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Waves Introduction: The classical wave equation A large number of physical phenomena starting from the vibration of strings in a musical intrument, behaviours of sound or light to dynamics of quantum mechanical particle are explained in terms of wave. Since the origin of wave phenomenon depends upo...

Waves Introduction: The classical wave equation A large number of physical phenomena starting from the vibration of strings in a musical intrument, behaviours of sound or light to dynamics of quantum mechanical particle are explained in terms of wave. Since the origin of wave phenomenon depends upon the physical situation under consideration we may adopt Max Born's approach regarding the definition of wave: A wave is a solution of a differential equation called wave equation. The whole physical explanation of a wave phenomenon is then attributed to the particular wave equation under consideration. We will begin our discussion with the so called classical wave equation in 1 + 1 dimension (1 space and 1 time) given as follows (1.1) where, the scalar field 'f ='f (x, t) is called a wave or wave function. v( I 0) is a real constant specific to the system under consideration. Eqn-1.1 tells that v has the dimension of velocity. Whether v indeed represents a velocity will be clarified in the next section. Remark :1. In some occasion the scalar field 'f is replaced by a vector field leading to a vector wave equation, a situation that we will soon come across while considering light as an electromagnetic wave. Equation-1.1 is readily extendable to 3 + 1 dimension replacing.,;a:2 by the Laplacian V 2 giving i v2'f - __!_ c 'f = o (1.2) v2 at2 It is obvious to note that the classical wave equation is linear and homogeneous. Solution of Classical Wave Equation (eqn-1.1) Let us introduce two new variables(;= x + vt and 17 = x - vt. For any function F=F(x, t) one can write aF aF dF= -dx+-dt ax at aF aF ax aF at a, = ax a, + at a, aF aF 1 aF -=-+-- a, ax v at. a _ a 1 a s· = dX + vdl· 1m1. 1ar1y, drj = dX - vdl· Hence, This gives a _ a 1a