equilibrium psitin at any time, t. s = maximum particle displacement r amplitude λ = distance ver which the wave begins t repeat k = π = a cnversin factr that relates the change in phase (angle) t a spatial λ displacement. We call k the wavenumber. When we let this wave begin t mve t the right with a speed, c, the psitin is shifted in the gverning equatin frm x t x-ct. π s( x,t) = s sin ( x ct) λ -

3 Belw is a picture f the same traveling wave shwn at sme later time, t. Traveling Wave at Sme Later Time, t s (m) ct x (m) Nw, if instead f taking a snap sht f the wave in the medium at tw different times, what if we had set a sensr smewhere in space maybe at x = 0 m, and recrded the wave s displacement ver time. The equatin gverning the wave wuld becme: where π πc π s 0,t ssin 0 ct ssin t ssin t s sin t λ λ T ( ) = ( ) = = = [ ω] T = perid Time t cmplete ne cycle. c = T λ = wave velcity Distance that wave energy travels per unit time. ω = π = a cnversin factr that relates the change in phase (angle) t a tempral T displacement. We call ω the angular frequency. 1 f = = frequency, is the inverse f the perid. It is the number f cycles per unit time T that pass the rigin. Nte that we have emplyed a similar strategy regarding the grup f cnstants in frnt f the time variable that we used when discussing the wavenumber, k. Since the wave repeats every π change in phase and that crrespnds t a time perid, T, angular frequency, ω=π /T, is nthing mre than a cnversin factr frm time t phase angle. The symmetry with wavenumber is striking causing many peple t identify the wave number as the special frequency and t specifically refer t angular frequency, ω, as the tempral frequency -3

4 T be clear, the speed f the wave c, is nt the speed f the medium. It is the speed f the wave disturbance envelpe and is ften called the phase speed. It is the speed yu wuld need t run next t the medium in rder t stay in phase with a pint n the disturbance. The speed f the medium is als called the particle speed and is fund by taking the derivative f the displacement with respect t time. u = s = particle speed Distance that the medium travels per unit time. t Nte that the average value f the particle velcity ver any cycle is zer. Traveling wave at x = 0 s (m) t (sec) Τ s 0 Putting these three pictures tgether, we have an expressin fr a traveling wave in a medium r mre cmpactly, π π s( x,t) = s sin x t λ T ( ) = [ ω ] s x,t s sin kx t We als have a new way f defining the speed f the wave. It makes gd sense that the wave speed is the distance the wave travels in ne cycle (the wavelength) divided by the time it takes the wave t cmplete ne cycle. It is a simple matter t substitute the frequency, f, fr the perid: -4

5 λ c= = fλ T The wave speed can als be calculated frm the angular frequency and the wavenumber: λ π ω c = = T π k We call waves mdeled using this result plane waves because in three dimensins the lcus f pints all having the same phase are planes. We call these planes wavefrnts and ften draw them as lines n a page separated by ne wavelength. In fact, the wavefrnts are actually parallel planes. We als find it cnvenient t shw the directin the wave is traveling using a ray which is cnstructed perpendicular t the wavefrnts. Spherical Wave Plane Wave Sund Waves When sund travels in a fluid, i.e a gas r a liquid, the displacement must be in the lngitudinal directin because fluids are pr at transmitting the shear frces necessary t sustain a transverse wave. Belw is a cartn f the lngitudinal displacement f a sund wave. We call the lcatins where the fluid is displaced int clumps f clsely spaced mlecules cndensatins (high density) and the lcatins where the fluid mlecules are sparsely spaced, rarefactins (lw density). The intermlecular frces tend t push ut n each ther at the cndensatins just as a cmpressed spring pushes back n a mass. The gas laws suggest the high density regins f a gaseus fluid are at a higher pressure (frce per unit area) and the lw density regins are at a lwer pressure. The same is true fr liquid fluids. -5

