Attribute of standing waves One of the attributes of a standing wave stated in the Tsokos Cambridge IB HL textbook is:
The amplitude of oscillation is different at different points on the string.
I do not understand this because by definition, amplitude is the maximum displacement of a wave. So how can amplitude of oscillation be different at different points on a string when there is only one given amplitude if given a string?
 A: When there is a standing wave on a string (e.g., a guitar string), there are actually two waves traveling in opposite directions: one left and one right.  Suppose the wavelength of each wave is equal to the length of the string.  In that case, at the middle of the string, the transverse displacement due to the left-moving wave is always precisely opposite to the transverse displacement due to the right-moving wave.  As a result, the string effectively holds still in the middle, while its left and right halves oscillate in opposite transverse directions.  
You can easily demonstrate this for yourself: Stretch a string (or a rubber band) between two stationary points.  Put one finger lightly at the precise center of the string, and with your other hand pluck the string at a point halfway between the middle and an end. It is possible to see by eye that the middle holds still while the two halves oscillate.
"Amplitude" in the textbook apparently refers to the maximum transverse displacement of the string -- at a given point on the string.
A: Let's consider the simplest case: a standing wave with a single oscillation frequency on a string of length $L$. Then the shape of the string $y(x,t)$ oscillates according to the following formula:
$$y(x,t)=A\sin\left(\frac{n\pi x}{L}\right)\sin\left(n\omega t-\delta\right)$$
for some overall amplitude $A$, some integer $n$, and where $\omega=\frac{\pi}{L}\sqrt{\frac{T}{\mu}}$ is the fundamental frequency of the string. Since the maximum value of $\sin(n\omega t-\delta)$ is 1, the maximum amplitude of the string as a function of $x$ is
$$y_{max}(x)=A\left|\sin\left(\frac{n\pi x}{L}\right)\right|$$
Note that there are certain points along the string at which $y_{max}=0$; those are called nodes, while the points at which $y_{max}=A$ are called antinodes. A standing wave of order $n$ (the "$n$th harmonic") will have $(n-1)$ nodes (not counting the ends of the string), and $n$ antinodes.
