Periodic Functions and Wave Phenomena

Why does Fourier analysis dominate signal processing, MRI, JPEG, and quantum mechanics? Because nearly every oscillating phenomenon you will ever encounter is, at the bottom, a superposition of sine and cosine waves. Before we can decompose a complicated signal into pure tones, we need to be fluent in the building blocks: what a period actually is, how to read amplitude and phase off an expression like $A\sin(\omega x + \phi)$, and how multiple waves combine. This section sets up that vocabulary so the rest of Chapter 13 has a place to land.

What "periodic" means

A function $f: \mathbb{R} \to \mathbb{R}$ is periodic with period $T > 0$ when $f(x + T) = f(x)$ for every $x$. If any such $T$ exists, the smallest one is the fundamental period. Every integer multiple of a period is also a period, so periods come in a discrete ladder $T, 2T, 3T, \ldots$. The fundamental period is the rung you cannot go below.

The two anchor examples are $\sin(x)$ and $\cos(x)$, both with fundamental period $2\pi$. Replacing $x$ with $nx$ compresses the graph horizontally by a factor of $n$, so $\sin(nx)$ has fundamental period $2\pi / n$. These compressed copies are the harmonics; the $n$-th harmonic fits exactly $n$ full oscillations into the original window $[0, 2\pi)$.

Amplitude, angular frequency, phase

The standard sinusoidal form $A\sin(\omega x + \phi)$ has three knobs:

• Amplitude $A$: the peak displacement above and below zero. Larger $A$ stretches the graph vertically; the function ranges over [-A, A].
• Angular frequency $\omega$: how many radians the argument advances per unit $x$. Period and angular frequency are reciprocals up to $2\pi$: $T = 2\pi / \omega$.
• Phase $\phi$: a horizontal shift in radians. $\sin(\omega x + \phi)$ reaches its first zero crossing at $x = -\phi / \omega$ instead of $x = 0$.

In physics and engineering you also see the ordinary frequency $f$, measured in hertz (cycles per second). It is related to $\omega$ by $\omega = 2\pi f$, and to period by $T = 1/f$. The conversion $\omega = 2\pi f$ is one of those identities you absolutely must be able to write down without thinking.

Use the function grapher to plot $\sin(nx)$ for several integer values of $n$. Notice that each curve crosses zero exactly $n$ times on the interval $[0, 2\pi)$, that is the visual signature of the $n$-th harmonic. Try adding $\sin(x) + \sin(2x)$: the combined wave keeps period $2\pi$ but the shape is no longer a clean sinusoid.

Superposition of waves

The principle of superposition says: if $f$ and $g$ both describe valid waves (solutions of a linear wave equation), so does $\alpha f + \beta g$ for any constants $\alpha, \beta$. For periodic functions, the practical consequence is that any sum $f(x) + g(x)$ of periodic pieces is periodic, with period equal to the least common multiple of the individual periods (if such an LCM exists). The whole edifice of Fourier series rests on this fact: complicated waves are linear combinations of simple ones, and that combination is itself a wave.

For example, $h(x) = \sin(x) + \tfrac{1}{3}\sin(3x) + \tfrac{1}{5}\sin(5x)$ is the start of the Fourier series for a square wave. Each term is periodic with a period that divides $2\pi$, so the sum is periodic with period $2\pi$.

• Music and acoustics: A note from a violin or trumpet is a fundamental sinusoid plus a stack of harmonics. The relative amplitudes of the harmonics are what make a violin sound different from a trumpet at the same pitch, this is "timbre," and it is literally a Fourier-amplitude fingerprint.
• Electrical engineering: AC mains power is a single sinusoid (50 Hz in Europe, 60 Hz in North America). Distortions and harmonics added by non-linear loads (LED drivers, switching power supplies) are what utilities measure when they complain about "harmonic pollution" on the grid.
• MRI scanners: The MR signal is acquired as oscillations at precisely chosen frequencies. The reconstructed image is the inverse Fourier transform of the measured wave amplitudes, section 13.4 makes this precise.
• Seismology: An earthquake record is a wave train. Spectral analysis (the Fourier amplitude vs. frequency plot) is how seismologists separate the slow surface waves from fast body waves, and how they estimate earthquake magnitude from amplitude at specific frequencies.

Pause and think: The function $h(x) = \sin(x) + \sin(\pi x)$ is a sum of two sinusoids. Is $h$ periodic? (Hint: the ratio $1/\pi$ is irrational. Could you find an LCM of $2\pi$ and $2$?)

Try it

• Predict first: what is the fundamental period of $\cos(5x)$? Then check with the formula $T = 2\pi / \omega$.
• Read off the amplitude, angular frequency, period, and phase shift of $g(x) = 3\sin(4x - \pi/2)$.
• The note A above middle C has ordinary frequency $f = 440$ Hz. Compute its angular frequency in rad/s and its period in milliseconds.
• Plot $\sin(x) + \sin(2x) + \sin(3x)$ on the function grapher. What is the period of the sum, and where is the first zero crossing?
• Write down the first three harmonics of a wave with fundamental period $T = 1$ second. What is the angular frequency of each?

A trap to watch for

The distinction between angular frequency $\omega$ (radians per unit $x$) and ordinary frequency $f$ (cycles per unit $x$) trips up almost everyone the first time. They are off by a factor of $2\pi$: $\omega = 2\pi f$. If a textbook says "the frequency is $5$ Hz," the angular frequency in the sinusoid argument is $10\pi$ rad/s, not $5$. Always check the units before you substitute into a formula, "Hz" implies cycles per second; bare "rad/s" implies $\omega$.

What you now know

You can read off the period, amplitude, angular frequency, and phase from a sinusoid. You know that adding two periodic functions produces another periodic function (provided their periods are commensurate), and you have seen visually that piling up harmonics builds up complicated wave shapes. The next section turns this observation into a rigorous decomposition theorem: any reasonable periodic function on [-\pi, \pi] can be written as an infinite sum of sines and cosines with computable coefficients.

Mark section complete →

References

• Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 13.
• Stein, E. M., Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press, ch. 1.
• Folland, G. B. (1992). Fourier Analysis and Its Applications. Wadsworth & Brooks/Cole, ch. 1-2.
• Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill, ch. 1.
• Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press, ch. 8 (orthogonality and periodic functions).