44 



RAY ACOUSTICS 



Because the sum of the squares of the direction 

 cosines equals unity, 



a' + 0' + y-= 1, 



the constant of proportionahty in the multiple pro- 

 portion above can be determined, and we obtain 



= [{T 



L\dx / \dy / \ dz / 



dx 



ay' 



(10) 



etc. 



By substituting these values of a,^,y into equation 

 (9) and squaring both sides, 





dz ) 



cl 



dy / ' \dz / c^{x,y,z) 

 If we define n, the index of refraction, by 



Co 



n{x,y,z) = 



c{x,y,z) ' 



equation (11) becomes 



—J = n-{x,y,z). 



(11) 



(12) 



(13) 



Equation (13), often called the eikonal equation, is 

 the fundamental equation of ray acoustics. It is a 

 partial differential equation satisfied by all functions 

 W which can define wave fronts according to equa- 

 tion (7). Initial conditions for equation (13) are 

 usually of the form that W has the value zero for all 

 points {x,y,z) on a particular surface. 



Once the solution W of equation (13) has been 

 found, the ray pattern can easily be drawn. The direc- 

 tion cosines of the rays at every point of space can 

 be computed from equation (10); more simply, if 

 equations (10) and (13) are combined, 



IdW IdW IdW 



n dx n dy n dz 



Later we shall eliminate the function W from equa- 

 tion (14), and derive a set of ordinary differential 

 equations, which together determine the course of 

 each individual ray. First, however, we shall give a 

 simple example illustrating how the ray pattern may 

 be calculated from the partial differential equation 

 (13) for the wave fronts. 



Let us consider the special case where the sound 

 velocity c depends only on the vertical depth co- 

 ordinate y. Thus, the sound velocity is assumed 

 constant everywhere on a particular horizontal plane. 

 We shall examine only the ray pattern in one vertical 



\dx J \d^/ 



plane, which we can take as the xy plane. Then equa- 

 tion (13) reduces to 



A2 



Sy) = ^'(2/)- (15) 



To find a simple solution of equation (15), we 

 assume that W{x,y) is the sum of a function of x and 

 a function of y. 



W(x,y) = Wiix) + W2{y). 



Substituting this expression into equation (15), we 

 obtain 



©"-a' = "'« 



dy 



To obtain a family of solutions, we put dWi/dx = k, 

 where k is an arbitrary constant. Then, the differ- 

 ential equation will be satisfied if 



dW2 / 



dy 



W. 



-jy^ 



n^iy) — k"^ dy. 



Therefore, in view of the assumed nature of W, the 

 equation (15) will be satisfied by all functions W de- 

 fined by 



'W{x,y) =kx-\- \ Vn'^iy) - k^ dy (16) 



where k is any constant. A particular choice of A; 

 corresponds to a particular solution W{x,y) and 

 therefore to a particular set of wave fronts (7). 



The direction cosines of the rays, corresponding to 

 this choice of k, can be calculated from equations 

 (14) and (16). 



k 

 a = - ; /3 



.f 



*i2 



We use these expressions to obtain the equations 

 y = y(x) of the soimd rays. If we denote by dy/dx 

 the slope of the direction of the ray at the point 

 (x,y), then 



dy ^^ 



dx a 



This equation integrates immediately to 



dy 



]/n'{y) _ 



1. 



=/: 



Y 



n\y) 

 k^ 



+ xo 



- 1 



(17) 



where Xq is an arbitrary constant. Regard the k as 

 fixed, and the Xo as variable; then equation (17) gives 

 an infinite set of curves which satisfy the definition 

 of rays. 



