548 XL HYDRODYNAMICS. 



&\. j p _2^Q == ^ ^o^ == ?o ~ 



from which we obtain 



222) P = 



This contains a part which is alternately positive and negative, and 

 also one which is always positive. If we seek the mean value of P 

 throughout the period, that is 



T 



P = f Pdt, T=~, 



o 



we easily find, since the mean of cos#, sin#, cos # sin #, is zero, 

 while the mean of sin 2 ^ is > 



223) P = A *f** a 



which is independent of the radius, as it should be. The mean 

 energy -flow per unit of time and per unit of area of the sphere is 



which is a measure of the intensity of the sound. Tihs decreases 

 as the inverse square of the distance. In order to give an idea of 

 the extremely small dynamical magnitudes involved in musical sounds, 

 it may be stated that measurements made by the author 1 ) showed 

 that the energy emitted by a cornet, playing with an average loudness, 

 was 770 ergs per second, or about one ten -millionth of a horse- 

 power, while a steam -whistle that could under favorable circumstances 



be heard twenty miles away emitted but - ? or one -sixtieth 



of a horse -power (see note, p. 153). 



2O5. Waves in a Solid. The equations of motion for an 

 elastic solid are obtained from the equations of equilibrium 144), 175 

 by the application of d'Alembert's principle in the same manner as 

 the equations of hydrodynamics were deduced from those of hydro- 

 statics. It will be convenient here to revert to the notation of 

 Chapter IX where u, v, w and 6 refer to displacements rather than to 

 velocities. Applying d'Alembert's principle we thus obtain 



225) p (X - g) + (I + ,*) fl + ^u = 0, etc. 



1) Webster, On the Mechanical Efficiency of the Production of Sound. 

 Boltzmann- Festschrift, p. 866, 1904. 



