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), 



*x) ........................... (10). 



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

 and we easily find, in real form, 



(11), 



ga cosh k (y + h) 

 6=^- ?.i 'cos(kx-at) ............ (12), 



or cosh kh 



with the same determination of <r 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, 



f x 

 ^gp I jf dx. 



Jo 

 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, 



Y dx. 



y j j/ = o 



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

 between cr and k, we obtain 



. cos 2 (trt + e) (2). 



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



