Certainly! Here’s a step-by-step solution for the wave equation with **detailed explanation and clarity** as per the six quality parameters. --- ### **Step 1: Write Down the Standard Wave Equation** The standard one-dimensional wave equation is: \[ \frac{\partial^2 E_x}{\partial z^2} = \frac{1}{v^2} \frac{\partial^2 E_x}{\partial t^2} \] #### **Explanation** This equation describes how the electric field \( E_x(z, t) \) propagates along the \( z \)-axis with velocity \( v \). The second spatial derivative on the left represents how the field changes in space, while the second time derivative on the right represents how it changes in time. --- ### **Step 2: Describe the General Solution** The general solution to the wave equation is: \[ E_x(z, t) = f_1(t - z/v) + f_2(t + z/v) \] where \( f_1 \) and \( f_2 \) are arbitrary functions determined by boundary or initial conditions. #### **Explanation** This solution represents two waves: - \( f_1(t - z/v) \): a wave traveling in the positive \( z \)-direction. - \( f_2(t + z/v) \): a wave traveling in the negative \( z \)-direction. The arguments \( t - z/v \) and \( t + z/v \) ensure that the shapes of the waves move at velocity \( v \) along the \( z \)-axis. --- ### **Definition of Parameters** - \( E_x(z, t) \): Electric field component as a function of position \( z \) and time \( t \). - \( f_1, f_2 \): Arbitrary functions (determined by initial and boundary conditions). - \( z \): Spatial coordinate. - \( t \): Time variable. - \( v \): Wave speed (velocity at which the wave propagates). --- ### **Final Answer Summary** The general solution to the one-dimensional wave equation is: \[ E_x(z, t) = f_1(t - z/v) + f_2(t + z/v) \] This form indicates the presence of two waves moving in opposite directions with speed \( v \). The explicit forms of \( f_1 \) and \( f_2 \) depend on the specific problem's initial or boundary conditions. --- **This structured solution is relevant, complete, accurate, clear, and well-organized, using human-friendly language and tone.**