6 In additin t describing sund waves in fluids by the displacement f the mlecules, we can als describe the wave by the velcity f the mlecules r the variatins in density and pressure. Acustic Presssure In the case f pressure, static pressure frm the height f the clumn f fluid abve the wave are always present. This frce is cnstant with time. In SP11 we learned hw t calculate this pressure, p, using the fllwing equatin: p= p +ρ gh where ρ is the density f the fluid and h is the height f the fluid clumn. The acustic pressure due t the cndensatins and rarefactins sits n this static pressure and scillates arund it due t the presence f the acustic wave mtin. While we culd cnsider the entire pressure variatin in describing an acustic wave, we will, by cnventin, instead cnsider nly the pressure variatin frm the static pressure. -6

7 static pressure Pressure in a Fluid pressure acustic pressure time We saw that simple harmnic mtin has a gverning differential equatin called the equatin f mtin whse slutin gives the psitin f a mass as a functin f time. In the case f a traveling wave, there is an analgus equatin whse slutin describes the medium s particle displacement as a functin f psitin and time. This partial differential equatin is knwn as the wave equatin. In the next sectin we will shw hw the wave equatin fllws directly frm sme fundamental Physics principles. -7

8 Sund Waves in a medium the wave equatin Initial Psitin V I p (x 1 ) A p(x ) V I x 1 x Later Psitin V F p (x 1 ) A p(x ) A V I V F s x 1 x s 1 T derive the ne-dimensinal wave equatin, let's lk at the mtin f a small vlume f fluid. We can relate its mtin t the spring-mass system frm the previus sectin. If we apply a pressure gradient t the fluid vlume, V I, (such as an acustic pressure frm an acustic wave) it will mve and cmpress the vlume f fluid. The pressure n the left face f the fluid blck is p 1 (x 1 ), while that exerted n the right face is p (x ). If there is a differential pressure, p, then the fluid blck might mve t the right, and, as the blck accelerates, it will change t vlume, V F. We will make sme assumptins regarding the mvement f the blck: 1. The prcess is adiabatic n heat is lst r gained by the presence f the acustic wave. This is a reasnable assumptin because fr acustic wave frequencies in the cean, the wavelength is t lng and thermal cnductivity f seawater t small fr significant heat flw t take place.. Changes in particle displacement f the fluid frm equilibrium are small. 3. The fluid clumn is nt defrmed (shear defrmatin) by differential pressure. -8

9 T fully describe the mtin f sund in the fluid frm first principles, we will examine three well knwn Physics laws Newtn s Secnd Law, an equatin f state, and cnservatin f mass. These laws, cupled with the assumptins abve prvide a rbust and pwerful mdel fr underwater sund. Newtn s Secnd Law Newtn's Secnd Law is custmarily used by examining the frces in a particular directin and then summing them as vectrs. In the case f ur fluid vlume, V I, the frces in the x directin are: ( ) ( ) F = p x A p x A= pa x 1 This net frce acrss the vlume is equal t the mass times the acceleratin f the vlume. The mass is fund by multiplying the initial density by the initial vlume ( x = x -x 1 ) m= ρ A x I The acceleratin in the x directin is the secnd time derivative f average displacement and a s = t x s s = + s 1 Substituting int Newtn s Secnd Law, F = ma becmes a x x s pa = ρia x and rearranging gives t p s =ρi r mre apprpriately x t p x s = ρ t In the final result, acustic pressure was used since the derivative f the static pressure is zer. Additinally, the instantaneus density and displacement fr an infinitesimally small vlume are substituted. -9

10 Equatin f State and Cnservatin f Mass Even thugh we think f liquids mstly as incmpressible fluids, in reality, they are nt. The Bulk Mdulus f Elasticity describes hw much the vlume f the liquid changes fr a given change in pressure. In equatin frm this is: ( ) ( ) ( ) p x p x1 pa B - VF VI VI V VI The significance f the negative sign in abve equatin is that when p a is psitive, then V F <V I and V is negative. Using slid gemetry we can develp an expressin t relate the acustic pressure t the displacement f the small vlume in the abve figure. Implied in this argument is the law f cnservatin f mass. We are nt allwing any f the medium t escape the vlume, nr are we allwing any additinal mass t seep in. V I F = A x ( ) V = A x+ s (Nte: 1 ( A x+ s A x) ( ) ( ) I s = s - s is a negative number) VF VI V s V = V = A x = x I Thus substituting in the last tw equatins and rearranging the definitin f the Bulk Mdulus f Elasticity: p a V = B V I s pa = B r mre crrectly x s pa = B x Substituting this last result int ur previus relatinship between pressure and displacement: pa s s s = ρ = B B = x t x x x -10

