Source on the Axis of a Cylindrical Tube. G57 



e ikct , this becomes 



V 2 <£ + P(/> = 0; (11) 



or, with the same coordinates as before, 



gf ♦;-£♦$+»»-* • • • <«> 



Considering only that part of the cylinder which lies on 

 the positive side of the plane £ = 0, we assume that <f> varies 

 as e~ m s z > The equation (12) then becomes 



£+5g+w-* 03) 



where k 2 = k 2 + m s 2 (14) 



The most general solution of (14) satisfying the conditions 

 of the problem is found, in the same manner as before, to be 



9 ~l2irika^ ^^2ira 2 m s {J (k s a)} 2 J ' ' ^ J 



where the summation extends over all values of k s which 

 satisfy the equation 



Jo'fcO = (16) 



Two cases arise in this problem according as the values of 

 rn s are or are not all real. First suppose k to be less than 

 the least value of k s (other than zero) which satisfies (16); 

 that is to say, we assume the period of the disturbing force 

 to be greater than that of the gravest mode of free radial 

 vibrations. The values of m s are then all real, and the solution, 

 written in real form, is 



,_ 1 rsin k(ct — z) ^ J (k s r)e~ ms ~ cos hct~\ ,^ . 



(p ~2^o?L I m s {J (k s a)} 2 J' ' U ] 



For large values of z the disturbance reduces practical^ 

 to the system of plane waves, 



*=jsi?»fc*f*-«> (1<s) 



The energy transmitted per unit time across unit area of the 

 wave front is then pc/Sir 2 a\ and the rate of emission of energy 



bv the source is therefore -r^-s. The rate at which energy 

 J 477 -or 



