mathematics

Rarefaction

Rarefaction is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Rarefaction rather than just read about it. In short: Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves (sound waves, for instance), rarefaction waves also exist in nature.

Key takeaways

Rarefaction belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rarefaction to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rarefaction from memory before moving on to harder problems.

Reference excerpt

Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves (sound waves, for instance), rarefaction waves also exist in nature. A common rarefaction wave is the area of low relative pressure following a shock wave (see picture). Rarefaction waves expand with time (much like sea waves spread out as they reach a beach); in most cases rarefaction waves keep the same overall profile ('shape') at all times throughout the wave's movement: it is a self-similar expansion. Each part of the wave travels at the local speed of sound, in the local medium. This expansion behaviour contrasts with that of pressure increases, which gets narrower with time until they steepen into shock waves.

Physical examples A natural example of rarefaction occurs in the layers of Earth's atmosphere. Because the atmosphere has mass, most atmospheric matter is nearer to the Earth due to the Earth's gravitation. Therefore, air at higher layers of the atmosphere is less dense, or rarefied, relative to air at lower layers. Thus, rarefaction can refer either to a reduction in density over space at a single point of time, or a reduction of density over time for one particular area. Rarefaction can be easily observed by compressing a spring and releasing it.

In manufacturing Modern construction of guitars is an example of using rarefaction in manufacturing. By forcing the reduction of density (loss of oils and other impurities) in the cellular structure of the soundboard, a rarefied guitar top produces a tonal decompression affecting the sound of the instrument, mimicking aged wood.

In philosophy Distinctions between rarefaction and condensation, or between "the rare" and "the dense", were important in the early development of western philosophy, for example in the thinking of the ancient philosopher Anaximenes, and in Aristotle's Physics.

See also Longitudinal wave P-wave Prandtl–Meyer expansion fan Rarefied gas dynamics

References

Illustrations

Rarefaction: An example of rarefaction is also as a phase in a sound wave or phonon. Half of a sound wave is made up of the compression of the medium, and the other half is the decompression or rarefaction of the medium. — An example of rarefaction is also as a phase in a sound wave or phonon. Half of a sound wave is made up of the compression of the medium, and the other half is the decompression or rarefaction of the medium.

Worked examples

Example 1 — a first encounter with Rarefaction

Start with the simplest possible case. Write down what Rarefaction claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rarefaction before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rarefaction ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rarefaction

In research: Rarefaction appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rarefaction in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rarefaction is common in secondary-school and first-year university syllabi. It links to neighbouring topics Acoustics, Conservation equations, Sound, so understanding it makes those chapters shorter.
In everyday life: Look for Rarefaction outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Rarefaction in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rarefaction means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rarefaction out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rarefaction in simple terms?

Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves (sound waves, for instance), rarefaction waves also exist in nature.

Why does Rarefaction matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rarefaction?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rarefaction.