168 DYNAMICAL THEORY OF SOUND 



61. Reflection. 



When there is a fixed barrier at the origin the general 

 solution is replaced, as in 24, by 



g = f(ct-a;)-f(ct + x) (1) 



Considering, for example, the region to the left of the origin, 

 the first term may be interpreted as representing a primary 

 wave-system approaching the barrier; the second term then 

 represents the reflected system. The latter has the same 

 amplitude at corresponding points ; the velocity j is reversed, 

 but the condensation s (= di~/dx) has its sign unchanged. We 

 have here, in its simplest form, the explanation of echoes. 



There is another case of reflection which it is important to 

 consider. Suppose that at one point (say x = 0) the condition 

 of unvarying pressure (s = 0) is imposed. We must have then, 



in 59 (9), 



F'(et)=f(ct), (2) 



which shews that the functions f, F must differ only by 

 a constant. Since this constant would merely represent a 

 displacement common to the whole mass, which is without 

 influence on the question, it may be ignored. We have then 



f = f(ct"x)+f(ct + x) t (3) 



where as before the first term may be taken to represent an 

 incident, and the second a reflected wave-system, in the region 

 lying to the left of 0. The velocity f is here reflected un- 

 changed, but the sign of s is reversed. The conditions would 

 be realized if the air were in contact at the plane as = with 

 a medium capable of exerting pressure, but destitute of inertia. 

 This is of course an ideal case, but the condition of invariable 

 pressure is approximated to in some degree at the open end of 

 a pipe. The present investigation has also an application to 

 the reflection of longitudinal waves at the free end of a rod 

 ( 43). 



The general problem of (direct) reflection at the common 

 boundary of two distinct fluid media is hardly more complicated. 

 The origin being taken in the boundary, a wave-system ap- 

 proaching from the left will give rise to a reflected wave on the 

 left and a transmitted wave on the right. We distinguish 