11 The One Dimensinal Wave Equatin Substituting the cnclusin frm cnservatin f mass and equatin f state int Newtn s Secnd Law results in the ne-dimensinal wave equatin that we can use t describe the displacement, s, frm their rest psitin f particles in a medium, with respect t time and psitin. This equatin is a partial differential equatin with a slutin that varies with time and psitin. As with the mass-spring system equatins, if we can find an equatin that satisfies this secnd rder differential equatin, the equatin culd be used t describe the mtin f the particles in the medium. Recall that: s ρ s = x B t One slutin that we will use was described abve as a plane wave and has the frm: ( ) = ( ±ω ) s x,t s sin kx t s 0 = amplitude f the scillatin r maximum displacement k = π/λ is the wave number ω = πf = π/t is the angular frequency ± determines the directin that the wave travels (+ is fr a wave traveling t the left, - is fr a wave traveling t the right) T check the validity f this slutin we must take the apprpriate secnd derivatives: Substitutin int the wave equatin s sin kx t sk sin kx t x s sin kx t s sin kx t t ( ω ) = ( ω ) ( ω ) = ω ( ω ) r ρ sk sin kx ω t = sω sin kx t B ( ) ( ω ) k ρ = ω B Rearranging and recalling that the speed f the wave, c = ω/k, -11

12 ω = k c B = ρ This is a fairly prfund result. It tells us that the plane wave slutin fr particle displacement is a gd slutin prvided the speed f the wave is nt arbitrary, but exactly equal t the square rt f the bulk mdulus divided by the density. When the bulk mdulus and density f water are used, a nminal value fr the speed f sund in water is 1500 m/s. This agrees with measured results. Had we used an equatin f state fr a gas instead f a liquid, we wuld have arrived at a similar result fllwing a similar prcedure. The plane wave slutin wuld still slve the wave equatin, but the wave speed wuld becme: γnrt c = m When typical rm temperature numbers are used, this results in a nminal speed f sund in air f 340 m/s. The rules f differential equatins make n statement abut the uniqueness f a slutin t the wave equatin. Many ther slutins exist as well. Had the slutin been expressed as a csine vice a sine, the wave equatin wuld still have been satisfied. Additinally, cmplex expnentials culd have been used as a slutin due t Euler s identity. s(x,t) = s e ( ωt) This expressin is really shrthand fr the real (r imaginary) part f the cmplex expnential. A Gaussian pulse f the fllwing frm als satisfies the wave equatin. ikx s(x, t) = s e kx ωt ωτ Additinally, if a certain frequency wave satisfies the differential equatin, all multiples r harmnics f that frequency must als wrk. ( ) = ( ± ω ) s x,t s sin nkx n t Rules fr differential equatins als specify that linear cmbinatins f slutins are als slutins. This is called the principle f superpsitin. A methd using the thery develped by a French mathematician named Furier will allw disturbances f almst any shape t be cnstructed using series f harmnic plane waves. These disturbances will still themselves be slutins t the wave equatin. -1

13 Alternate Views fr Describing an Acustic Wave The Pressure Field S far, we have viewed sund mving in a fluid as a harmnic traveling wave, cnsidering nly particle displacements. This is nt a unique view. Just as an electrmagnetic wave can be seen as an scillating electric field r and scillating magnetic field, s t can a sund wave be seen as an scillating pressure field, an scillating velcity field r an scillating density field. Of curse, the fundamental difference remains that the electrmagnetic wave is always a vectr field, while the sund wave in a fluid is generally a scalar field. Using the slutin fr the wave equatin, s( x,t) s sin( kx t) = ω, we can find the equatins fr tw f these fields. First the we will find the acustic pressure. Previusly we fund the relatinship f the acustic pressure p a, and the displacement f the small vlume frm the equatin f state. Using this we get: ( ) s pa x,t =-B x ( ) s0sin kx- wt pa( x, t ) = -B = -Bs0kcs kx- wt x 0 ( ) The first imprtant bservatin abut the pressure field relative t the displacement field is that they are 90 degrees ut f phase with each ther. This means that when the particle displacement f the medium is at a maximum, the acustic pressure is at a minimum. Additinally, when the displacement is zer, the maximum acustic pressure is: pamax = Bsk =ρc sk By cnventin, acusticians prefer nt t use an engineering mdulus, B, instead substituting B=ρc. Alternate Views The Velcity Field and Specific Acustic Impedance The particle velcity is nt the wave velcity. The speed that the wave travels is a functin f the medium and is a cnstant. The speed f sund, c, is given by the equatins: B λ ω c= = = fλ= ρ T k The particle velcity f the medium, n the ther hand tells us hw fast the mlecules in the fluid are mving. It is fund by simply taking the time derivative f the equatin describing the psitin f the medium, the plane wave slutin. s u( x,t) = t s0 sin( kx ωt) u ( x, t) = = s0ωcs( kx ωt) t where u = s ω= s ck max

