constant velocity c, along positive direction x. If the wave is 

 travelling in a negative direction, (3.2) becomes 



q> = f (x + ct) . (3.3) 



4, The equa t ion of non-dispersive wave motion 



If the disturbance is constant over all points of a plane 

 drawn perpendicular to the direction of propagation of a wave, 

 it is called a plane wave, and the plane is called a wave front. 

 The wave front moves perpendicular to itself with the velocity 

 of propagation c. Let us consider a plane wave in two dimensions. 

 If x:y = l:m is the direction of propagation, where 1 and ra are 

 the direction cosines of the normal to each wave front, then the 

 equation of the wave front in two dimensions is 



Ix + my = const, (4.1) 



so that at any moment t, cp is constant for all x and y which 

 satisfy (4,1). It is clear then, that 



9 = f(lx + my - ct) (4.2) 



is a function which fulfills all these requirements and therefore 



represents a plane wave travelling with velocity c in the direction 



l:m. 



2 2 

 Since 1 and m are direction cosines, 1 + m = 1 and it can 



be verified that 9 satisfies the differential equation 



cjx dy c c)t 



