292 BELL SYSTEM TECHNICAL JOURNAL 



dU/dy, dU/dz. However, the value of the derivative ds/dt prevaiUng 

 at the moment when the wavelet starts is simply the derivative d U/dt, 

 which we may as well write dU/dt — it makes no diflference. 



Our definition of wave-motion may now be stated more rigorously. 

 A quantity 5 is said to be propagated by waves, if its value at the 

 origin at the moment t is determined by the values of U, dU/dx, dU/dy, 

 and dU/dz over any surface enveloping the origin. 



We now turn to another and more familiar definition of wave-motion, 

 which shall presently be shown to fall as a special case under this one. 



The Wave-Equation 



There is a very celebrated differential equation of mathematical 

 physics, known as "the wave-equation" par excellence. Any theory 

 which culminates in this equation is designated as a wave-theory. 

 The foundation of the theory of sound is the proof that the excess of 

 pressure in the air over its average value is subject to this equation. 

 The elastic-solid model of the luminiferous aether was partially suited 

 to explain the phenomena of light, because the compressions and the 

 distortions of an elastic solid conform to the wave-equation. The 

 electromagnetic theory of light was born when Maxwell discovered 

 interrelations between electric and magnetic fields, out of which 

 by transformation a wave-equation could be formed. Undulatory 

 mechanics is based upon an equation of this type which emerges during 

 the process of setting and solving the classical equations of motion. 



This wave-equation is : 



d^s d^s , ^"-^ \ _ d~s .,. 



To demonstrate why it is called a wave-equation and what is the 

 physical meaning of the constant c, it is customary to make a drastic 

 simplification by assuming that the function s depends only on one co- 

 ordinate. Such is the case, for instance, when 5 stands for the trans- 

 verse displacement of an endlessly long taut string initially parallel to 

 the axis of x; likewise, when it stands for the excess of the pressure of 

 the air over its average value, and this excess is constant over every 

 plate normal to the x-direction — a condition known as that of "plane 

 waves." Then the wave-equation assumes the form: 



There are infinitely many solutions of this equation, and among them 