14 It is ntewrthy that fr a plane wave, the particle velcity and the particle displacement are 90 degrees ut f phase, but that the velcity and acustic pressure are in phase. We can als find the maximum particle velcity frm the characteristics f the wave. In yur electrical engineering classes, yu were intrduced t a quantity called impedance. It was the rati f the driving frce in a circuit, the vltage, t the rate at which charge passes by a pint in the circuit, the current. V Zelectric % % I By analgy, the driving frce in an acustic wave is the pressure and the rate at which particles in the medium pass a particular pint is the velcity. It is n accident that we define the specific acustic impedance as the rati f the pressure t the particle velcity. p(x, t) z u(x,t) Fr the case f a plane wave we have fund expressins fr bth the pressure and velcity fields. p ρc s0k cs( kx ωt) z = =ρc u s ck cs kx ωt 0 ( ) The specific acustic impedance relates the characteristics f a sund wave t the prperties f the medium in which it is prpagating. Nminal values fr the density, ρ, and the wave speed, c, fr water are ρ = 1000 kg/m 3 and c = 1500 m/s. D nt be cnfused int thinking that specific acustic impedance is always the prduct f density and the speed f sund. This is nly true fr a plane wave. Fr ther gemetries, fr instance a spherically spreading wave, the specific acustic impedance is a different expressin even in the same fluid. Mre n Cntinuity f Mass The Density Field When mtivating the wave equatin, it was mentined that the mass in ur test fluid vlume was nt changing. Specifically, the initial mass in psitin I is the same as that in psitin F. ρ V =ρ V I I F F Recalling ur expressin fr the equatin f state and substituting, ρi VI VI VF VI ρ F ρ I ρi ρ F ρf ρ I pa = B = B = B 1 = B B VI VI ρi ρf ρi We find that the fractinal change in density, ρf ρ I is directly prprtinal t the pressure. ρi This fractinal change in density is called a cndensatin variable. It is ften written, ρ ( x,t) ρ p = a = skcs ( kx ω t) ρ B 0-14

15 We have develped fur different descriptins fr a traveling acustic plane wave; particle displacement, particle velcity, acustic pressure and fractinal change in density. Particle displacement is 90 degrees ut f phase with the ther three, but all fur discriptins travel with the same wave speed and have the same perid and wavelength. All fur can be used t prperly mdel acustic effects. Energy in a Sund Wave Missing in the discussin f wave equatins and their slutin is any mentin f energy. We started the semester with a review f simple harmnic (sinusidal) mtin. The reasn we did this shuld be apparent t yu by nw. As a plane wave traverses any medium, all specific particle lcatins underg simple harmnic mtin as the wave passes by. Because f this, we can use the basic SP11 equatins fr kinetic and ptential energy f the medium. The nly mdificatin is t replace mass with density s as t calculate energy density r energy per unit vlume. This is a lgical mdificatin since the medium carrying the wave is cntinuus. It wuld make n sense t identify a particular piece f mass, nr the ttal mass. The equatins fr kinetic and ptential energy density in a simple harmnic scillatr are respectively as fllws 1 ε K = ρ u 1 1 khkes mω s 1 ε P = = = ρω s V V Since we have equatins fr particle displacement and particle velcity, we can simply substitute these int the abve ε K = ρ u = ρ s cs( kx t) scs ( kx t) ω ω = ρω ω ε P = ρω s = ρω ssin( kx ω t) = ρω ssin ( kx ω ) t It shuld be clear that the ttal energy is the sum f the ptential and kinetic energy and that when the kinetic energy is maximum the ptential energy is zer and vice versa. The questin f hw the energy is partitined depends n when yu ask the questin. The average energy in a simple harmnic scillatr is calculated using the fllwing definitin fr a peridic functin: 1 T T () () f t 0 f t dt Fr kinetic and ptential energy we find that since the time average f sin () t cs () t ε = ρω s cs ( kx ω t) = ρω s = ρu ε = ρω s sin ( kx ω t) = ρω s = ρu 4 4 K max P max θ = θ =, 1-15

