# What is the equation of partial standing wave

What is the equation of a partial standing wave which is formed by adding two opposite traveling waves with different amplitudes: $$y(z,t) = A\cos(\omega t + bz) + B\cos(\omega t - bz)$$ .

This is not a pure standing wave. Its magnitude must be a function of $$z$$ i.e. $$f(z)$$, but I do not know if it is of the form $$y(z,t) = f(z) \cos (\omega t + bz)$$ i.e like an Amplitude modulated traveling wave.

Assuming that $$A_2 = A_1 + \Delta A > A_1$$ we can always write \begin{align} y &= A_1 \cos(wt + k z) + A_2\cos(wt - k z) \\ &= A_1 \big[ \cos(wt + k z) + \cos(wt - k z) \big] + \Delta A \cos(wt - k z) \end{align} The first part with amplitude $$A_1$$ builds a standing wave, the second part with amplitude $$\Delta A$$ is simply superimposed.
The following image shows an incident wave (red) with amplitude $$A_2=2$$, which is only partially reflected on a mirror. The reflected wave (green) has amplitude $$A_1=1$$. The blue wave is the superposition. The horizontal axis is the position axis, which you called $$z$$ (I use $$x$$ instead).
• Either you can do the calculation using $cos(\varphi) = Re\{e^{i \varphi}\}$, or you just substitute the standard expression for the standing wave. From the text and from the image I hope it is clear that there do not exists nots as in the equal amplitude superposition case. May 8, 2020 at 20:14