SIMPLE-HARMONIC WAVES. DIFFRACTION 249 



to a source - 29</3n . 8S in infinite space. The corresponding 

 velocity-potential at a point P is 



where r denotes the distance of 8S from P. Integrating over 

 all the elements &S of the plane, we have 



1 rr^^ a -ikr 



which is the required formula. 



The motion to the right of the plane x = is also determinate 

 when the value of </> at every point of the plane, and thence 

 the pressure, is given, these two quantities being connected by 

 the relation p p + p<j> = p Q + ikcp<f>. Suppose for a moment 

 that in an otherwise unlimited medium we have a thin massless 

 membrane occupying the plane x = 0, and that on each element 

 of this a normal force X per unit area is exerted, which is 

 adjusted so as to produce the actual periodic pressure, and 

 therefore the actual value of $, on the positive face of the 

 membrane. By the theorem of 78 (15), the effect for an 

 element BS, will be equivalent to a double source, and the 

 corresponding velocity-potential at a point P will be 



(2) 



The variable parts of the pressures on the two faces of the 

 membrane, viz. + p(f> = ikcp<f>, must balance the force X, so 

 that X = %ikcp$. Substituting in (2), and integrating over the 

 plane x = 0, we obtain 



* * 



The structure of the integrals in (1) and (3) recalls the 

 process by which " Huygens' principle " is applied in optics to 

 find the disturbance at any point P in terms of "secondary waves" 

 supposed to issue from the various elements of a wave-front. 

 There was at one time much discussion as to the exact character 

 to be assigned to these secondary waves, more especially as to 

 the law of intensity in different directions. We now recognize 



