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Ch8 - CHAPTER 8 ACOUSTIC SOUND WAVES 8-1 Eguwnfil’lm We...

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Unformatted text preview: CHAPTER 8 ACOUSTIC SOUND WAVES 8-1 Eguwnfil’lm We now discuss wave motion in compneAALbfic fifiuidé (gaéeé). The passage of these so—called acoubiia or bound waves through a gas is accom- panied by oscillatory motion of gas particles in the direction of wave propagation. Hence these waves are composed of compressions and rarefac— tions and are longitudinal. The density p of the medium is no longer constant but is related to the pressure p through an equation 06 Aiute. The fluid (gas) is assumed to be aniécéd, and the motion is taken to be inaotaiionafl. The continuity equation of fluid mechanics, which expresses the condi— tion of oonéenvaiion 06 maAA, is given by __.+ V'(QZ) = 0 , (8.1) The balance or conéchvation 06 Zincah momentum is specified by the equation& 06 motion, which for this case neglecting body forces reads h l _ _.v , 8.2 p P ( ) where 2p is the gaadicnt of the pressure p, and %E~is the matcniafl iime deniuaiiue which is the derivative with respect to time holding the particle constant. By definition Dale; . _ at +y—Z! ’ (8.3) t 8—2 or in scalar form DvX BVX Bvx BVX Bvx Dt = 3t + VX 3x + Vy 8y + V2 32 v s For this particular wave phenomena we assume that the density change is small and is given by p = 00(1 + e) , (8-4) where p0 is a reference undisturbed density and 8 << 1. Furthermore we also assume that the velocity components and their derivatives are small; i.e., of order 8. Hence terms like v 'Y.X and Epfiv are of order 82, and may be neglected. Consequently (8.1) and (8.2) reduce to 7+05Z'31=0 8 (8.5) l__l _t— p—v-P Now 3 W=wm—$w mm Let c2 = §B~, (8.7) . . 2 . . and assume that according to our first order theory, c is independent of the spatial variables. 8V Differentiating (8.5)l with respect to time and eliminating 5: between (8.5)2 yields 82 c2 —%=—fl-b;@} at V0 2 1 2 No!2 =pcv o {;}_p .__.v _ — } 2 p = szzp _ _c__ W012 2 But the term — §-!yp|2 is of order 82; hence we have 2 2 l 8 v p = —§-——§ . (8.8) C St Note that p in equation (8.8) could actually be replaced by the perturbation density 8. It can also be shown that the pressure p, and the poteniiafl fianciéon ¢ (where y_= — y¢) also satisfy the wave equation, i.e., ’ 2 2 l 3 E V P ='_§ 2 C a: (8.9) v2¢ =-l§-§—9 . 2 c at &2 EggATION OF STATE In order to properly interpret solutions to equations (8.8) orl(8.9), we must have an equation of state, i.e., p = p(p). For simplicity we consider here the case of an deafi gab. We furthermore assume that the compressions and rarefactions take place so rapidly that they may be regarded as heucfléibfic * adiabatic phoccAAcA , i.e., no heat transfer. For this case we have p = KpY , (8.10) Where K is some constant determined by p0 and p0, and c y = —£—> 1 (8.11) Cv where cp and CV are the Apecéfiic heaib of the gas at constant pressure and constant volume. Using (8.10) in (8.7) gives c2 = KypOY'1(1+e)Y‘l (8.12) y_l YPO =KYp =b— a o 2': . Newton, who originally considered this problem, erroneously assumed iAoiheflmafi conditions and took p = Kp. l l l l g j 8—4 where we have assumed 8 << 1. Now for an deafi gab RT p = g- o , (8.13) where R is the unwell/sow, ga/s comrant, T is the abeoflwtc IQme/La/CU/Le and M is the moKecuflah weight. For this case C = M32" , (8.14) M 8.3 SflCK WAVES It can be seen from equation (8.12) that the velocity of propagation of acoustic waves can be greater in compressions than in rarefactions. Hence an original harmonic pulse can become changed through an oughtahing efificct producing a steepening wave front (see figure below). x t+At \' pa»~—- steepening \ wave front The same situation occurs in a long tube filled with high pressure gas in part of the tube. membrane high pressure low pressure i i i i i i 5 8—5 When the membrane is suddenly broken, wave motion starts down the tube. However because the pressure is much higher behind the wave, disturbance signals in the high pressure region travel faster and catch up with the original wave front. In this way the wave front steepens and the wave amplitude grows. This steepening and amplitude growth eventually produces a new type of wave called a éhoch wave. Shock waves are also produced by forcing the medium to move fiaétcn than the local speed of sound c. The following figure demonstrates this \ effect by illustrating the pressure distribution of a moving point source. (0} , (b) /Moch Cone Zone of Silence Action Zone of Silence SHence (c) (d) r . Pressure field produced by a point source of disturbance moving at uni— farm speed leftwards. (a) Incomprcssible fluid (V/c = O). (b) Subsonic motion (V/c = V2). (0) Transonic motion (V/c = 1). (d) Efip/Jersonzif motion, illustrating Kurman’s three rules of supersonic flow 6 = . Most types of wave motion, including shock waves, may be studied by considering the wave to be a paopagaiing Aingufian Auafiace across which a jump discontinuity of a particular variable exists. For the shock wave case, the pneAAune, denéiry and veflociiy each suffer a jump discontinuity across the wave front. The discontinuities of p, p and v are interrelated, and are expressed in equations called éhoch condiiionb. These shock con— ditions are a consequence of the basic dynamical equations of continuity and motion, and the equation of state of the particular media (gas). Bringing in the conservation of energy, additional relations called Rankine-Hugonidi ncflatéoné may be found. The set of shock conditions for an ideal, inviscid adiabatic gas is 7‘: given by 2[pl(V - an>2 — «(p11 [[9]] = ”fif— 2 20 [p (V—v ) -Yp] [[pH = —-1—-—1——-—E-—-————L—«2— (8.15) ml + (Y — 1>pl<v — V1.9 2[pl(v — vn1>2 — yplm ”En“ ‘ T—>—<“’”‘>‘“ ’ where H D means "the jump of"; e.g., [[¢]]= $2 — $1 , with subscripts 2 and l referring to behind and in front of the wave, respectively. V is the propagation speed of the shock wave, vn is the velocity of the gas normal to the shock front, and n is the unit normal vector to the wave front. pl’p19an wave front >‘c T. Y. Thomas, ”Mathematical Foundations of the Theory of Shocks in Gases”, Jour. Math. Anal. & Applic.,_g§, 595, 1969. ...
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