Simple Nature - Light and Matter

significant amount of heat to flow from the hot areas to the coldareas. (This is analogous to insulated compression or expansionof a heat engine; in general, a compression or expansion of thistype, with no transfer of heat, is called adiabatic.) The pressureand volume of a particular little piece of the air are therefore relatedaccording to P ∝ V −γ . The cycle of oscillation consistsof motion back and forth along a single curve in the P-V plane,and since this curve encloses zero volume, no mechanical workis being done: the wave (under the assumed ideal conditions)propagates without any loss of energy due to friction.The speed of sound is also related to γ.p. 375.See example 13 onMeasuring γ using the “spring of air” example 24Figure c shows an experiment that can be used to measure theγ of a gas. When the mass m is inserted into bottle’s neck,which has cross-sectional area A, the mass drops until it compressesthe air enough so that the pressure is enough to supportits weight. The observed frequency ω of oscillations about thisequilibrium position y o can be used to extract the γ of the gas.c / Example 24.ω 2 = k m= − 1 m= − A m= − A2mdFdydPdy∣∣yo∣dPdV∣yo∣∣VoWe make the bottle big enough so that its large surface-to-volumeratio prevents the conduction of any significant amount of heatthrough its walls during one cycle, so P ∝ V −γ , and dP/ dV =−γP/V . Thus,ω 2 = γ A2mP oV oThe Helmholtz resonator example 25When you blow over the top of a beer bottle, you produce a puretone. As you drink more of the beer, the pitch goes down. Thisis similar to example 24, except that instead of a solid mass msitting inside the neck of the bottle, the moving mass is the airitself. As air rushes in and out of the bottle, its velocity is highestat the bottleneck, and since kinetic energy is proportional to thesquare of the velocity, essentially all of the kinetic energy is thatof the air that’s in the neck. In other words, we can replace m with332 Chapter 5 Thermodynamics