lecture 7

Lecture 7
Solution of the General Wave Equation
Eq. (21) can be written as:
?^2 E=1/c^2 (?^2 E)/(?t^2 ) ……… (22)
where E is a function of x,y,z, and t, c is a positive constant, will be proven later to be the speed of the wave.
For simplicity we will consider the wave propagating in only a single spatial direction, the z direction, and consequently eq. (22) becomes:
(d^2 E(z,t))/(dz^2 )=1/c^2 (?^2 E)/(?t^2 ) ……… (23)
We will assume that the wave function E(z,t) is separable and thus can be written as:
E(z,t)=E(z)E(t)=E_z E_t …. (24)
Thus eq. (23) can be written as:
c^2/E_z (d^2 E_z)/(dz^2 )=1/E_t (d^2 E_t)/(dt^2 ) ………… (25) (How?)

In order to satisfy the equation - both sides must be equal to the same constant, which we will arbitrarily denote as -?^2. This leads us to the following two equations:
(d^2 E_z)/(dz^2 )+?^2/c^2 E_z=0 …………. (26)
(d^2 E_t)/(dt^2 )+?^2 E_t=0 …………. (27)
These equations are of a familiar form and have the following solutions respectively:
E_z=C_1 e^(i(?/c)z)+C_2 e^(-i(?/c)z) …………. (28)
E_t=D_1 e^i?t+D_2 e^(-i?t) …………. (29)
where C_1,2 and D_1,2 are constant can be determined by the boundary conditions.
The general solution of the wave equation can be written as:
E(z,t)=Ce^(±i((?/c)z??t)) …………… (30) (What does ±stand for?)

Considering a wave travelling in one direction (from left to right) one gets:
E(z,t)=Ce^(-i(k_z z-?t)) …………… (31)
where
k_z=?/c ………….. (32)