502 - « - 



Where 2d is the distance between the surfaces. 



Such a velocity potential satisfied Laplace's equation and the ''ouhdary conditions at both surfaces. 

 But is not physically permissible because it diverges. This difficulty was pojnted out by Herring. it 

 can, however, be overcome as follows:- 



|f we consider the region in «*iich r < d, we may transform the image terms (Involving spherical 

 harmonics about the origins ± n d) to series of spherical harmonics about the centre of the bubble. we 

 then get, by well-known formulae:- 



A^ A PjCcose) 2A^ ^ (1) 2A^ r^ P rco50) ^ (1) 



r P d 2 (n) d^ j (n^) 



+ similar series. 



A A 

 we now strike out the terms which vary with the time only, such as — , — | etc., as we can always 



d d^ 

 subtract any term depending on the time only from a velocity potential without altering its physical meaning. 



In particular, the new velocity potential will now be finite everywhere (owing to the disappearance of the 

 terms making up the series S i I and will still satisfy both Laplace's equation, and the boundary condition 

 at the rigid surfaces. This velocity potential must therefore be the appropriate one for our problem, 

 we treat this velocity potential by writing down the conditions that the pressure and normal velocity should 

 be continuous at the surface of the bubble R = a + b P (cos 6) * 



The final form of the velocity potential is 



A A/^ (COS 6) 2A„ r^P (COS 9) X ^ _(^ 



* - 7 * p ^ \ („3, O (,5, (3) 



1 

 we notice that the perturbing term is of thp order of "TJ . so will be fairly sensitive to the 



distance apart of the pl».tes. If we regard -iv and b as smiU quantities, and neglect their powers and 



d^ 2 



products, we get:- 



, . 3A, . ■ 808 , 



*o ' ^ a.— ir = b + 2a bj + — j- 3^ a (u) 



From the equation for continuity of normal velocity. Substituting these values in the expression 

 for the pressure, we obtain finally:- 



3 ., 

 pg - gz = a a + -■ a-" (5) 



(we have neglected the higher order perturbing terms). 



fc, * 3A b, - -a b, ■^ -^ - ( ~ ) = " <«) 



i3 



Where Pg is the pressure in the bubble, and egz the total pressure in the water outside. Equation 

 (5) is of exactly the same form as the equation for a charge in an infinite sea, showing that, to a first 

 approximation, neither the period nor the energy of the bubble will be affected by the presence of the two 

 rigid surfaces, (This result is due to the absence of terms involving Legendre co-efficients of odd order). 

 We know that the effect of a single rigid surface is both to attract the bubble and to increase the period, 

 whereas the introduction of a second rigid surface seems to cancel both effects. Indeed, we see from 

 Lieutenant Campbell's results that for a bubble 12 inches from a single rigid surface the first oscillation 

 takes 26 milliseconds, whereas for a bubble midway between two rigid surfaces 2U inches apart, the first 

 oscillation takes only 23 milliseconds in agreement both with theory and with the period of a bubble right 

 away from rigid surfaces. 



Equations 