16 This shws that n average, the kinetic energy f a plane wave and the ptential energy f a plane wave are the same, each being exactly ne half the ttal energy f the harmnic scillatr. The ttal average energy density f the wave is then, 1 ε = ε K + ε P = ρ u m ax p p Using the acustic impedance, u = z = ρ c allws us t write the ttal energy in terms f maximum pressure. 1 pamax ε = ρc Acustic Intensity Acustic intensity, I, is defined as the amunt f energy passing thrugh a unit area per unit time as the wave prpagates thrugh the medium. As we described in SP11, energy mved per unit time is pwer which has units f Watts. Intensity then must have units f Watts/m. Pwer Wrk 1 Frce displacement I = = = Area time Area Area time I = [ Pressure velcity] This unit analysis suggests acustic intensity can be calculated frm the prduct f acustic pressure and particle velcity. I= p u where a ( ) ( ) ( ω ) p = p sin kx- ω t and u = u sin kx- t a amax max p but u = ρ c p I x,t = a ( x,t) ρc One imprtant thing t nte is that since the acustic pressure is a time-varying quantity, s is the intensity. We will use a mre meaningful quantity, the time average acustic intensity. The average intensity f an acustic wave is the time average f the pressure ver a single perid f the wave and is given by the equatin: I p a = ρc Since the time averaged acustic pressure is written: p 1 a = pamax I 1 p = amax ρc, the average acustic intensity can be -16

17 This result lks remarkably similar t the average energy density f a traveling plane wave. In fact this is nt accidental. If yu cnsider a plane wave as a cylinder f length cdt and crss sectin A, the ttal energy in this cylinder, de, wuld be the prduct f the energy density and the vlume. de = ε Acdt Rearranging we see an alternative expressin fr average acustic intensity, cdt A I 1dE = = ε Adt c Since 1 p ε = ρc amax, the average acustic intensity is again I 1 p = amax. ρc This result is pleasing in that it agrees with an analgy suggested earlier between vltage and pressure. In yur electrical engineering class, yu learned that electric pwer was vltage squared divided by impedance. Average pwer was fund using 1 V P = max Z Nw we have fund that average acustic pwer per unit area is simply acustic pressure squared divided by specific acustic impedance. P 1 p I = = amax A z This sheds light n why the mdifier specific precedes acustic impedance. By analgy, specific acustic impedance, z, must be acustic impedance Z, divided by area. T further make use f electrical engineering backrund, time averaged pressure may als be determined by: p = p a rms p p = p = therefre: I max a rms p p = = ρc ρc max rms Lastly, frm this pint further, unless therwise nted, when we refer t the intensity f the wave, we actually mean the time-averaged intensity. -17

