Waves transfer energy from one place to another without permanently moving matter along with it. Understanding wave properties is the foundation for everything else in this unit, from how sound travels to how musical instruments work.

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Fundamental Wave Characteristics

A wave has five core properties you need to know:

Wavelength ( $\lambda$ ) is the distance between two consecutive identical points on a wave, such as crest to crest or trough to trough. It's measured in meters.
Amplitude is the maximum displacement of the wave from its rest (equilibrium) position. A taller wave has a larger amplitude and carries more energy.
Frequency ( $f$ ) is the number of complete wave cycles that pass a fixed point per second. It's measured in hertz (Hz), where 1 Hz = 1 cycle per second.
Period ( $T$ ) is the time it takes for one complete wave cycle to occur, measured in seconds.
Wave speed ( $v$ ) is how fast the wave moves through a medium, measured in meters per second (m/s).

Wave Structure Components

When you look at a diagram of a transverse wave, you'll see specific parts labeled:

Crest is the highest point above the equilibrium (rest) position.
Trough is the lowest point below the equilibrium position.
Nodes are points where the wave has zero displacement from rest. These show up especially in standing waves.
Antinodes are points of maximum displacement in a standing wave, the opposite of nodes.

The equilibrium position is the horizontal center line of the wave. Amplitude is always measured from this line to a crest (or trough), not from crest to trough. This is a common mistake on tests. If a question gives you the full height from crest to trough, you need to divide by two to get the amplitude.

Relationships Between Wave Properties

These properties connect through specific relationships:

Wavelength and frequency are inversely related when wave speed stays constant. If frequency goes up, wavelength gets shorter, and vice versa. Think of it this way: if more cycles pass per second, each cycle must be shorter in length.
Period and frequency are inverses of each other:

$T = \frac{1}{f}$

So a wave with a frequency of 5 Hz has a period of $\frac{1}{5} = 0.2$ seconds.

Wave speed depends on both wavelength and frequency:

$v = \lambda f$

For example, a wave with a wavelength of 2 m and a frequency of 10 Hz travels at $2 \times 10 = 20$ m/s.

Amplitude affects energy but not speed. A wave with higher amplitude carries more energy, but amplitude does not change wave speed, frequency, or wavelength. These are independent properties.

Fundamental Wave Characteristics, Waves | Boundless Physics

Types of Waves

Classification by Vibration Direction

The two main types are defined by how particles move relative to the wave's direction of travel.

Transverse waves oscillate perpendicular to the direction the wave travels. Picture shaking a rope side to side: your hand moves up and down, but the wave pulse travels horizontally along the rope. Water surface waves and all electromagnetic waves (like light) are transverse.

Longitudinal waves vibrate parallel to the direction the wave travels. Instead of crests and troughs, they have compressions (where particles are pushed close together) and rarefactions (where particles are spread apart). Sound waves in air are the most common example. If you push and pull a stretched Slinky along its length, that's a longitudinal wave.

One thing that trips people up: longitudinal waves still have wavelength, frequency, and amplitude. The wavelength is just measured from one compression to the next (or one rarefaction to the next), and the amplitude corresponds to how tightly the particles are packed during compression.

Classification by Medium Requirements

Mechanical waves need a physical medium (solid, liquid, or gas) to travel through. They transfer energy by causing particles in the medium to vibrate. Water waves, sound waves, and seismic waves are all mechanical. No medium, no wave.
Electromagnetic (EM) waves do not need a medium. They can travel through the vacuum of space because they consist of oscillating electric and magnetic fields. Light, radio waves, microwaves, X-rays, and gamma rays are all EM waves. They all travel at $3 \times 10^8$ m/s in a vacuum.

Fundamental Wave Characteristics, Waves and Wavelengths – Psychology

Wave Behavior at Boundaries

When waves encounter boundaries or obstacles, four things can happen:

Reflection is when a wave bounces off a surface. An echo bouncing off a canyon wall is a classic example.
Refraction is when a wave changes direction as it passes from one medium into another, caused by a change in wave speed. A straw looking "bent" in a glass of water is a visual example of refraction. The wave itself doesn't stop; it just changes speed and bends as a result.
Diffraction is when waves bend around obstacles or spread out after passing through narrow openings. You can hear someone talking around a corner because sound waves diffract. Diffraction is more noticeable when the opening or obstacle is close in size to the wavelength.
Interference happens when two or more waves overlap in the same space. Constructive interference occurs when waves add together (crests align with crests), producing a larger wave. Destructive interference occurs when they cancel out (crests align with troughs), producing a smaller wave or no wave at all.

Wave Mathematics

Wave Equation and Its Applications

The core equation for waves is:

$v = \lambda f$

where $v$ is wave speed (m/s), $\lambda$ is wavelength (m), and $f$ is frequency (Hz).

You can rearrange it to solve for any variable:

To find frequency: $f = \frac{v}{\lambda}$
To find wavelength: $\lambda = \frac{v}{f}$

Example: A sound wave in air travels at 340 m/s and has a frequency of 680 Hz. What is its wavelength?

Write the rearranged equation: $\lambda = \frac{v}{f}$
Plug in values: $\lambda = \frac{340}{680}$
Solve: $\lambda = 0.5$ m

This equation works for all wave types, whether mechanical or electromagnetic.

Frequency and Period Calculations

Frequency and period are reciprocals:

$f = \frac{1}{T}$ and $T = \frac{1}{f}$

Example: A wave completes 200 cycles in 10 seconds. What are its frequency and period?

Frequency: $f = \frac{200}{10} = 20$ Hz
Period: $T = \frac{1}{20} = 0.05$ seconds

That 0.05 seconds means each individual cycle takes just fifty milliseconds to complete.

Energy and Intensity Relationships

Wave energy is proportional to the square of the amplitude. That means if you double the amplitude, the energy increases by a factor of four ( $2^2 = 4$ ). If you triple the amplitude, energy increases by a factor of nine ( $3^2 = 9$ ). This is why a tsunami, with its enormous amplitude, carries so much more energy than a small ripple.

Intensity describes how much energy a wave delivers per unit area. As you move farther from a wave source, intensity decreases according to the inverse square law: doubling your distance from the source cuts the intensity to one-quarter. Tripling your distance cuts it to one-ninth. This is why a speaker sounds much quieter when you walk across a room away from it.

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