202, 203] REFLECTION AT WALL. 545 



209) S ~-^= l a l'(x-ai), 



so that 



u = as. 



Thus the velocity of the particle is in the same direction as that of 

 the wave where there is condensation, or s is positive, in the opposite 

 direction where there is rarefaction, or s is negative. 



2O3. Echo. Organ -pipes. Suppose there is a rigid wall whose 

 equation is x = l. The velocity of a particle normal to the wall 

 must be zero, so that 



210) u = F 1 '(x-af) + F 2 '(x + ai) = Q 

 when x = I, or 



211) ' FJ (I - at) + jy (I + af) = 



for all positive values of i. Thus one of the functions is determined 

 by the other. Put 



I -f at = y, 



so that our equation 211) is 



212) FJ(y)--Fl(*l-y), 



a differential equation connecting F and F 2 , the integral of which is 



213) F 2 (y) = F 1 (2l-y) + C 



for y^l. Since the velocity depends only on the derivative of qp, 

 the value of C is immaterial, and we will put it equal to zero. 



The equation F 2 (y) = F 1 (21 y) indicates that the curve 

 representing _F 2 is the geometrical reflection in the wall of that 

 representing F ly in other words the function F% represents a wave 

 travelling to the left, which after x at is greater than I represents 

 the motion on the left of the wall, the values of cp at points a 

 certain distance to the left of the wall being the same as they would 

 have been at the same distance to the right of the wall had the 

 direct wave gone on unchanged. Since the values of u depend on 

 the derivative of (p according to x, the velocity changes sign in the 

 reflection. This must be the case for the condition producing reflection 

 is that u = at x = Z, so that the wave coning to the left must 

 have a velocity equal and opposite to that of the wave going to the 

 right. If there is a wall at x = as well as at x = I, the wave is 

 reflected in that also, so that the motion consists in the continual to 

 and fro motion of the original disturbance. The motion at any point 



is periodic in the time We may accordingly develop the motion 



, Dynamics. 35 



