Question

(10 pts) Consider a stationary sound wave in a moving gas with velocity, a, relative to the wave, as show in the figure below. The applicable equations are: • Continuity: Piui = p2u2 • Momentum: Pi + pui = P2 + puz p + dp T + dT p+dp Derive an expression for the speed of sound applicable to any gas. For a calorically perfect gas, state two expressions for the speed of sounda a Sa+ da

Answer 1

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The process is reversibly isentropic because of the properties of air and due to one assumption that no interaction of heat takes place. For detail refer to image:

o ore Ptap THAT et de atda By continuity equation Here Here fa=(1+df) (atda) 16 = fat edat adet afda (small) pro = -ade da to By momentum equation. 1 Pt 1,0 = Pattu Potta² = (P+ap) & (H+df) (at da? &t fou tad + (4 + df) (a²+ då taada? 6 e da trafda ta de + dfda+zadade 0 = 2af data² dp top o = deta de taal da as from e da tade o = dp + a² af + 2al-adf)

O = OP talde - 2004 ap = aat assumption slightly grodient very small - Friction negligible, - heat conduction negligible - Adiabatic, Creversible). These assumption Que not genocually assumption they are actually happen due to peroperties of air. Hence, ea is univeusally applicable. Now far isentropie e = c. de acred Px de acerrae Fare ideal gas P = PRT. expression To - Jura expression a