Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/430

App. II. A. fictitious pressure of the gas; this at once gives us Laplace's correction.

In this case the assumption is obviously that the amplitude is small, for otherwise the tangent $$EF$$ no longer approximates sufficiently to the actual curve.

The rationale of Laplace's correction may also be studied from the direct examination of the conditions. If we suppose in an adiabatic gas that a small isolated compression wave be constrained to move with the velocity proper to the gas obeying Boyle's law, the pressure during the reflection of the wave will be in excess of the momentum the wave communicates, to the extent that an adiabatic compression pressure is greater than the Boyle's law pressure for a given change of density. For small amplitude this is in the relation of $$\gamma$$ to unity. Obviously the wave must travel faster to supply the momentum necessary to equalise, and since the momentum communicated per unit time varies as the square of the velocity, the velocity must be multiplied by $$\sqrt{\gamma} .$$

The question of the behaviour of an adiabatic wave of sensible amplitude is one of great complication that yet awaits a general solution. The compression regions are always endeavouring to move faster and the rarefaction regions slower than the mean velocity. From the present standpoint this is evidently due to the pressure increase becoming proportionately greater than the density increase (Fig. 161), and vice versâ, thus destroying the necessary balance between the pressure reaction and the communication of momentum by which it is maintained. The more usual and equally correct point of view is to attribute the difference of velocity of different portions of the wave to the difference of temperature of its parts.

Where we have a train of waves in a gas following the adiabatic law, it has been shown that there must be a pressure increase due to the energy that enters the thermodynamic system. Where the train is continuous, as in the Kundt's tube, no complication arises from this cause, but where we are dealing