PLANE WAVES OF SOUND 173 



vibrations of given frequency (n/27r). Thus if a prescribed 

 vibration 



f = J. cos(nt + e) .................. (9) 



be maintained at x = 0, and if the tube be closed at x = I, the 

 motion of the gas is given by 



sin - cos (nt + e). . . .(10) 



c 



The amplitude becomes abnormally great, even when we take 

 account of dissipative forces, if sin (nl/c) = 0, or I = %m\, where 

 m is integral. This is the principle of a method due to Kundt 

 (1868) by which the velocity of sound in various gases can be 

 compared by small-scale experiments. The wave-lengths are 

 found by measuring the distances between the nodes, whose 

 position is indicated by the heaping up of lycopodium powder 

 previously scattered in the tube. The vibrations are excited 

 in the two tubes (containing the two gases to be compared) by 

 disks fitted to the two ends of a longitudinally vibrating rod. 

 If the end x= I is open, the formula (10) is replaced by 



g= ^008^=^ 008 (lit + 6), ,..(11) 



cos (nl/c) c 



and the condition of strongest resonance is cos (nl/c) = 0, or 

 / = Jm\, where m is an odd integer. 



The preceding investigations would apply also to the 

 vibrations of a column of water, or other liquid, contained 

 in a tube, provided the material of the tube were absolutely 

 rigid. In practice, however, the yielding of the walls has 

 an appreciable effect; the potential energy corresponding to 

 a given strain (dg/dx) of the fluid is diminished, and the wave- 

 velocity is lowered. The fact was observed by Wertheim (1847), 

 but the true explanation is due to Helmholtz (1848). The 

 question has been further investigated by Korteweg (1878) 

 and the present writer. Owing to the much greater velocities 

 ( 44) of elastic waves in solids such as glass or steel, as 

 compared with the sound- velocity in water, the stresses in 

 the walls adjust themselves so rapidly that it is legitimate 

 to assume that the deformation of the tube has the statical 

 value corresponding to the instantaneous distribution of 



