378 SURFACE WAVES. [CHAP. IX 



The possibility of progressive waves advancing with unchanged form is of 

 course limited, theoretically, to the case of uniform depth ; but the foregoing 

 numerical results shew that practically a variation in the depth will have no 

 appreciable influence, provided the depth everywhere exceeds (say) half the 

 wave-length. 



We remark, finally, that the theory of progressive waves may 

 be obtained, without the intermediary of standing waves, by 

 assuming at once, in place of Art. 217 (1), 



The conditions to be satisfied by P are exactly the same as before, 

 and we easily find, in real form, 



rj = asm(kx at} (11), 



qa cosh k(y + h) 



&amp;lt;p = Vi - cos (KM at] (12), 



a cosh kh 



with the same determination of a as before. From (12) all the 

 preceding results as to the motion of the individual particles can 

 be inferred without difficulty. 



219. The energy of a system of standing waves of the simple- 

 harmonic type is easily found. If we imagine two vertical planes 

 to be drawn at unit distance apart, parallel to xy, the potential 

 energy per wave-length of the fluid between these planes is, as in 

 Art. 171, 



Substituting the value of 77 from Art. 217 (7), we obtain 



%gpa?\ . sin 2 (at + e) (1). 



The kinetic energy is, by the formula (1) of Art. 61, 



Substituting from Art. 217 (8), and remembering the relation 

 between &amp;lt;r and k, we obtain 



2 X . cos 2 (at + e) ..................... (2). 



The total energy, being the sum of (1) and (2), is constant, 



