Honors Classical Physics PHY141 Lecture 31 Sound Waves Please set your clicker to channel 1 Lecture 31 1
Example Standing Waves A string of mass m = 00 g and length L = 4.0 m is stretched between posts with tension F T = 80 N. 80 Q: Calculate the wave speed v: v FT 40 m/s 0.0 4.0 Q: calculate the various standing waves that may live on the string L 8.0m n, n 1,,3,... n 8.0,4.0,.67,.0,... m n n Q: what are the frequencies of emitted by the vibrating string? v v 40 m/s fn n n, n 1,,3,... fn 5,10,15, 0,... Hz L 8.0m n Q: give the wave shape and calculate the maximum transverse acceleration for n=1, A=1 mm; n x y1( x, t) Asin( k1x)sin( 1t) Asin sin( ft 1) 0.001sin(0.79 x)sin(31.4 t) L d y A 1 sin( k1x)sin( 1t) A max 1 A4 f1 1 m/s dt max Lecture 31
Sound Waves Sound waves are PRESSURE waves, i.e. traveling pressure variations, caused by longitudinal motion of molecules of the medium (air/gas/fluid/solid) around their individual equilibrium positions. For a gas: when molecules aggregate together, pressure goes up, where they disperse, pressure goes down with respect to ambient pressure. The longitudinal wave propagating down a slinky is a good model example: a large series of balls interconnected with tiny massless springs will propagate a longitudinal wave like in a slinky Pressure (change) is proportional to the fractional volume change; the medium-dependent proportionality constant is the Bulk Modulus B: Δp B (ΔV/V) (units Pa); i.e. Hooke s Law for gases, fluids, solids! For a longitudinal sound wave propagating into the +x-direction we have (see book): ΔV/V = [Sy(x+Δx,t) Sy(x,t)]/(SΔx) where S is an arbitrary coss sectional area of the air moving because of the longitudinal displacement y of the molecules around their equilibrium positions. Thus: ΔV/V = Δy/Δx with y = Acos(kx ωt) B = Δp/(ΔV/V) = Δp/(dy/dx); and: Δp(x,t) = BkAsin(kx ωt) Maximum: Δp max = BkA Lecture 31 3
Speed of Sound The medium in which the sound wave (= pressure/compression wave) propagates is characterized by Bulk Modulus B [N/m ] and density ρ= m/v [kg/m 3 ]. The only way to make a velocity out of this is: v (B/ρ) t can be shown that in fact: v = (B/ρ) For sound in air, treating air as an ideal gas and the pressure waves propagating adiabatically (no flow of heat): pv γ = C = constant (see later in book!), γ Air = 1.40, and: dp dv d dv CV M is the molar mass (mass per mole) of air, and T is the absolute temperature in Kelvin Then: v = (B/ρ) = (γrt/m) = (1.40 8.31 93/0.088) = 344 m/s B dp dv V CV p 1 nrt V CV V mrt MV p V RT M Lecture 31 4
B Speed of Sound Waves v 343 m/s (T=93K) The medium in which the sound wave (= pressure/compression wave) propagates is characterized by B [N/m ] and density ρ= m/v [kg/m 3 ]. The only way to make a velocity out of this is: v (B/ρ) t can be shown that : For sound in air, treating air as an ideal gas and the pressure waves propagating adiabatically (no flow of heat; Q=0): dp dp dcv nrt mrt RT V V CV p dv V dv adiabatic: dv V MV M pv C where M is the molar mass (mass per mole) of air, and T is the absolute temperature in Kelvin Then: v B RT M varies like T! (T is temperature here; in K!) v B 1.408.319 0. 088 343 m/s Lecture 31 5
the speed of sound in air depends on A. The frequency of the sound B. The wavelength of the sound C. The ambient temperature 100% 0% 0% A. B. C. Lecture 31 6
Sound ntensity P/Area (Δp max ) ntensity at the position of the receiver-of-sound is defined as the average power (P) emitted by the (pointlike) source that is received per unit of Area (A) at the receiver: P F v dy ( x, t pxtv (, ) ) y( xt, ) pxt (, ) Area Area dt BkAsin( kxt) A sin( kxt) Bk A kx t P Area 1 Bk A sin ( ) Bk A kxt 1 B v sin ( ) A 1 pmax B Bk A 1 overbars overbar indicate indicates Average Lecture 31 7
Sound ntensity Level β(10 db)log 10 (/ 0 ) Our hearing is very sensitive: young ears are sensitive to sound intensities of only 0 = 10 1 W/m, the threshold of hearing, at the best frequency around 1000 Hz. The pain level is about 1 W/m thus, our ears have a very large dynamic range of 10 1! Sound ntensity Level is defined as a logarithmic base-10 scale, with unit db, the deci-bel tenth of a Bel, after A. Bell, defined as: ( ) 10 dblog10 with 0 10 1 W/m hearing goes from 0 db (=10 1 W/m ) to 10 db 0 Alexander Graham Bell (1847 19) Lecture 31 8
Example a 0% efficient, 00 W (electrical) loudspeaker is blasting at nominal power: ( ) 10 db log What is the distance at which reach pain level? (Assume the speaker radiates sound uniformly in the 1 frontal hemisphere only) P Area 000.0W ( r ) 1 W/m pain 1 pain rpain 40 W / ( ) rpain.5 m What distance do need for only 0.1 W/m? P Area 000.0W ( r ) 0.1 W/m 0 pain 10 10 W/m 1.0 W/m 0 r 1040W ( ) r 8 m What is the Sound Level difference between the two intensities above? 1 ( 1) ( ) 10log 10log 10log 1 0 1 0 0 0 NOTE: This is simplified: it ignores the non-point source character of the loudspeaker, and effects from absorption and reflection off nearby surfaces 10log10 10log 10 db Lecture 31 9
Bosch 36W column loudspeaker polar pattern Monsoon Flat Panel speaker: (5 db grid) 400 Hz: Real Loudspeakers 1 khz: 3 khz: 7 khz: Lecture 31 10