BENG 221: Mathematical Methods in BioengineeringLecture 18Wave Transmission of Pressure and SoundReferencesHaberman APDE, Ch. 12.http://en.wikipedia.org/wiki/Wave_equationhttp://en.wikipedia.org/wiki/Longitudinal_waveBENG 221 Lecture 18 M. Intaglietta Transmission of waves in gases. Sound. Strings present the transmission of transverse waves. In gases waves are transmitted longitudinally. We will analyze the transmission of waves in tube where gas displacements are made by a piston. Moving the piston creates a compression that travels forward. If the piston is quickly retracted then there is a wave of rarefaction that also travels along the tube. Consider an element of gas in the tube located between x and xx+∆ where the gas has an equilibrium pressure . As the wave advances the element of gas oscillates about its equilibrium position. The coordinate y is used to describe displacements of gas from its equilibrium position. The displacement of the left side of the element of gas has coordinate y and that on the right side 0pyy+∆. Pressure on the left side is p and on the right side is pp+∆. For a very thin slice pressure in the displaced gas is , which is also the pressure on the left side face, and the pressure on the right side face is . The forces acting on the element of gas are obtained by multiplying by the area of the tube A. The net restoring force acting on the displaced gas is 0pp+0pp p++∆pA−∆. If 0ρ is the density of the gas at the equilibrium pressure then the mass of element is 0p0Axρ∆ leading to the equation of motion: Note that x gives the position of the gas molecules at rest (therefore while is uniform in 0px∆) while y gives the position of displaced molecules and p is not uniform in . In the case illustrated since > y∆y∆x∆ we are dealing with a rarefaction wave. ()200 02()( )dypp pp pA pA Axdtρ+−++∆ =−∆=∆ 12201dy pdt xρ∆=−∆ and at the limit for very small x∆: 2201ytxρ∂=−∂∂p∂ (23) The volume in its equilibrium position is Ax∆. In the displaced position the coordinate of the right face is xxy y+∆ + +∆ while the coordinate of the left face is x + y. Therefore the length of the displaced element is given by the difference of these two coordinates or xy∆+∆ and the change in length, and therefore the change in volume is Ay∆. Consider the general definition of compressibility k: 1 change in volumekoriginal volume change in pressure=− Note that the compressibility of a gas can be derived from the perfect gas equation pVRT= where 2211dV RT dV pRTand kdp p V dp RTp p=− =− = = Referring this definition to our development: ()()001AyykAxpppp∆∆=− =−∆∆+−x therefore: 1 ypkx∆=−∆ and in the limit: 1 ypkx∂=−∂ (24) and in view of (23) 2221pyxkx∂∂=−∂∂ Therefore substituting in (23) we are led to the one dimensional wave equation for the transmission of longitudinal perturbations: 222201ydtkdxρ∂=∂y The velocity of propagation, by analogy to the wave equation for strings (lateral displacements) is given by: 01vkρ= The bulk modulus B is the reciprocal of the compressibility, in other words, the pressure required to induce a volume change relative to the total volume. This quantity is the equivalent to the Young’s modulus Y for linear changes (stress required o induce a change in strain). Therefore a general expression for the velocity at which waves travel in a materials is: 0Bvρ= Pressure variation in a sound wave From the development of the propagation velocity of lateral displacement (waves) in a string we found that a disturbance is propagated with a velocity ν, where in this case L = wave length, and A = displacement amplitude 2cos ( )nyA x tLπν=− If we know the displacement as a function of time we can compute the pressure by differentiating with respect to x since: (,)yxt 1 ypkx∂=−∂ in view of (24) which leads to: 322sin ( )dy Axtdx L Lππν=− − and therefore: 22sin ( )ApxtkL Lππν=− Since 01kνρ= 2022sin ( )ApxtLLπρ νπν⎡⎤=⎢⎥⎣⎦− (25) The term within brackets represents the maximal pressure amplitude P while A is the maximal displacement. Wave velocity & thermodynamics of perfect gases The previous equation (25) shows that: (2022sinpA)xtLLππννρ∂=∂− (26) therefore using the thermodynamic relation: where thenMnM RTpV nRT n RT pMVMρρ== = = therefore in terms of maximum values (26) is also equal to: 2pRTMνρ∂==∂ a result derived by Newton, which underestimates the actual speed by about 15%. The more correct formulation was give by Laplace, who realized that the compression and relaxation in the sound wave is too rapid for allowing constant temperature (isothermal conditions), and that the actual conditions were adiabatic, i.e., no heat transfer due to the high speed at which compression and rarefaction occur, leading to the expression: RTMνγ= 4where pvCCγ= which is the ratio of specific heat at constant pressure vs. constant volume, usually about 1.4. Pressure dispersion The analytical derivation does not include a mechanism for the attenuation of the pressure amplitude, which occurs due to refraction and absorption of the pressure wave. This can be accounted for the expression: xoAAeα−= where α is a parameter that characterizes the viscous effects in the medium, or the conversion of mechanical energy in the wave into thermal energy. Pressure waves that originate from a point source decay naturally at the rate of -60 log R db, where R is the ratio of radial distances, and for cylindrical sources at the rate of -40 log R. The dispersion of pressure can also be described by the diffusion equation: 2pKpt∂=∇∂ Intensity of sound waves Waves propagate energy. The intensity I of a traveling wave is defined as the average rate energy is transported by the wave per unit area across a surface perpendicular to the direction of propagation. Also intensity is the average power transported per unit area. The energy associated with a travelling wave is in part potential, associated with the compression of the medium, and kinetic relate to particle velocity. By analogy to the dynamics and