I am trying to understand how the ratio between the amplitudes of two standing waves on a transmission line of length $x$ change as a function of the voltage reflection coefficient, $\Gamma$
Background to the problem: There is an incident forward wave from the transmitter, travelling towards the antenna feed-point along the transmission line as that can be described as a function of time and length down the transmission line as:
\begin{equation*} V_F = A \cos{\left( 2\pi ft - \frac{2\pi}{\lambda}x \right)} \end{equation*}
And the reflected wave which is exactly the same as the forward wave, just multiplied by the reflection coefficient $\Gamma$ \textit{and it is obviously travelling in the opposite direction} to the forward wave - so the propagation constant $\frac{2\pi}{\lambda}$ will be positive here hence:
\begin{equation*} V_R = \Gamma A \cos{\left( 2\pi ft + \frac{2\pi}{\lambda}x \right)} \end{equation*}
It can therefore be said that the superposition resultant wave along the transmission line as a function of time and distance is the summation of these two waves where:
\begin{equation*} V_F + V_R = A \cos{\left( 2\pi ft - \frac{2\pi}{\lambda}x \right)} + \Gamma A \cos{\left( 2\pi ft + \frac{2\pi}{\lambda}x \right)} \end{equation*}
Taking the $A$ out for simplification:
\begin{equation} \boxed{ V_F + V_R = A \Bigg\{ \cos{\left( 2\pi ft - \frac{2\pi}{\lambda}x \right)} + \Gamma \cos{\left( 2\pi ft + \frac{2\pi}{\lambda}x \right)} \Bigg\}} \label{eqn: wave-equation} \end{equation}
This Equation is the final expression for the wave-equation of a resultant EM wave travelling along a transmission line for various values of $\Gamma$ for a given frequency.
The Figure shows a plot of the Wave-Equation for a certain value of the reflection coefficient. Can you see \textbf{two} waves here? There appears to be a major and minor wave that will appear to \lq stand\rq~ in time at very high frequencies. I want to know how to find the amplitude of the major standing wave AND the amplitude of the minor standing wave as a function of $\Gamma$. Once I have achieved this, finding the standing wave ratio as a function of the voltage reflection coefficient will be easy. I know what the answer should be - it should be this:
\begin{equation*} \begin{split} \text{Major Standing Wave Amplitude} &= 1 + \lvert \Gamma \rvert\\ \text{Minor Standing Wave Amplitude} &= 1 - \lvert \Gamma \rvert \end{split} \end{equation*}
The question I have is how to I achieve this mathematically from the Wave-Equation presented above? I suspect this may be PDE related or more straightforward?
I would greatly appreciate any help anyone could give me with this,
Kindest Regards,
Adrian.