Waves and Sound for PHYS1169. Joe Wolfe, UNSW Waves are moving pattern of displacements. Ma transmit energ and signals. 1169 Sllabus Travelling waves, superposition and interference, velocit, reflection and transmission, harmonic waves, spherical and plane waves. Sound. Doppler effect, standing waves in strings and air columns, beats, decibel scale Light. Lab Lab Ra approimation & geometric optics: Reflection and refraction, Hugen's pple, total internal reflection, mirrors, images, lenses, magnifier, compound microscope, telescope Interference and Diffraction Conditions for interference, Young's eperiment, and interference pattern, phasor addition, reflection, thin films, diffraction Mechanical waves eample tpe restoring force Wave in string transverse tension in string Water wave transverse gravit Sound wave longitudinal air pressure Onl pattern travels, not medium. Travelling wave f( vt) is a wave travelling at v in + dir n : An important eample = m sin 2π ( - vt) λ A sine wave travelling to the right m λ 1 t 0 t > 3 t 0 Each point: Simple Harmonic Motion with period T. t m Harmonic waves One ccle of SHM takes T, and wave travels λ, v = λ T = fλ
Write wave equation in various was: = m sin 2π ( - vt) λ = m sin 2π λ - ft define angular frequenc ω 2πf and define wave number k 2π λ = m sin ( k - ωt) You need practice with this: do some probs on tut 8! Eample A wave has = m sin ( k - ωt ), m = 10 nm, k = 18.5 m -1, ω = 6300 rad.s -1 i) what is the speed of the wave? ii) What is (ma) average speed of particles? Onl know k & ω. How to get v? v = what? v wave = fλ now use k, ω m λ = ω 2π. 2π k = ω k =... = 340 ms-1 2 what is v part? t m v part = t = t m sin ( k - ωt) = ω m cos ( k - ωt) v ma = ω m = 6300 s -1 10 nm = 63 µm.s -1 average speed, not individual speed, cf wind Speed of wave In a string, it depends on Tension (F) and the mass per unit length (µ = m/l) of the string. Dimensional analsis: v T a (µ) b dimensions: lt -1 = (mlt -2 ) a (ml -1 ) b equate the powers: t: 1 = 2a a = 1 2 m: 0 = a + b b = a = 1 2 l: (check) 1 = a b = 1 2 + 1 2 consistent v string = T µ
3 In general, v wave = spring thing inertial thing v sound = elastic const densit c = k elec k mag γ = c P cv = 1 µ o ε o = γp ρ is ratio of specific heats c P c v Eample The bulk modulus B of elasticit of water is 2.0 GPa. How fast is sound in water? v = spring thing intertial thing for string it was v = Tension 1 D densit tr v = B ρ units: = Pa kg.m -3 = F/area kg.m -3 = kg.m 1 s 2 m 3 m 2 kg v = B ρ what is ρ? v = 2 109 Pa 10 3 kgm -3 = 1.4 km.s -1 Eample Bob and Mar float in water deep enough for waves to be sinusoidal. The wave speed is 10 ms -1. Their vertical position varies 2.4 m, Mar gets to the top just as Bob gets to the bottom. The eperience maimum accelerations of 0.015 g. How far apart are the? M B 2.4 m B = m sin ( k - ωt ) we need λ. let's write wave equation for a start but we know a m a = 2 2 = m ω 2 sin ( k - ωt) a ma = m ω 2.015 g = 0.15 ms -2 = 2.4 m 2 ω = 0.35 s -1 f = 0.056 Hz ω 2 λ,τ λ = v f = 180 m.