18 Review and definitin: 1. displacement (s) distance fluid particle mves frm equilibrium (meters). perid (T) time required t cmplete ne cmplete scillatin (secnds) s 3. particle velcity (u) displacement/time (meters/secnd) = t β ω 4. wave speed (c) speed the wave frnt is mving (meters/secnd) where c = = = λf ρ k 5. frequency (f) 1/T (Hz r 1/secnd) 6. angular frequency (ω) πf (radians/secnd) 7. wave lengths (λ) distance between same amplitude pints f tw successive wave frnts (meters) 8. wave number (k) π/λ (1/m) 9. wave frnts surface ver which all particles vibrate in phase 10. acustic ray a vectr perpendicular t the wave frnt pinting in the directin f prpagatin at ne specific 11. static pressure (p s ) pressure f envirnment minus any changes due t sund wave (Pa r N/m ) 1. acustic pressure (p a ) pressure fluctuatins due t presence f wave mtin f particle displacement (Pa) 13. instantaneus pressure (p tt ) static plus acustic pressure at any ne instant 14. plane waves small segment f a spherical wavefrnt at a lng distance frm the surce 15. rms pressure (p rms ) = pa rt mean square value f the acustic pressure (Pa) 16. Intensity (I) = p a max/ρc = p rms/ρc 17. acustic impedance (z) p/u =ρc = ρω/k 18. Bulk Mdulus f Elasticity (B) prvides relatinship between change in pressure t change in vlume f unit f fluid 19. density (ρ) mass cntained in a unit vlume f fluid (kg/m 3 ) -18

19 Prblems 1. A sund wave prpagates a pint abut 50 meters belw the surface f a calm sea. The instantaneus pressure at the pint is given by: p = 6x sin( 400πt ), where t is in secnds and p in Pascals. a) What is the value f static pressure at the pint? b) What is the value f maximum (r peak) acustic pressure at the pint? c) What is the rt-mean-square acustic pressure? d) What is the acustic pressure when t=0, 1.5,.5, 3.75, 5.00 millisecnds? e) What is the average acustic intensity f the sund wave? (The density f the water is 1000 kg/m 3 and the sund speed is 1500 m/sec.) f) What is the intensity level, L, in db re 1 µpa?. A plane acustic wave is prpagating in a medium f density ρ=1000 kg/m 3. The equatin fr a particle displacement in the medium due t the wave is given by: 6 s = ( 1x10 ) cs( 8πx 1000πt ), where distances are in meters and time is in secnds. a) What is the rms particle displacement? b) What is the wavelength f the sund wave? c) What is the frequency? d) What is the speed f sund in the medium? e) What is the value f maximum (r peak) particle velcity? f) What is the value f maximum acustic pressure? g) What is the specific acustic impedance f the medium? h) What is the bulk mdulus f the medium? i) What is the acustic intensity f the sund wave? j) What is the acustic pwer radiated ver a 3 m area? 3. A plane acustic wave is prpagating in a medium f density ρ and sund speed c. The equatin fr pressure amplitude in the medium due t the wave is given by: p = p0 cs( kx ωt), where p 0 is the maximum pressure amplitude f the sund. π a) Shw that the equatin abve can be written in the frm, p = p0 cs ( x ct). λ b) Shw that maximum pressure amplitudes (cmpressins) can be fund at the fllwing lcatins in space: x=nλ+ct where n= 0, 1,, 3, c) Shw that maximum pressure amplitudes (rarefactins) can be fund at the fllwing lcatin in space: x=(n+1/)λ+ct, where n = 0, 1,, 3, 4. A plane acustic wave travels t the left with amplitude 100 Pa, wavelength 1.0 m and πx 3000π frequency 1500 Hz; p1 = 100Pacs + t, while anther plane wave travels t m sec the right with amplitude 00 Pa, wavelength ½ m and frequency 750 Hz: πx 1500π p = 00Pacs + t. m sec -19

20 a) Find the rms average ttal pressure. Yur answer will nt depend n distance x. (Hint: rms average pressure P ttal b) If p1 = p0 cs( kx ωt) and p = p0 cs( kx ω t +φ ), where the <> symbl dentes a time average. 1, find the rms average ttal pressure. 5. Given the fllwing equatin fr an acustic wave, riginating frm a surce in the cean 5 π π ( ) [ 160] p x, t = 8x10 Pa sin x t 13 m sec Determine the fllwing: a) The wavelength b) The rms pressure f the wave c) What is the frequency f the wave? d) The time averaged intensity f the acustic wave 6. If the particle displacement can be fund t be: 6 π π ( ) [ 160] s x, t = 6x10 m cs x t 13 m sec a) What is the value f the peak particle velcity? b) What wuld be the maximum acustic pressure if the Bulk Mdulus f Elasticity f the medium were.0x10 9 N/m? 7. If a pressure pulse frm a small explsin in water is knwn t be equal t ( ) t 0.1sec p = 1000Pa e at x = 0 a) Cnstruct a slutin t the wave equatin fr the pulse prpagating t the right. This expressin must be in the frm f a functin f x and t. b) Sketch p(x,t) frm part a) fr time t = 0, t = 0.1 s, and t = 0. s. 8. If an acustic pressure pulse in water at x = 0 is knwn t be p p() t = where τ = 1 millesec, p = 1 Pa t 1+ τ a) Find a wave expressin fr the pressure pulse traveling in the x-directin t the left. b) Find an expressin fr the intensity f the wavefrm fund in part a). 9. What is the speed f sund in yards per secnd in: a) air? b) water? -0

