Now take 2 waves, $y_1$ and $y_2$. For instance, each wave might be from dropping 2 stones in the water at different locations. At some 3rd location, each wave will arrive and cause a displacement. The displacement of the 1st wave will be given by $y_1 = A\sin(\omega_1 t)$. Then the 2nd wave arrives, and it will also displace the water by $y_2 = A\sin(\omega_2 t)$, where $\omega_1 \ne \omega_2$ ($f_1\ne f_2$). The total displacement of the water at that location will be given by $$y_{tot} = y_1 + y_2\nonumber$$. That the 2 waves add linearly is not obvious!

We can study the function $y_{tot}$, and to do so let's employ a few tricks that makes the math easy. Define the average frequency $\bar f$ and the difference between the 2 frequencies $\Delta f$: $$\bar f \equiv \half (f_1 + f_2)\label{yav}$$ $$\Delta f \equiv f_1 - f_2\label{dy}$$ We can use these to solve for $f_1$ and $f_2$, and you would get $$f_1 = \bar f + \half\Delta f \nonumber$$ $$f_2 = \bar f - \half\Delta f \nonumber$$ Now we can add the 2 functions together: $$\begin{align} y_{tot} &= y_1 + y_2\nonumber \\ &= A\sin(2\pi f_1) + A\sin(2\pi f_2)\nonumber \\ &= A\big[\sin(2\pi[\bar f + \half\Delta f]t) + \sin(2\pi[\bar f - \half\Delta f]t)\big]\nonumber \\ \end{align}\nonumber $$ So each $\sin$ term is a function of 2 angles: $\sin(2\pi\half\Delta f t + \pi\bar{f}t)$. We then use the trig formula: $$\sin(a\pm b) = \sin(a)\cos(b)\pm\cos(a)\sin(b)\nonumber $$ and rewrite $y_{tot}/A$ (to make the latex simpler) substituting $\bar\omega=2\pi \bar f$ and $\Delta\omega = 2\pi\Delta f$ as $$\begin{align} y_{tot}/A =&\sin([\bar \omega + \half\Delta\omega]t) + \nonumber\\ &\sin([\bar\omega - \half\Delta\omega]t)\nonumber \\ =& \sin(\bar{\omega}t)\cos(\half\Delta\omega t) + \cos(\bar{\omega}t)\sin(\half\Delta\omega t) +\nonumber \\ &\sin(\bar{\omega}t)\cos(\half\Delta\omega t) - \cos(\bar{\omega}t)\sin(\half\Delta\omega t) \nonumber \\ =& 2\cos(\half\Delta\omega t)\sin(\bar{\omega}t)\nonumber\\ \end{align}\nonumber $$ This gives us the formula $$y_{tot} = 2\cos(\half\Delta\omega t)\sin(\bar{\omega}t)\label{sum}$$ for the sum of two waves that have different frequencies.

Now let's imagine that $f_1$ and $f_2$ are close, so that $\delta f\ll \bar{f}$. We can then see that $y_{tot}$ looks like a wave with amplitude $A' = 2\cos(\half\Delta\omega t)\nonumber$ modulated by $\sin(\bar\omega t)$, since the amplitude $A'$ is almost constant over time periods where the wave is being modulated. For example, if $y_1$ and $y_2$ were two audio tones, say $440$Hz and $442$Hz, what you would here if you played them togehter would be a $441$Hz tone that modulated at $\half\Delta\f=1$Hz. This is "beating", and $\Delta f=2$Hz in this example would be the "beat frequency", the difference between the two frequencies. This is how musicians can tune their instruments: they have a reference (like a pitchfork) play a 440Hz tone, and play the same note on their instrument, and listen for the modulation. When the modulation goes to zero, $\Delta f=$ means that the instrument is in tune with the pitchfork reference tone. Of course if $\Delta f$ is not much much less than $\bar f$, you won't hear the "beating", but will just hear to cord of the 2 tones.

The phenomema of beats can be applied to variations in space as well. For instance, if you have two picket fences that have the very thin pickets and the same space between pickets (so that the width of the pickets is less than the space between pickets), if you put the fences next to each other, having the pickets line up gives you the best view. But if you move the fences relative to each other, you are changing the relative spatial phase of the periodicity of the fences, and when you try to look through, you will see some interference that depends on that relative phase. For instance, in the simulation below, you will see a set of green and blue lines with equal width and spacing, and below in red, the sum of the two. With the mouse, grab the green set of lines and move it left or right. You can see the "interference" between the two sets of lines in the red lines. This interference is called a "Moire" pattern, and comes from the phenomena of adding spatially periodic things.

That example above has both green and blue lines with the same wavelength. In the simulation below, you can also change each wavelength, You can see the Moire interference pattern (red) between two vertical lines (green and blue). You can increase or decrease the pitch (distance between vertical lines) of each of the two sets, and you can change the relative angle between the two sets by changing the angle θ of the 2nd set relative to the first.

As you change the spacing, you will see the interference pattern change. The green spacing, and the blue spacing, constitute a different "wavelength", with a corresponding "frequency" given by $f=1/\lambda$. Taking the difference between the two frequencies will give you the beat frequency $\Delta f$ of the moire pattern, and $1/\Delta f$ will be the wavelength of the moire interference. To check, you can draw horizontal line between two points, say between two maxima of the moire interference pattern, and the length will be reported in the table. It should agree pretty closely with the reported beat wavelength. Note that as the difference in wavelengths between the green and the red become large, you no longer see the beat phenomena of a small wavelength modulating a larger wavelength. Beating is only useful (and interesting!) if the difference in periodicities are small compared to the periodicity itself.