UVA PHYS 3040 - Sound, hearing and the human voice

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Sound, hearing and the human voiceIn this chapter we discuss the basic physics ofsound transmission in air, how sound vibrations arereceived by the ear, and how we generate soundswith our voices. This is a rich area of study, and byno means fully understood even today.1. Sound propagation in gasesThe key equations we shall need from our previousstudy of fluids are conservation,∂ρ∂t + ∇ ⋅ (ρv→) = 0and Euler’s equation1ρ ∂v→∂t + v→ ⋅ ∇ v→ = −∇pwhere we have neglected viscosity and externalforces (such as gravity).Imagine the gas has equilibrium pressure p_, densityρ__, and zero macroscopic velocity2. Then we maypicture a sound wave as a propagating train of smallexcursions from equilibrium δp, δρ and v→. Ignoringterms of second or higher order in small quantities,we reduce the conservation law and Euler’s equa-tion to∂δρ∂t + ρ__ ∇ ⋅ v→ = 0andρ__ ∂v→∂t = −∇δp .Taking the time derivative of the first, and thedivergence of the second equation, and eliminat-ing δv→ between them, we obtain∂2δρ∂t2 = ∇2δp .This is almost a wave equation—in fact it is onesince almost certainly δp is proportional to δρ.However if we want to find out what the wavevelocity is (that is, the speed of sound) so we cancompare with experiment, we need to know theconstant of proportionality between δp and δρ.Such a relation follows from knowledge of thethermodynamics of the gas. Sir Isaac Newton as-sumed that the temperature remains constant dur-ing the fluctuations of density and pressure in asound wave. That is, he assumed Boyle’s Law,pV=constant orδ (pV ) = 0,which yields the relationδpp_ = δρρ__ .In fact, Newton was wrong—the fluctuations areadiabatic (sometimes called isentropic), meaningno heat flows in or out of the small volumes wehave been considering. Under these conditions wehave3δQ = 0 = δU + p δVwherePhysics of the Human Body 87Chapter 10: Sound, hearing and the human voice1. That is, Newton’s second law of motion,2. Since the molecules move quite rapidly, but randomly, there is no overall flow.3. This is conservation of energy, which Newton did not know about—in fact the concept of energy had notyet been invented!U = f2 RTis the internal energy of the gas (per gram-mol).Here R is the gas constant, 8.3 J/mol/oK and T theabsolute temperature. The number f is the numberof (active) degrees of freedom of the gas mole-cules4. Thus, e.g., monatomic gases have f=3,diatomic gases have f=5, and tri-(or more)atomicgases have f=6. Now if we combine the equation for energy con-servation with the equation of state of a perfectgas,pδV + Vδp = RδTwe havef2 (pδV + Vδp) + pδV = 0and sinceδVV ≡ − δρρwe see that2f + 1 δρρ = δppor in other words,δp = γ p_ρ__ δρwhere γ is the ratio of specific heats at constantpressure and volume:γ = cpcV = 2f + 1 .When the above relation is substituted into ourprevious equation we get∂2δρ∂t2 = γ p_ρ__  ∇2δρ .This is a wave equation—a solution of the formδρ = δρ0 f(x − ut)satisfies it, with u (the speed of sound) given byu2 = γ p_ρ__  .We now compute the speed of sound in air. Fromthe perfect gas law the preceding formula can bere-expressed asu = γ RTµ1⁄2 .where µ is the molar mass (grams per gm-mol).Hence the speed of sound in dry air at 20 oC isu = (1.4 × 8.3 ×2930.029 )1⁄2 = 343 m/sec .Since Newton’s theory omitted the factor γ (5⁄3 formonatomic gases and 7⁄5 for diatomic ones) theobserved speed of sound in air (a mixture of dia-tomic gases) is about 20% greater than his predic-tion.Newton was convinced of the general correctnessof his ideas, and proposed several hypotheses toexplain the discrepancy; chief among them, thathe hadn’t considered humidity (which he thoughtwould increase the speed of sound5). Because hedid this, Broad and Wade accuse Newton of fraud6.88 Physics of the Human BodySound propagation in gases4. The number of active degrees of freedom depends on temperature. At higher temperatures the vibrationalmodes of the molecules can be excited, whereas at temperatures below 75 oK the rotational modes of hy-drogen are no longer excited.5.…since sound travels faster in water than in air. Actually water vapor is triatomic so this would decrease γ,thereby decreasing the speed of sound relative to dry air.6. W. Broad and N. Wade, Betrayers of the truth (Simon and Schuster, New York, 1982).What Newton did was not scientific fraud: in factit is entirely appropriate for theorists to suggestalternative hypotheses and new lines of investiga-tion when their numbers are a bit off. In accusinghim of fraud, therefore, Broad and Wade displaytheir ignorance of how science is actually pursued,as well as their inability to distinguish good sciencefrom bad.We now clear up several outstanding issues beforeleaving the theory of sound in gases. First, since∂δρ∂t + ρ__ ∇ ⋅ v→ = 0andρ__ ∂v→∂t = −∇δp ,we see that if the sound wave is propagating in thex direction,δρ = δρ0 f(x − ut) ,then the velocity field will bev→ = x^ u δρ0ρ__ f(x − ut) .This means that the displacement field (of the airfrom its equilibrium position) points along thedirection of propagation. We call such a wavelongitudinal7.The next issue is, “How good was the adiabaticapproximation” (in which we neglected heat flowfrom one parcel of gas to its neighbor)? We cananswer this by considering the equation of heatconduction,cp ∂T∂t + κ ∇2 T = 0 ,where cp is the specific heat at constant pressureand κ is the thermal conductivity.This is like the diffusion equation which we havealready studied. The time scale for heat to diffusea distance λ (that is, a wavelength of sound, whichis the characteristic distance scale) isτH ~ cpλ2κand we must compare this with the period of thesound wave,τS = 1ν = λu .That is, we want the ratioτHτS ~ cpκ λ u .For air κ ≈ 25×10−3 W/m2/ oK and the (volumet-ric) specific heat iscp ≈ f2 + 1 RΩwhere Ω is the molar volume, 0.0224 m3 at STP.Thus for sound of frequency 20 KHz, (withλ ≈ 2 cm) we findτHτS ~ 2.6×105 .In other words, the time for heat to be transmittedfrom one small parcel of air to another is muchgreater than the time for the pressure and densityin the parcels to oscillate. Hence we may neglectheat transmission.The third


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