21 Lessn Traveling Waves Types Classificatin Harmnic Waves Definitins Directin f Travel Speed f Waves Energy f a Wave Waves Types f Waves Mechanical Waves - Thse waves resulting frm the physical displacement f part f the medium frm equilibrium. Electrmagnetic Waves - Thse wave resulting frm the exchange f energy between an electric and magnetic field. Matter Waves - Thse assciated with the wave-like prperties f elementary particles. Requirements fr Mechanical Waves Sme srt f disturbance A medium that can be disturbed Physical cnnectin r mechanism thrugh which adjacent prtins f the medium can influence each ther. Classificatin f Waves Transverse Waves - The particles f the medium underg displacements in a directin perpendicular t the wave velcity Plarizatin - The rientatin f the displacement f a transverse wave. Lngitudinal (Cmpressin) Waves - The particles f the medium underg displacements in a directin parallel t the directin f wave mtin. Cndensatin/Rarefractin Waves n the surface f a liquid 3D Waves 1

22 Lessn Sund Waves Harmnic Waves Transverse displacement lks like: s (m) At t = s 0 λ x (m) π s( x) = s0 sin x λ s (m) Let the wave mve Traveling Wave ct x (m) s( x,t) s0 sin π = ( x ct) λ Standing at the rigin Transverse displacement lks like: s (m) At x = Τ s t (sec) π πc π s( 0,t) s sin ( 0 ct) s sin t s sin t λ λ T = = = Phase Velcity That negative sign distance mved in ne cycle c = λ f time required fr ne cycle = T = λ Wave mving right π π s( x,t) = s sin x t λ T Wave velcity is a functin f the prperties f the medium transprting the wave Wave mving left π π s( x,t) = s sin x+ t λ T s ct s ct -s -s

23 Lessn Wave number Angular frequency Alternate ntatin π π s( x,t) = s0 sin x t λ T ( ) = [ ω ] s x,t s sin kx t π k = λ π ω= T λ λ π ω c = = = T π T k Definitins Amplitude - (s ) Maximum value f the displacement f a particle in a medium (radius f circular mtin). Wavelength - (λ) The spatial distance between any tw pints that behave identically, i.e. have the same amplitude, mve in the same directin (spatial perid) Wave Number - (k) Amunt the phase changes per unit length f wave travel. (spatial frequency, angular wavenumber) Perid - (T) Time fr a particle/system t cmplete ne cycle. Frequency - (f) The number f cycles r scillatins cmpleted in a perid f time Angular Frequency - (ω) Time rate f change f the phase. Phase - (kx - ωt) Time varying argument f the trignmetric functin. Phase Velcity - (v) The velcity at which the disturbance is mving thrugh the medium Tw dimensinal wave mtin Spherical Wave Plane Wave static pressure Acustic Variables s( x,t) = s sin[ kx ωt] pttal = pstatic + pa acustic pressure θ =θ i r Displacement s ParticleVelcity u = t Pressure ρ( x,t) ρ Density ρ Cndensatin = Cmpressin Rarefactin = Expansin A micrscpic picture f a fluid The Wave Equatin Initial Psitin V I A p(x p (x ) 1) V I x 1 x Later Psitin - V F p(x A ) p (x 1) V I s x 1 x s 1 V F A Assumptins: Adiabatic Small displacements N shear defrmatin Physics Laws: Newtn s Secnd Law Equatin f State Cnservatin f mass Newtn s Secnd Law/ Cnservatin f Mass Equatin f State/ Cnservatin f Mass PDE Wave Equatin pa s = ρ x t s pa = B x s ρ s = x B t 3

