192 DYNAMICAL THEORY OF SOUND 



or, keeping only the real part, 



(8) 



Tl fit 



a result which is easily verified. When fty is moderately large 

 the value of u reduces practically to the first term, which is 

 the same as if there had been no friction. The rigid boundary 

 accordingly acts as a drag only on a thin stratum ; for example 

 when y Sir/ft the velocity falls short of its value at a great 

 distance from the surface by about one part in 535. 



In actual problems of acoustics (relating for example to 

 vibrations in pipes) the force pX per unit volume is replaced 

 by the negative pressure-gradient dp/da, and we have of course 

 changes of density to take into account, but the results have 

 a similar interpretation. The linear magnitude 



h = 2ir/l3 = )/(4eirp.2irln) ............... (9) 



may be taken to measure the extent to which the dragging 

 effect penetrates into the fluid. With the previous data its value 



in centimetres is about 1'29/N*, where N is the frequency; thus 

 for N = 256 we find h = "80 mm. 



We may apply the above investigation to obtain an estimate 

 of the effect of viscosity on the wave-velocity in a tube, on the 

 supposition that the diameter is small compared with the 

 wave-length but large compared with the quantity h. The 

 tangential stress on the fluid at the boundary y = is, in the 

 case of (7), 



by (9), the time-factor e int being understood. The total tan- 

 gential force exerted by the walls of a cylindrical tube of radius 

 a on the contained air may therefore be equated to 



per unit length, where p denotes the mean pressure over the 

 section (?ra 2 ). Hence if u be the mean velocity, we have, 

 calculating the forces on the air contained in an element Bx 

 of the length, 



9 du -dp . /n ., , dp 



7rp a? = - ira? + $(I-i)ha^-, 



r dt dx * ^ dx 



