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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.

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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). fig

The superposition does not look like a standing wave. Instead, it looks like a wave that is travelling with a periodically changing velocity.

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  • $\begingroup$ Thanks for the answer , but what i wanted was an exact expression. For example the magnitude of such a partial standing wave is sqr ( 1 + (Gama)^2 + 2*( Gama) * cos( bz)) . This is the function f(z) i was referring to. What i wanted is what is multiplying this f(z)? For a pure standing wave f(z) = sin(bz) which is multiplied by cos (wt) and there is no cos ( wt + bz) , that is why it is a pure standing wave. What should multiply f(z) for the partial standing wave? Should it be of the form cos ( wt) or cos ( wt + bz) ? $\endgroup$ May 8, 2020 at 19:48
  • $\begingroup$ 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. $\endgroup$
    – Semoi
    May 8, 2020 at 20:14
  • $\begingroup$ @Semoi Can you post a link to the applet used for this simulation? $\endgroup$
    – nasu
    Dec 11, 2020 at 16:41

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