24 Lessn Slutins t differential equatins Guess a slutin Plug the guess int the differential equatin Yu will have t take a derivative r tw Check t see if yur slutin wrks. Determine if there are any restrictins (required cnditins). If the guess wrks, yur guess is a slutin, but it might nt be the nly ne. Lk at yur cnstants and evaluate them using initial cnditins r bundary cnditins. The Plane Wave Slutin ( ) = ( m ω ) s x,t s sin kx t s ρ s = x B t ρ sk sin kx ω t = sω sin kx ωt B ( ) ( ) k ρ = ω B ω B = c = k ρ General rule fr wave speeds Bulk mdulus B Sund Speed c = = density ρ Lngitudinal wave in a lng bar Lngitudinal wave in a fluid c = Elastic Prperty Inertial Prperty Yung's mdulus Y c = = density ρ Bulk mdulus B c = = density ρ Air Bulk Mdulus 1.4(1.01 x 10 5) Pa Density Speed 1.1 kg/m m/s Variatin with Temperature: Air Seawater Sea Water.8 x 10 9 Pa 106 kg/m m/s v ( T) m s ( ) v T.051T T s 3 m Example Alternate Slutins A plane acustic wave is prpagating in a medium f density ρ=1000 kg/m3. The equatin fr a particle displacement in the medium due t the wave is given by: 6 s = ( 1x10 ) cs( 8πx 1000πt ) where distances are in meters and time is in secnds. What is the rms particle displacement? What is the wavelength f the sund wave? What is the frequency? What is the speed f sund in the medium? ( ) = ( ±ω ) s x, t s cs kx t s(x,t) = s e ( ) = ( ± ω ) s x,t s sin nkx m t s(x,t) = s e ( ωt) ikx kx ωt ωτ 4

25 Lessn Superpsitin Superpsitin Waves in the same medium will add displacement when at the same psitin in the medium at the same time. Overlapping waves d nt in any way alter the travel f each ther (nly the medium is effected) Furier s Therem any cmplex wave can be cnstructed frm a sum f pure sinusidal waves f different amplitudes and frequencies Particle Displacement Particle Velcity Pressure Density Alternate Views ( ) = ( ±ω ) s x,t s sin kx t s u = = s0ωcs kx ωt t ( ) s pa x, t = -B = - c s0kcs kx- wt x ( ) ρ ( ) ( x,t) ρ ρ p = a = skcs kx ω ρ0 B ( t) Audible Infrasnic Ultrasnic Pitch is frequency 0 Hz 0000 Hz < 0 Hz >0000 Hz Middle C n the pian has a frequency f 6 Hz. What is the wavelength (in air)? 1.3 m Specific Acustic Impedance Like electrical impedance Acustic analgy Pressure is like vltage Particle velcity is like current Specific acustic Impedance: Fr a plane wave: 0 Z electric V % % I p(x,t) z u(x,t) ( ωt) ( ω ) p ρc s0k cs kx z = =ρc u s ck cs kx t Energy Density in a Plane Wave ε K = ρ u = ρ s cs( kx t) scs ( kx t) ω ω = ρω ω ε P = ρω s = ρω ssin kx ω t = ρω ssin kx ωt 1 1 p ρ u = ρc amax max εk ( ) ( ) εp 5

26 Lessn Average Energy Density ε = ρω s cs ( kx ω t) = ρω s = ρu ε = ρω s sin ( kx ω t) = ρω s = ρu 4 4 K max P max 1 1 p ρ u = ρc 1 ε = ε + ε = ρ u 1 pamax Or ε = ρc amax max K P max εk εp Average Pwer and Intensity de = ε Acdt A cdt de P = = ε Ac dt P 1 1 p 1 I = = ε c= ρ cu = = p u A ρc amax max a max max Instantaneus Intensity ( x,t) pa I( x,t) = pa ( x,t) u( x,t) = = z u( x,t) z Pwer Wrk 1 Frce displacement I = = = Area time Area Area time I = Pressure velcity [ ] V Z P= = ZI = VI Rt Mean Square (rms) Quantities p p = p = therefre: I max a rms p p = = ρc ρc max rms 6