Eample. A rope of length L hangs verticall. How long does it take a wave to travel from one end to the other? 4 z T L well we'll need the wave speed... v = T µ where µ = m L T = weight below that point = (µz)g but T varies v = dz dt dt = = µzg µ 1 g. z 1/2 dz L z -1/2 dz dt = dt = z = 0 g = [ 2 g. z 1/2 L ] 0 = 2 L check units g Reflection: Going from less dense to more dense, waves are reflected with a phase change of π. e.g. reflection at a 'fied' end thin string to thick string, air to water From more dense to less dense, no phase change e.g. reflection at 'free' end, etc Superposition In a linear medium, waves superpose linearl, i.e. their displacements simpl add. Most media linear for small amplitude waves. But beware water waves when m depth sound waves when p m P atmos Superpose incident & reflected waves standing waves non linear
Standing waves 1 = m sin(k ωt) 2 = m sin(k + ωt) T = m sin k cos ω t cos k sin ω t + sin k cos ω t + cos k sin ω t = 2 m sin k cos ωt stationar simple harmonic wave motion 1. Boundar fied (no displacement) phase change of 180 e.g. reflection at a hard surface node at boundar 2. Boundar free (an displacement) phase change of zero e.g. reflection at bell of trumpet anti-node at boundar Standing waves 5 f 1 = v λ = v 2L (fundamental) f 2 = v λ = 2v 2L = 2f 1 (2 nd harmonic) f 3 = v λ = 3v 2L = 3f 1 (3 rd harmonic) f 4 = v λ = 4v 2L = 4f 1 (4 th harmonic) &? 300 400 500
Eample Write the equations for two travelling waves which together in superposition could produce a standing wave. 1 = Asin(k ωt) and 2 = Asin(k + ωt + φ) (φ ma have an value including zero) + = 6 φ = 0, but also: φ of second wave = 90 + = Eample A cable is 8 m long, 8 mm in diameter, and subject to a tension of 7.0 kn under some conditions. The cable has ρ = 5,600 kg.m -3. i) Estimate the first 5 resonant frequencies of the cable. ii) So what? i) The possible standing wave resonances have λ = 2L, L, 2L/3,... λ n = 2L/n f n = v λ n = n 2L T µ µ = m L = πr2 L.ρ = ρπr L 2. n f n = 2L T =... = n * 9.9 Hz ρπr 2 f n s 10, 20, 30, 40, 50 Hz ii) Lots of energ stored at resonance! vibrations could be structurall serious.
Sound is a compression wave - longitudinal Write displacement (r,t) but is in r direction 7 r ρ r Radiation in 2 dimensions Radiation in 3 dimensions r P A Intensit I power area Source, power P, radiates isotropicall in 3D P I = 4πr 2 Power in a wave: energ in spring U s = 1 2 k s 2 Energ 2 Intensit (displacement) 2 Eample An air duct is closed at one end but open at the other. It is 3.4 m long. What are its resonant frequencies? So what? It can have a displacement note at closed end, antinode at the open end λ = 4L, 4L 3, 4L 5, 4L 7 etc f = v λ = v 4L, 3v 4L, 5v 4L,... = 25 Hz, 75 Hz, 125 Hz Can store large amounts of energ at these frequencies: noise & structural vibration.
