# Waves

Mathematics I · Physics · https://tryals.app/learn/mathematics-i/waves

## Traveling Disturbances

A **wave** is a disturbance that travels through space, carrying **energy** without transporting matter. A cork bobs on water without drifting forward.

Key quantities include **wavelength** $\lambda$ (crest-to-crest distance), **frequency** $f$ (oscillations per second in Hz), and **period** $T = 1/f$. The fundamental **wave relation** connects them:
$$v = f\lambda$$

| Type | Oscillation Direction | Example |
| --- | --- | --- |
| **Transverse** | Perpendicular to travel | Light, string |
| **Longitudinal** | Along travel direction | Sound |

**Common Pitfall:** Never confuse **frequency** with **period**. They are **reciprocals** ($f = 1/T$): high frequency means a **short** period.

## Wave Behaviors

The signature behavior distinguishing waves from particles is **superposition and interference**: when two waves overlap, their displacements add.

| Interference Type | Alignment | Result |
| --- | --- | --- |
| **Constructive** | Crests align | Larger amplitude |
| **Destructive** | Crest meets trough | Cancellation |

Other hallmarks include **diffraction** (bending around obstacles) and the **Doppler effect** (frequency shift from relative motion).

**Common Pitfall:** Thinking a wave carries **matter** along with it. The medium oscillates locally while only energy moves onward.

## Practice questions

6 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A wave carries both energy and matter across the room.

**Answer:** False

**Why:** False. The medium oscillates about a fixed position while the disturbance propagates, the cork bobs and stays put. Only energy and momentum move across. That is why sound crosses a room without a draught, and why wave speed is set by the medium rather than by how hard the source was struck.

Page: https://tryals.app/practice/mathematics-i/waves/a-wave-carries-both-energy-and-matter-across-the-room

### 2. Sort each wave by the direction in which the medium oscillates relative to the direction of travel.

**Answer:**

- Transverse: Light, A wave on a plucked guitar string, A ripple travelling across a water surface
- Longitudinal: Sound in air, A compression pulse sent along a slinky pushed end-on

**Why:** Light, string waves and surface ripples oscillate perpendicular to their travel; sound and a slinky compression oscillate along it. The distinction is physical, not cosmetic: only transverse waves can be polarised, which is one way we know light is transverse.

Page: https://tryals.app/practice/mathematics-i/waves/sort-each-wave-by-the-direction-in-which-the-medium-oscillates

### 3. The relation v = fλ says a wave's speed is frequency times wavelength. When a sound passes from air into water its speed rises sharply. Which of the other two changes, and why?

A. The wavelength, since frequency is set by the source.
B. Neither changes; the relation v = f lambda only holds within a single medium.
C. The frequency, since the new medium supports faster oscillation.
D. Both change by the same factor, keeping their product proportional to the new speed.

**Answer:** A. The wavelength, since frequency is set by the source.

**Why:** At the boundary the air molecules push the water molecules at exactly the rate they are oscillating, they cannot push at any other, so frequency carries across unchanged and λ must grow with v. Sound is about four times faster in water, so its wavelength is four times longer. The same reasoning explains refraction, and why colour does not change underwater.

Page: https://tryals.app/practice/mathematics-i/waves/the-relation-v-f-says-a-waves-speed-is-frequency-times

### 4. An ambulance siren drops in pitch as it passes. Nothing about the siren changed. What did?

A. The amplitude, which the ear interprets as a change in pitch
B. The siren's output frequency, which the driver adjusts
C. The rate at which wavefronts reach you, bunched or stretched
D. The wave's speed through the air, which depends on the source's motion

**Answer:** C. The rate at which wavefronts reach you, bunched or stretched

**Why:** The source chases its own wavefronts, bunching them ahead and stretching them behind, so the received frequency differs from the emitted one while the wave speed is unchanged. The same effect red-shifts distant galaxies, which is how we know the universe is expanding.

Page: https://tryals.app/practice/mathematics-i/waves/an-ambulance-siren-drops-in-pitch-as-it-passes-nothing-about-the

### 5. Complete the statement about what a wave carries.

**Answer:** As water waves pass beneath it, a floating cork **bobs up and down roughly in place**, because a wave transports energy rather than matter.

**Why:** The medium oscillates locally while the disturbance and its energy move on. This is precisely what separates a wave from a stream of particles: sound carries the energy of a shout across a room without carrying any air across it.

Page: https://tryals.app/practice/mathematics-i/waves/complete-the-statement-about-what-a-wave-carries

### 6. A high-frequency wave has a SHORT period, not a long one. Why must the two run opposite ways?

A. Because period depends on wavelength rather than on frequency
B. Because high-frequency waves travel faster and so finish sooner
C. Because the relationship holds only for waves in a vacuum
D. Because frequency is cycles per second and period seconds per cycle

**Answer:** D. Because frequency is cycles per second and period seconds per cycle

**Why:** The units settle it: $T = 1/f$. Many cycles per second means little time for each, and the reciprocal relation is forced by what the two quantities count.

Page: https://tryals.app/practice/mathematics-i/waves/a-high-frequency-wave-has-a-short-period-not-a-long-one-why-must