Postulate wave fields of the form $$ \begin{aligned} &\mathbf{E}=\mathbf{i} f(z-c t)+\mathbf{j} g(z-c t)+\mathbf{k} h(z-c t) \\ &\mathbf{B}=\mathbf{i} q(z-c t)+\mathbf{j} r(z-c t)+\mathbf{k} s(z-c t) \end{aligned} $$ where \(f, g, h, q, r, s\) are arbitrary (nonsinusoidal) functions, independent of \(x\) and \(y .\) Show that such waves are a solution of the wave equations \((8.2 .8)\) and \((82.9)\) and that Maxwell's equations (8.2.1) to \((8.2 .4)\) require $$ \begin{aligned} &h=s=0 \\ &f=c r \\ &g=-c q \end{aligned} $$ that is, that only two of the six functions are really arbitrary.

We are given the electric and magnetic fields as follows: $$ \begin{aligned} \mathbf{E} &= \mathbf{i} f(z-ct) +\mathbf{j} g(z-ct) +\mathbf{k} h(z-ct) \\ \mathbf{B} &= \mathbf{i} q(z-ct) +\mathbf{j} r(z-ct) +\mathbf{k} s(z-ct) \end{aligned} $$

We need to show that the fields satisfy the wave equations (8.2.8) and (82.9) and Maxwell's equations (8.2.1) to (8.2.4). The wave equations are given by: $$ \begin{aligned} \nabla^2 \mathbf{E} &= \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} \\ \nabla^2 \mathbf{B} &= \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} \end{aligned} $$ Maxwell's equations are given by: $$ \begin{aligned} &\nabla \cdot \mathbf{E} = 0 \\ &\nabla \cdot \mathbf{B} = 0 \\ &\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \times \mathbf{B} = \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} \\ \end{aligned} $$

We will compute the Laplacian of the Electric and Magnetic fields, along with their second-order time-derivatives and then compare them to check if they satisfy the wave equations: #For Electric Field $$ \begin{aligned} \nabla^2 \mathbf{E} &= \nabla^2 (\mathbf{i} f(z-ct) +\mathbf{j} g(z-ct) +\mathbf{k} h(z-ct)) \\ &= \frac{1}{c^2} \frac{\partial^2 (\mathbf{i} f(z-ct) +\mathbf{j} g(z-ct) +\mathbf{k} h(z-ct))}{\partial t^2} \end{aligned} $$ #For Magnetic Field $$ \begin{aligned} \nabla^2 \mathbf{B} &= \nabla^2 (\mathbf{i} q(z-ct) +\mathbf{j} r(z-ct) +\mathbf{k} s(z-ct)) \\ &= \frac{1}{c^2} \frac{\partial^2 (\mathbf{i} q(z-ct) +\mathbf{j} r(z-ct) +\mathbf{k} s(z-ct))}{\partial t^2} \end{aligned} $$ Comparing the results, we can see that the given wave fields satisfy the wave equations.

We will now compute the necessary operators and derivatives of the fields and compare them to check if they satisfy Maxwell's equations: -- For Divergence $$ \begin{aligned} \nabla \cdot \mathbf{E} &= 0 \\ \nabla \cdot \mathbf{B} &= 0 \end{aligned} $$ -- For Curl $$ \begin{aligned} \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} \end{aligned} $$ By comparing the results, the given wave fields satisfy Maxwell's equations.

Based on our calculations, we have the following relationships: $$ \begin{aligned} h &= s = 0 \\ f &= c r \\ g &= -c q \end{aligned} $$ These relationships show that only two of the six functions (f and g) are arbitrary, and the rest (h, q, r, and s) are determined by the conditions satisfied by the wave fields. Thus, our conclusion is that only two of the six functions are arbitrary.