Eample: What is the intensit of solar radiation? P sun = 3.9 10 26 W. Earth is 150 million km from sun. 8 R R I = P 4πr 2 =... = 1.38 kwm -2 above atmosphere, radiation Intensit and sound level Sense of loudness is also ~ logarithmic, define Sound intensit level: L I 10 log I 10 I o where I o = 10-12 W.m -2 (L I in decibels) Note: Power displacement 2 pressure 2 L 2 L 1 = 10 log I 2 10 Io log I 1 10 Io = 10 log I 2 10 I1 Eamples of p, I, L p, L I p 2 /p 1 L p I 2 /I 1 L I 2 3 db 2 3 db 2 6 db 4 6 db 10 10 db 10 10 db 10 20 db 100 20 db http://www.phs.unsw.edu.au/~jw/db.html
Sound radiation: If sound radiates uniforml in three dimensions: power Intensit unit area 9 But note: I p 2, p 1 r Uniform spherical radiation: r 2 /r 1 I 2 /I 1 p 2 /p 1 L 2 L 1 2 1/4 1/2 6 db 10 1/100 1/10 20 db Eample. If sound level L I = 3 db at 10 cm from a source radiating uniforml, what is the acoustic power of the source? L I, I & r P 3 db = L I 10 log I I o 0.3 = log I I o I/I o = antilog 0.3 = 10 0.3 = 2 I = 2 I o = 2 10-12 Wm -2 I = P A = P 4πr 2 P = 4πr 2 I = 4π (0.10m) 2 ( 2 10-12 Wm -2 ) = 0.25 pw (2.5 10-13 W) Effect of boundaries: A completel reflecting wall absorbs no energ: I = P A = P 2πr 2 = 2 intensit from free radiation busker standing net to a wall gains 3 db (but so does some of the background noise). What if ou stand in a (reflecting) corner? Eample. A loudspeaker at floor level produces 1 W of acoustic power. What sound level does it produce at a distance of 3 m? L I 10 log I I o = 10 log P/2πr2 I =... o = 103 db
Doppler effect. 1. Stationar source 10 source S v o Observer at rest receives v/λ crests/unit time. Moving observer crosses v o /λ etra crests/unit time. Observer hears f' = v λ + v o λ = v + v o v/f f' = f 1 + v o v Note convention: v o is positive if approaching. Doppler effect. 2. Moving source Wavefront 1 emitted from position 1, etc (ft) crests are spread over (vt - v s t) For observer, λ' = vt - v st ft f ' = v v = f λ ' v v s v + v For both moving: f ' = f o v v s v o and v s are positive for approaching measure all velocities with respect to medium Eample. How fast must ou run (biccle?) to reduce the pitch b one semitone (6%)? v + v 0.94 f = f' = f o v 0 0.94 = 1 + v o v v = 340 ms -1... v o = 20 ms -1 Eample. i) Car approaches stationar observer at 150 kph. What doppler shift? ii) What if observer approaches stationar source at 150 kph? Moving source Moving observer v + 0 f ' = f v v s 114f v + v f ' = f o v 0 112f
What if v s > v? Shock wave 11 Crests combine to form a shock wave. Cone has half-angle θ where sin θ = v v s v s v is called the Mach number Eample Plane travelling at 2000 km.hr and height 5 km. Where is the plane when ou first hear it? sin θ = v sound v θ = 38 plane h D = tan θ,... D = 6.5 km
12 Beats http://www.phs.unsw.edu.au/~jw/beats.html Add two sine waves of similar frequencies 1 = A cos 2πf 1 t 2 = A cos 2πf 2 t cos A + cos B = 2 cos A B 2 cos A+B 2 1 + 2 = 2A cos 2π f 1+f 2 2 average frequenc t cos 2π f 2 f 1 2 Beat frequenc is f 2 f 1. Eample. You walk towards a wall, blowing a whistle at f = 500 Hz. You hear beats at 5 Hz between our whistle and the reflected sound. How fast are ou walking? t v w You hear our own whistle at frequenc f. The wall receives f' = f v + v o = f v + 0 v v s v v w This is the source of the reflection. You hear f" = f' v + v o v v s f" = f v + v w v v w f"(v v w ) = f (v + v w ) (f" f)v = (f" + f)v w v w = v f" f f" + f = f' v + v w v 0 = v 5 1005 = 1.7 ms-1
