Peaks, Troughs, Wavelength

A peak of a wave, more commonly known as a crest, is the highest point of a wave.

A trough of a wave is the lowest point of a wave.

The wavelength of a wave is defined to be the distance over which the wave's shape repeats. It is generally abbreviated $\lambda$, and the unit of wavelength is the same as the unit for length, which is meters.

Due to the nature of a wave, the wavelength can be seen as the distance between two points of the same phase on the wave. For example, the wavelength is the distance between two adjacent crests or two adjacent troughs.

Let us look at an example: consider the graph of the function $y=\sin x$. This is a perfect example of a wave.

The highest points on the wave occur when $\sin x=1$. This occurs when $x=2k\pi +\frac{\pi}{2}$, for integer $k$. Thus, the peaks are all points of the form $\left(2k\pi +\frac{\pi}{2},1\right)$ for all integers $k$.

Similarly, the troughs occur when $\sin x=-1$. This occurs when $x=2k\pi -\frac{\pi}{2}$, for integer $k$. Thus, the troughs are all points of the form $\left(2k\pi -\frac{\pi}{2},-1\right)$ for all integers $k$.

Finally, to find the wavelength, think about how long it takes for the sine function to repeat itself. We know that $\sin (x+k)=\sin x$ for all $x$ if and only if $k$ is an integer multiple of $2\pi$. Thus, the smallest distance over which the wave repeats itself is $2\pi$, so the wavelength is $2\pi$.

This example is quite simplistic, but it provides a general understanding of these three essential properties of a wave.

In general, always be on the lookout for these distinguishing elements whenever you are dealing with waves.