The Calculus Diaries - Jennifer Ouellette [91]
It is not a big-wave day when I have my outing—good news for me, as a beginner, because the waves are smaller and the waters less crowded with hard-core surfers. One of the keys to surfing is choosing the right incoming wave. This is not an easy call to make; ocean wave dynamics are pretty complex. That’s why surf forecasters rely on real-time meteorological data from satellites to locate the biggest waves. Avid surfers get pretty adept at eyeballing the incoming waves to identify the most promising (by size, by when they’re likely to break, and so forth) and also at estimating how fast those waves will be traveling by the time they reach the surfer. But to a novice like me, they all look the same, and it’s tough to predict when they’ll crest and break.
Calculus comes into play when analyzing the waves themselves. All types of waves have three basic properties: wavelength, frequency, and amplitude. In the case of sound waves, the distance between compressions determines the wavelength. Objects that vibrate very quickly create short wavelengths because there is very little space between the compressions, creating a high-pitched sound. Objects that vibrate very slowly create long wavelengths because the compressions are spaced further apart. Frequency measures how many crests, or compressions, occur within one second; the measurement of this speed of vibration is called a Hertz (Hz), and 1 Hz is equivalent to 1 vibration per second. A sound wave’s amplitude, or range of movement, determines the volume (loudness) of the sound.
Ocean waves have these properties, too. Generally, waves are measured according to height (measured from trough to crest), wavelength (measured from crest to crest), and period (the interval between the arrival of consecutive crests at a fixed point), corresponding in turn to amplitude, wavelength, and frequency. Mathematically, waves are described as periodic functions: They repeat over regular intervals, forming a series of crests and troughs over time. Graph a periodic function on a Cartesian grid, and you get the signature ideal sine wave, the “face” of a periodic function:
The above sine wave represents the function sin(x), the mathematical idealization of a wave. The cosine, or cos(x), is the complement of the sine. It looks very much the same, except everything is shifted slightly to the left along the x axis, such that the cosine wave appears to start at its maximum while the sine wave starts at zero on the graph.
Any wave can be thought of as a sine or cosine, merely shifted by different amounts. Mathematicians simply call such waves sinusoids. We can glean quite a bit of information about a waveform from its “face.” The number of crests and troughs we can count in a given period of time, such as one second, gives us the frequency of the sinusoid. A large number of crests and troughs means it is a high-frequency wave; a low number of crests and troughs means it is a low-frequency wave. If we multiply x by a number, like 2, we increase the frequency of the wave (above), described by the function sin(2x). This means that the periods between crests and troughs will be shorter, giving us a wave with a higher frequency.
We can also adjust the amplitude—how strong/loud the wave is—by multiplying the function by another number, such as 2, giving us the function 2sin(2x). The resulting graph on page 236 shows a sine wave with higher crests.
The sine and cosine are the simplest waveforms, equivalent to pure musical notes, or a light wave of a single color. Different kinds of waves can interfere with each other, mixing together to form more complex waveforms.