Wave going into positive direction has the equation:

$$p(x)=\sin(x+\omega t)$$

Into the negative direction:

$$n(x)=\sin(x-\omega t)$$

Their sum:

\begin{align} p(x)+n(x)&=\sin(x+\omega t)+\sin(x-\omega t) \\ &=\sin(x)\cos(\omega t) - \cos(x)\sin(\omega t) + \sin(x)\cos(\omega t) - \cos(x)\sin(\omega t)\\ &=2 sin(x)cos(\omega t) \end{align}

As you can see, the wave on the bottom is a standing sine wave, whose amplitude is multiplied by the original frequency of the moving waves.

Wave going into positive direction has the equation:

$$p(x)=\sin(x+\omega t)$$

Into the negative direction:

$$n(x)=\sin(x-\omega t)$$

Their sum:

\begin{align} p(x)+n(x)&=\sin(x+\omega t)+\sin(x-\omega t) \\ &=\sin(x)\cos(\omega t) + \cos(x)\sin(\omega t) + \sin(x)\cos(\omega t) - \cos(x)\sin(\omega t)\\ &=2 sin(x)cos(\omega t) \end{align}

As you can see, the wave on the bottom is a standing sine wave, whose amplitude is multiplied by the original frequency of the moving waves.

Wave going into positive direction has the equation:

$p(x)=sin(x+\omega t)$

Into the negative direction:

$n(x)=sin(x-\omega t)$

Their sum:

$p(x)+n(x)=sin(x+\omega t) + sin(x-\omega t) =$

$sin(x)cos(\omega t) - cos(x)sin(\omega t) + sin(x)cos(\omega t) - cos(x)sin(\omega t) =$

$2 sin(x)cos(\omega t)$

As you can see, the wave on the bottom is a standing sine wave, whose amplitude is multiplied by the original frequency of the moving waves.

Wave going into positive direction has the equation:

$$p(x)=\sin(x+\omega t)$$

Into the negative direction:

$$n(x)=\sin(x-\omega t)$$

Their sum:

\begin{align} p(x)+n(x)&=\sin(x+\omega t)+\sin(x-\omega t) \\ &=\sin(x)\cos(\omega t) - \cos(x)\sin(\omega t) + \sin(x)\cos(\omega t) - \cos(x)\sin(\omega t)\\ &=2 sin(x)cos(\omega t) \end{align}

As you can see, the wave on the bottom is a standing sine wave, whose amplitude is multiplied by the original frequency of the moving waves.