Are standing waves in water exact sine waves? In many physics demonstrations waves in water are used to illustrate principles of standing waves and wave propagation. After such demonstrations, classes tend to move on to basic problems illustrating the water wave (or sometimes wave propagation in stretched rope) and use the sine wave to describe the up-down motion of the fluid column. 
Is the sine wave just an elementary teaching model of a more complex motion (like a triangle/sawtooth/etc) or or does the water ACTUALLY move up and down exactly as described by a sine/cos function?
 A: The sine function is just an idealized way to approximate wave motion, and indeed suitable for teaching the basic principles of how waves propagate, reflect and interfere with one another to create standing waves, but as with any real physical system, including the motion of waves, the closer you look the more you see non-ideal behavior. 
For surface waves like what's observed in the oceans, far from the shore, if you were to track a small particle in the wave you would see that it makes a circular orbit as the wave's energy passes through the water as a medium. The orbit diameters become smaller and smaller as you move deeper away from the surface until they essentially vanish - where the energy is no longer felt. Thus the water molecules are not only moving up and down, but also back and forth in the direction of the wave as the wave's crest and trough move by, but are not carried by the wave. The only thing that is carried by the wave is the energy. The energy that was transferred by the wind from a far distance.
A: Books generally teach sine or cosine waves because according to Fourier thereom any wave can be written as linear combination of sine or cosine waves.
FOURIER THEOREM
A mathematical theorem stating that a periodic function f(x) which is reasonably continuous may be expressed as the sum of a series of sine or cosine terms (called the Fourier series), each of which has specific amplitude and phase coefficients known as Fourier coefficients.
http://www.sfu.ca/sonic-studio/handbook/Fourier_Theorem.html
But generally waves in the demostrations are not always sine or cosine waves.
