Wu 



x= Ut - |c„ |t = Ut(l -e) sin26i, 



^'' (66) 



z = c t = -ut (1 - e) sin 6 cos 6 , 



in which use has been made of (49) (with cp = w) and (64), and 



e = /3V(k2 + /32) = (U/3/N)' , (67) 



which is generally small. Eliminating 9 in (66), we obtain the locus of the sta- 

 tionary waves produced by the disturbance at Q as 



x2+ z2 = Ut (1 - e)x, (68) 



which is a circle with diameter joining the points x= o and x = ut (l- e). This 

 result is shown in Fig. 5 (with ut normalized to 1); also shown in Fig. 5 is the 

 slope of constant phase lines for several wave elements. 



Next we consider the three dimensional case. The equations analogous to 

 (66) are: 



X = ut - |cg I t = ut |l - cos^cp [(1- e) cos20+ e]i , 



y = c t = Ut sin cp | cos cp | [(1-e) cos ^6' + e] , (69) 



By 



z = c t = -Ut I cos cp| (1- e) sin 9 cos , 



in which e is given by (67). Upon elimination of 6 and cp from the above, we 

 obtain 



Ut (Ut - x) 



(Ut- x)2 + y^J 



[x(Ut- x)-y2] , (70) 



which is the equation of a closed surface bounded in the sphere y^+ z^ = x(Ut - x). 

 It is seen that the waves are nearly horizontal directly above and below the 

 origin, and vertical along the inner rim of the surface near x = ut . 



We proceed to treat a few typical problems analytically in order to achieve 

 an overall understanding of the analytical features of the problem and the physi- 

 cal arguments presented above. 



DIPOLE RADIATION 



In order to avoid the indeterminancy of the quasi- steady solution, we con- 

 sider a corresponding initial value problem of a plane flow so that the vertical 

 velocity component w satisfies (see Eq. (34)) 



K-N'(t I- 5)] "= ^'^ sT [''.Kx)KOlH(t)e-'-'. (71) 



where D = B/Bt, v^ is the Laplacian in (x,z), n^ again given by (38b), and H(t) 



566 