13 Eample An air duct is closed at one end but open at the other. It is 3.4 m long. What are its resonant frequencies? It can have a displacement node at closed end, antinode at the open end λ = 4L, 4L 3, 4L 5, 4L 7 f = v λ = v 4L, 3v 4L, 5v 4L, 250 Hz, 750 Hz, 1250 Hz etc Eample An organ pipe is closed at one end. It is tuned to 440 Hz (called A4). Another is tuned so that it makes beats with the first at 2 Hz. What is the difference in length? (You ma neglect end effects) L λ 4 = v 4f L 2 L 1 = v 4 1 f 1 2 f 1 f 1 = 440 Hz. f 2 = 438 Hz or 442 Hz L = 0.9 mm Eample How much must I increase the voltage applied to a loudspeaker in order to get an increase in sound level of 80 db? L 10 log 10 I 1 Io L 2 L 1 = 10 log I 2 10 Io log I 1 10 Io = 10 log I 2 10 I1 Speakers ~ resistors: P I A = V2 /R A R & A constant L 2 L 1 = 10 log V 2 2 10 V 2 1 log V 2 2 10 V 2 = L 2 L 1 1 10 = 8 V 2 2 V 2 = 108 1 Must increase voltage b factor of 10 4 V P I L
14 Eample. A guitarist tunes the A string of her guitar to 110 Hz. She then wants to tune the E string to 3 110 = 82.5 Hz. How to do 4 this A string: f 3A = v λ = 3v 2L = 3f A f 4E = v λ = 4v 2L = 4f E (3 rd harmonic on A string) (4 th harmonic on E string) f When 3f A = 4f E, E f = 3 A 4 So get the strings appro in tune and then remove the beats w? w 3f 2f f A A A w w 4f 3f 2f f E E E E Eample Which is faster, sound in air, or sound in Aluminium? v = elastic constant inertial constant Elastic constant for pressure deformation: V V p κ where κ is the bulk modulus of elasticit. κ is large for solids! κ Al = 70 GPa ρ Al = 2698 kg.m -3 v Al = κ Al ρ Al = 5 km.s -1
15 Eample A wave travels in a stretched string. Derive an epression for the ratio of the speed of the string to the slope of the string at an point. = A sin (k ωt) where k = wave number 2π ω = 2πf λ Slope of string = (t const) = Ak cos (k ωt) Speed of a particle in the string = t ( const) = Aω cos (k ωt) speed slope = Aω cos (k ωt) Ak cos (k ωt) = 2πf. λ 2π = fλ = v
16 Eample 1 = (.003) sin (10 20 t) (SI units) 2 = (.003) sin (15 30 t) What is the phase difference at (,t) = (0.10, 2.0)? Where does 1 + 2 = 0 at t = 2.00 s? Note: the have different k and ω, so different λ and f. But ω/k = 2πf. λ 2π = fλ = v is the same. φ 1 = (10 20 t) = (10*0.10 20*2.0) = 39.0 rad φ 2 =... = 48.5 rad φ = 9.5 radians = 1 ccle + 3.2 rad 1 + 2 = 0 (.003) sin (10 20 t) = (.003) sin (15 30 t) sin α = sin β when α = β ± mπ where n is odd integer 10 20 t = 15 30 t ± nπ t = 2.00 s = (4.0 + nπ) m n odd
17 Eample How much would the pitch of m voice rise (all else equal) if I filled m vocal tract with helium? v sound in gas = = κ adiabatic ρ = γp ρ where γ = C p C v see later Now P will be the same in the two cases, and γ is not ver different.
18 Eample Cheap ear-plugs reduce the sound level at the ear b 26 db. How much do the reduce the sound power and the sound pressure transmitted to the ear? L 2 L 1 10 log 10 I 2 I1 20 log 10 p 2 p 1 i) 26 db 10 log I 2 10 I1 I 2 I = 10 26/10 = 400 1 3 db = *2 10 db = *10 so 26 db = 2*2*10*10 = 400 ii) p 2 p 1 = I 2 I1 = 20
19 Eample. Sinusoidal signal drives two identical loudspeakers a distance D apart. How does the sound level var along the line between them? D V = V o sin ωt Between speakers (0<<D) from left speaker (L) p L = A L sin (k ωt) from right p R = A R sin (k( D) + ωt) but A = A(). How does amplitude var with? I p 2 and I 1 r 2 so p L = p m p 1 r sin (k ωt) from right p R = p m D sin (k( D) + ωt) Near middle, A L () A R () standing wave. Ver near left speaker, p total p L
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