546 XI. HYDRODYNAMICS. 



in a series of normal vibrations as in 46. If we take the particular 

 solution of equation 109) given in equation 115) of that section, 

 and write 



213) (p = Acoslcxcosnt, 



where n = It a, we have 



u = Y- = Ak sin kx cos nt, 



l dtp A n -, 

 s = -- 2^-=J.-^ cos kx sin nt. 



CL G v d 



Every particle oscillates with a simple harmonic motion with an 

 amplitude sin ~kx, and we have a pure tone. The compression also 



A 7. 



varies harmonically with an amplitude - coskx. Thus the maximum 



pressure occurs at points of no motion, such points being called 

 nodes. These occur where Jcx = m, where r is any integer, or 



X == k' IT' IT' '"' ^ e wave 'l en g^ being %=., the nodes are 

 separated by distances The condensation s follows a similar 



law, but vanishes half way between the nodes where the motion is 

 a maximum. The regions between the nodes are called loops. The 

 maximum changes of s are at the nodes. 



As there is no motion at the nodes, but only changes of pressure, 

 we may place reflecting walls there and apply the theory to the case 

 of a stopped organ -pipe, whose length is accordingly any number of 

 half wave-lengths. If the ends of the pipe are at x = and x = l, 

 we have 



7 * 7 7 7 IT 11 



l = rj Kl = rit, K = -p? 



i I 



and the frequency is determined by 



n ak ar 



Consequently the possible frequencies for a simple harmonic vibration 

 of a stopped pipe are in the ratio of the integers 1, 2, 3, etc. 



For a pipe open at the end the condition is that the pressure 

 is that of the external air, that is, there is a loop. Thus a pipe 

 open at both ends has its length equal to an integral number of 

 half wave-lengths, and has the same harmonics as a closed pipe. 

 Opening one of the holes in a flute produces a loop, so that the 

 tones of a flute are produced by the column of air between the 

 mouth -piece and the first open hole. 



For a pipe open at one end and stopped at the other, the length 

 is equal to an odd number of quarter wave-lengths, so that the 

 frequencies are proportional to the odd integers 1, 3, 5, etc. 



