16.9 Waves

Wave Characteristics and Types

Waves transfer energy from one place to another without permanently displacing the matter they travel through. That single idea connects everything from the sound of a guitar string to light arriving from the sun. This section covers the core properties of waves, how to calculate wave velocity, the difference between transverse and longitudinal waves, and how waves interact with each other and their surroundings.

Characteristics of Waves

Every wave can be described by four key properties: amplitude, wavelength, frequency, and period.

Amplitude measures the maximum displacement of a wave from its equilibrium (rest) position. It's measured in distance units like meters. For sound waves, a larger amplitude means a louder sound. For light, a larger amplitude means a brighter beam.

Wavelength ($\lambda$) is the distance between two consecutive identical points on a wave, such as crest to crest or trough to trough. Wavelength is also measured in distance units. For visible light, wavelength determines color: red light has a wavelength around 700 nm, while violet light is around 400 nm.

Frequency ($f$) is the number of complete wave cycles that pass a fixed point each second. The unit is the Hertz (Hz), where 1 Hz = 1 cycle per second. Higher frequency means more cycles per second. For sound, higher frequency corresponds to a higher pitch. For electromagnetic waves, higher frequency means more energy per photon.

Period ($T$) is the time it takes for one complete wave cycle to pass. It's the reciprocal of frequency:

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

A wave with a frequency of 50 Hz has a period of $T = \frac{1}{50} = 0.02$ seconds. A longer period means slower oscillation.

Wave Velocity Calculation

Wave velocity ($v$) is the speed at which a wave moves through a medium. The fundamental relationship is:

$v = \lambda f$

To calculate wave velocity:

1. Identify the wavelength ($\lambda$) of the wave.
2. Identify the frequency ($f$) of the wave.
3. Multiply them together. The result is the wave speed in the appropriate units (typically m/s).

Example: A sound wave in air has a frequency of 440 Hz (the note A above middle C) and a wavelength of about 0.78 m. Its velocity is:

$v = (0.78 \text{ m})(440 \text{ Hz}) = 343 \text{ m/s}$

That matches the known speed of sound in air at room temperature (~20°C). Light in a vacuum travels at $v = 299{,}792{,}458$ m/s regardless of frequency.

You can also rearrange the equation to solve for wavelength or frequency:

• $\lambda = \frac{v}{f}$
• $f = \frac{v}{\lambda}$

Transverse vs. Longitudinal Waves

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

Transverse waves have particle motion perpendicular to the direction the wave travels. Think of shaking a rope side to side: the wave moves along the rope, but each piece of rope moves up and down. The high points are called crests and the low points are called troughs.

• Light and all electromagnetic waves
• Waves on a vibrating string (guitar, violin)
• Seismic S-waves (secondary waves) during earthquakes

Longitudinal waves have particle motion parallel to the direction the wave travels. Instead of crests and troughs, they have compressions (regions where particles are pushed close together) and rarefactions (regions where particles are spread apart). Picture pushing and pulling one end of a slinky along its length.

• Sound waves in air, liquids, and solids
• Seismic P-waves (primary waves) during earthquakes
• Pressure waves in fluids (like water hammer in pipes)

Some waves are a combination of both. Surface water waves are a good example: if you watch a floating cork as a wave passes, it moves in a roughly circular path, combining up-and-down (transverse) and back-and-forth (longitudinal) motion.

Wave Interactions

When waves encounter boundaries, other waves, or obstacles, several important behaviors can occur.

Interference happens when two or more waves overlap in the same medium. Their displacements add together at every point. If two crests line up, the result is a larger displacement called constructive interference. If a crest lines up with a trough of equal size, they cancel out, producing destructive interference.

Standing waves form when two waves of the same frequency travel in opposite directions through the same medium (for example, a wave on a guitar string reflecting back on itself). The result is a pattern with fixed nodes (points that never move) and antinodes (points of maximum displacement). The string appears to vibrate in place rather than carry a traveling wave.

Reflection occurs when a wave hits a boundary and bounces back. The law of reflection states that the angle of incidence equals the angle of reflection, measured from a line perpendicular to the boundary.

Refraction is the change in direction of a wave as it passes from one medium into another where its speed is different. A familiar example: light bending as it enters water, which is why a straw in a glass looks bent at the surface.

Diffraction is the bending of waves around obstacles or through openings. It's most noticeable when the size of the obstacle or opening is comparable to the wavelength. This is why you can hear someone talking around a corner (sound wavelengths are on the order of meters, similar to doorway sizes) but you can't see around the corner (visible light wavelengths are hundreds of nanometers, far smaller than the doorway).