Spectral analysis -- summing different wavelengths

Sinusoidal waves play an important role in understanding wave phenomena, both in our perception of them as biological systems ourselves, and in how we can use them as scientists to probe aspects of the world around us.

We perceive wave phenomena as light and sound, two of the main senses we use in making sense of the world we live in. Pure sinusoidal waves of light are perceived as a "color of the rainbow" and pure sinusoidal waves of sound are perceived of as pure musical tones. But most wave signals are produced by sources that contain a mix of frequencies.

How sinusoidal waves combine to form a complex signal is a field of mathematics called spectral analysis since it allows one to take a signal containing many different frequencies and split it apart, the wave a prism splits the light from the sun into a spectrum. (In math, this is referred to as Fourier Analysis after Joseph Fourier, the developer of the technique.)

In studying traveling waves, we saw that we could make any shape $f(x)$ move by simply shifting the $x$ to $f(x-v_0t)$ or $f(x+v_0t)$.  So if we want to see how we can add sinusoidal waves, we don't have to work with the two-variable sine wave $\sin{(kx-ωt)} = \sin{(k(x-v_0t)}$, we can just work with $\sin(kx)$ or $\sin(ωt)$ and set it into motion after we're done. 

We saw in our study of beats, that adding two nearby frequencies gave an oscillation that had an interesting shape — a fast oscillation occurring within an envelope of a slow oscillation. If we add more frequencies, we can get more complex shapes.

One way of getting a sense of how this works is to explore adding together oscillations with different frequencies using a PhET simulation, Fourier: Making Waves. One example of adding together different amount of waves with difference frequencies is shown in the figure below. On the left is shown the sum 

$$f(t) = \sum_{n=1}^{\infty} A_n \sin{(\omega_n t)}$$

where $ω_n = nω_1$ and $ω_1$ is the fundamental frequency. On the top is shown the sum and on the bottom is shown the values of $A_n$.

Given the signal shown at the top, the amounts of each angular frequency (or frequency, or wavelength) that went into making up the signal is called a spectral analysis and the figure showing the amount of each frequency is called the spectrum.

Spectral analysis has a number of different important applications, for example:

• We can use it to measure the temperature of a distant object by doing a spectra analysis of the light it emits.
• We can use it to measure the chemical composition of a substance that emits or transmits light.
• We can use it to understand what makes musical instruments sound different.
• We can use it study the sounds emitted by animals to tell something about the structure of their sound producing apparatus and their communication strategies. 

Measuring temperature through a spectral analysis of light

One of the drivers of the early development of the quantum theory was the observation that every object emits electromagnetic radiation (light / photons) by virtue of its temperature. The shape of the spectrum is characteristic of the temperature of the object. The spectrum of light emitted by the sun is shown as a function of the wavelength in the figure at the right. (Such a spectrum is called a blackbody spectrum since an ideal blackbody is both a perfect emitter and absorber.) The yellow curve shows the spectrum of light emitted by the sun that arrives at the earth. The shape (where is peaks) allow us to measure the temperature of the surface of the sun.

Measuring chemical composition through a spectral analysis of light

As discussed in the page, Quantum oscillators -- discrete states, because of the wave nature of electrons, atoms and molecules only can be in certain specified states of particular energies. This is very analogous to the standing waves (or "normal modes") that are allowed on a fixed string. As a result, due to energy conservation, only specific frequencies of light can be absorbed or emitted by particular chemicals — and those frequencies are characteristic of those chemicals.

If a gas of a chemical is heated or a spark passed through it, the molecules will, with some probability, be bumped up into particular excited states. Those excited states will then emit photons of light, giving off light of characteristic colors.

If a continuous spectrum of light is passed through a gas of chemicals, each chemical will absorb its characteristic frequency producing a black line on the spectrum. This absorption spectrum allows us to identify what atoms and molecules are present in the sun's atmosphere and to measure the chemicals in our own. (The element Helium was first discovered on the sun by this method.) The particular dips (red curve) in the solar spectrum observed on the ground on the earth allows us to measure the composition of the upper atmosphere. And looking at light passing through a smoke plume from a factory allows us to measure what chemicals are being emitted in the plume! 

Understanding the difference in musical instruments through a spectral analysis of sound

When a musical instrument plays a note — say middle C on a piano (256 Hz) — it not only emits sound waves of that frequency but also of other frequencies, particularly the higher modes we saw when we looked at standing waves on a string (f₀, 2f₀, 3f₀, 4f₀, etc.). Different instruments produce different mixes of the associated frequencies and different decays times for the production of the sound, and this is what is responsible for their distinctive sounds.

A lovely video demonstrating the spectral analysis of musical tones is Anna-Maria Hefele's video showing how one can sing multiple notes at the same time! It's called overtone or throat singing and is a large part of the musical culture of Tuva in central Asia. 

Using spectral analysis of the sounds produced by animals

Spectral analysis of sounds are also used to describe speech, particularly by humans and dolphins. Since typical sound waves oscillate very fast compared to the time of a speech sound, we can break the sound up into many different time blocks each having many many oscillations. In each of those blocks we can do a spectral analysis. This results in a description of how the mix of frequencies change as the sound is made. (Such a process is often referred to as a spectrogram - or more technically, as a fast Fourier Transform or FFT.) 

The spectrogram at the right shows an analysis of a dolphin's signature whistle. At each time bin, the vertical line above it shows the amount of each frequency present by the color (more is red, less is blue). This particular spectrogram shows three "chirps" — tones of quickly rising frequency — of about 0.3 seconds each. This particular spectrogram was made as part of a study to classify dolphin signature whistles in order to determine whether they were inherited or learned, and if learned, from whom?

Workout: Spectral analysis

Joe Redish 4/11/17