Lecture 18. Sound Waves: Intensity, Interference, Beats and Doppler Effect.

Transcription

1 Lecture 18 Sound Waes: Intensity, Interference, Beats and Doppler Effect.

2 Speed of sound Speed of soun in air, depends on temperature: = ( T ) m/s where T in C

3 Sound intensity leel β = 10log I I 0 with I 0 = W/m 2 Units: decibels log is base 10 Threshold of human hearing: W/m 2 β = 0 Aerage whisper: W/m 2 β = 20 decibles Normal conersation: 10-6 W/m 2 β = 60 decibels Lawn mower: 10-3 W/m 2 β = 90 decibels Threshold of pain: 1 W/m 2 β = 120 decibels Twice the decibels does NOT feel twice as loud!

4 ACT: Ear plugs A set of ear plugs is rated at 30 db, i.e. they reduce the sound intensity by 30 db. By what factor is the sound intensity reduced? A) 1/8 B) 1/30 C) 1/1000

5 Normal modes (Harmonics or oer tones) Which standing waes can I hae for a string of length L fixed at both ends? I need nodes at x =0 and x =L Nodes x 0,,,... 2 L,,... n 2 2 for n 1,2,... 2L n n for n 1,2,... DEMO: Normal modes on string Allowed standing waes (normal modes) between two fixed ends Mode n = n-th harmonic

6 Standing waes on a String fixed at both ends speed of wae on string = F/m Fixed ends: Nodes (no displacement) Fundamental frequency (1 st harmonic) L = 1 /2, 1 =2L f 1 =/ = /2L 2 nd harmonic L = 2 f 2 = / = /L = 2f 1 3 rd harmonic L = 3 3 /2 =2L/3 f 3 = / = 3 / 2L = 3f 1 All harmonics are allowed f n = nf 1 = n (/2L) n= 1,2,3,

7 Reflections/standing waes in pipes Reflection of sound waes against a surface Consider a sound pulse (air moes to the right and back to initial position) traeling along a pipe toward a closed end: A closed end is a fixed end Incoming pulse s No displacement: s = 0 Wae must be Inerted (s becomes s) Reflected pulse s s s x x

8 Reflection of sound at an open end s Pulse traels out into open air Oscillation back from a larger slice moes more air into pipe s and increases pressure and causes a wae to propagate back in (a reflection!) (and another wae is transmitted outside) Beyond the open end of the pipe, ariations in the pressure must be much smaller than pressure ariations (gauge pressure) in pipe. Just beyond the open end, p 0

9 Boundary conditions for sound Open end gauge pressure = 0 always node air displacement amplitude is maximum Closed end air displacement = 0 always node maximum (absolute) gauge pressure amplitude

10 Pipe open at both ends DEMO: Organ pipes Open pipe: antinode at both ends; =speed of sound Fundamental mode: 1 st Harmonic L = 1 /2 1 = 2L f 1 =/ 1 = /2L Next 2 nd harmonic (pipe contains a full waelength) L = 2 2 = L f 2 =/ 2 = /L = 2f 1 2 nd harmonic 3 rd harmonic L = 3 3 /2 3 = 2L/3 f 2 =/ 3 = 3(/2L) = 3f 1 f n = λ n = n 2L = nf 1 All harmonics are present :n=1,2,3,4, Note: the gauge pressure wae in the pipe open at both ends looks like that of a spring fixed at both ends:

11 Stopped Pipe closed at one end and open at the other closed end: node; open end: anti-node Fundamental mode L= 1 /4 1 =4L (quarter waelength in pipe) f 1 = / 1 = /4L Next harmonic 3 rd harmonic L= 3 3 /4 3 =4L/3 f 3 = / 3 = 3/4L= 3f 1 5 th harmonic L= 5 5 /4 5 =4L/5 f 5 = / 5 = 5(/4L) = 5f 1 f n = λ n = n 4L = nf 1 n=1,3,5,7, Only odd harmonics are present

12 What is the speed of Standing wae? The Wae is standing, speed is zero! But wait, = (F/μ), how come it is zero? Standing wae is result of the interference of two identical waes of opposite elocities: ± = ± F μ string

13 Standing Waes (superposition of two waes traeling in opposite directions)

14 When a wae reflects at a fixed/free end, the reflected wae (of same speed opposite direction) interferes with incident wae.

15 Interference for sound (3D) Let S 1, S 2 be two sources that emit spherical sound waes in phase. S 1 d 1 P At point P: s 1 ( r 1, t) = s 1max cos(kd 1 ω t) s 2 ( r 2, t) = s 2max cos(kd 2 ω t) S 2 d 2 Phase difference = 2π λ ( d 2 d 1 ) This is what matters Destructie interference k (d 2 d 1 ) = n odd π d 2 d 1 = n odd λ 2 = ( n ) λ Constructie interference k (d 2 d 1 ) = n een π d 2 d 1 = n een λ 2 = n λ

16 Interference in real life? Your stereo equipment does not seem to produce these minimum intensity spots many frequencies at the same time multiple reflections on walls, ceiling, furniture DEMO: Interference

17 Beats Consider two harmonic waes meeting at x = 0. Same amplitudes, but 2 = The displacement ersus time for each is shown below: A cos( t ) 1 A cos( t ) 2 C (t ) = A(t ) + B (t ) Destructie interference Constructie interference

18 Beats (math) DEMO: Beats Acos(ω 1 t) + Acos(ω 2 t) = 2A cos cos(ω L t)cos(ω H t) where ω L = ω 1 ω 2 2 and ω H = ω + ω cos( L t ) Beat 1 Beat 2 Beat 3 Note: What you actually hear (beats) has frequency: f beat = 2f L = f 1 f 2

19 Doppler Effect

20 Doppler math: moing source Speed of sound is constant. t = 0 S (source) Source emits S Listener (ear) perceies L λ L = λ S + S T t = T S T S = + S f L f S f S f L = f S + s Front of wae emitted at t = 0 L =T L Note: Since then, S > 0 away from listener S < 0 toward listener

21 Doppler math: moing listener (sound) L (listener) λ S = T L + L T L where λ S = f S t = 0 S and T L = 1 f L T L f S = f L + L f L t = T L f L = + L f S L T L Note: Since then, L > 0 toward source L < 0 away from source

22 Moing source and moing listener f L = + L + S f S L, S > 0 in direction from listener to source ( > 0 always) To get signs correct 1) sketch the situation, including a few waefronts 2) decide whether obsered waelength or period will be shorter or longer 3) use this to guide whether frequency increases, decreases 4) keep in mind speed of sound does not depend on what the source or obserer is doing (Book's method): Just consider the direction from L to S as positie to be your sign reference! Then compare elocities of L and S with it.

23 Shock waes What if the source (a plane, for instance) is moing almost at the speed of sound? Air piles up here. High pressure and density in front of plane source Large aerodynamic drag (plane pushes on air, air pushes back) Sound barrier

24 Supersonic speeds And what if source >? Points of constructie interference are along the red lines (sides of a cone) BIG amplitude there! When this cone touches the ground source > a person on the yellow line hears a ery loud sound (sonic boom)

25 Mach number t source t θ source > sin θ = t source t = 1 S sin θ = S Mach Number