1.4 Wave Motion and Types of Waves

Wave motion and its characteristics

Wave motion is the mechanism by which energy travels through a medium (or through empty space) without permanently displacing the matter it passes through. Understanding how waves propagate and what quantities describe them is foundational for everything else in this unit, from standing waves to interference patterns.

Fundamental concepts of wave motion

When a wave moves through a medium, individual particles don't travel with the wave. Instead, they oscillate around their equilibrium positions, passing energy to neighboring particles. The energy propagates in the direction of wave travel, but the matter stays put.

Think of a crowd doing "the wave" at a stadium: each person stands and sits in place, yet the disturbance travels around the entire arena. The same principle applies to water waves, sound waves, and waves on a string.

Key characteristics of waves

Every wave can be described by a handful of measurable quantities:

• Amplitude (A): The maximum displacement of a particle from its equilibrium position. Amplitude is directly tied to energy; doubling the amplitude quadruples the energy carried by the wave (since energy is proportional to $A^2$).
• Wavelength ($\lambda$): The distance between two consecutive identical points on the wave (e.g., crest to crest or compression to compression). Measured in meters.
• Frequency ($f$): The number of complete oscillation cycles per second, measured in Hertz (Hz). A higher frequency means the wave oscillates more rapidly.
• Period ($T$): The time it takes for one full cycle to complete. Related to frequency by $T = \frac{1}{f}$, measured in seconds.
• Wave speed ($v$): How fast the wave pattern moves through the medium. Determined by the medium's properties, not by the wave's amplitude or frequency alone.

Mathematical relationships and wave equations

The core speed relation ties wavelength, frequency, and speed together:

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

You can rearrange this to find any one quantity if you know the other two. For example, the frequency-wavelength relationship is:

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

The classical wave equation governs how the displacement $y(x,t)$ evolves in space and time:

$\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}$

Any function of the form $y = f(x \pm vt)$ satisfies this equation, meaning disturbances that maintain their shape while traveling at speed $v$ are valid solutions.

Types of waves: Transverse vs Longitudinal

The distinction between transverse and longitudinal waves comes down to the direction particles oscillate relative to the direction the wave travels.

Transverse waves

In a transverse wave, particles oscillate perpendicular to the direction of propagation. Picture shaking a rope side to side: the wave travels horizontally along the rope, but each segment moves up and down.

• Creates crests (highest points) and troughs (lowest points).
• Examples: waves on a string, electromagnetic waves, S-waves in seismology.
• Can be polarized, meaning the oscillation can be restricted to a single plane.

A transverse wave on a string is commonly written as:

$y(x,t) = A \sin(kx - \omega t)$

where $A$ is amplitude, $k = \frac{2\pi}{\lambda}$ is the wave number (spatial frequency), and $\omega = 2\pi f$ is the angular frequency.

Longitudinal waves

In a longitudinal wave, particles oscillate parallel to the direction of propagation. A classic example is sound: air molecules are pushed together (compressions) and pulled apart (rarefactions) along the same axis the sound travels.

• Creates alternating regions of compression (high density) and rarefaction (low density).
• Examples: sound waves in air, compression waves in a slinky, P-waves in seismology.
• Cannot be polarized, since the oscillation direction already matches the propagation direction.

The displacement form looks identical in structure to the transverse case:

$s(x,t) = s_0 \sin(kx - \omega t)$

Here $s_0$ is the maximum displacement of a particle from equilibrium along the propagation axis.

Comparison and unique properties

Both types exhibit reflection, refraction, diffraction, and interference. Some waves combine both motions: surface waves (like ocean waves or seismic Rayleigh waves) cause particles to move in elliptical paths that have both transverse and longitudinal components.

Properties of waves: Mechanical vs Electromagnetic

Waves can also be classified by whether they need a medium to travel through.

Mechanical waves

Mechanical waves require a physical medium (solid, liquid, or gas) to propagate. The wave energy transfers through particle-to-particle interactions governed by Newton's laws.

• Speed depends on the medium's density and elasticity. For example, sound travels faster in steel (~5,960 m/s) than in air (~343 m/s) because steel is far more elastic.
• Cannot travel through a vacuum.
• Lose energy over distance due to internal friction in the medium.
• Examples: sound waves, water waves, seismic waves, waves on a string.

Electromagnetic waves

Electromagnetic (EM) waves consist of oscillating electric and magnetic fields that regenerate each other as they propagate. They don't need a medium at all.

• Travel at the speed of light in vacuum: $c \approx 3 \times 10^8 \text{ m/s}$
• In a material medium, they slow down by a factor of the refractive index $n$: $v = \frac{c}{n}$
• Described by Maxwell's equations.
• Carry both energy and momentum (which is why light can exert radiation pressure).
• The EM spectrum spans from radio waves (long $\lambda$, low $f$) to gamma rays (short $\lambda$, high $f$).

Common wave properties

Despite their different natures, both mechanical and EM waves share these behaviors:

• Reflection at boundaries
• Refraction when entering a new medium
• Diffraction around obstacles or through openings
• Interference when multiple waves overlap (superposition)
• Doppler effect when the source or observer is in motion
• Ability to form standing waves under the right boundary conditions

Wave propagation in different media

How a wave behaves depends heavily on the medium it's traveling through and what happens at the boundaries between media.

Factors affecting wave propagation

Two medium properties dominate wave speed:

• Elasticity: More elastic media restore displaced particles faster, so waves travel faster. This is why sound is much faster in solids than in gases.
• Density: In many cases, higher density slows waves down, though the relationship depends on the wave type. For a string, $v = \sqrt{T/\mu}$ where $T$ is tension and $\mu$ is linear mass density.
• Temperature also matters. Sound in air speeds up with temperature because warmer air molecules move faster and transmit disturbances more quickly.

At boundaries between media, three things can happen: reflection (wave bounces back), transmission (wave continues into the new medium), and refraction (transmitted wave bends due to a speed change).

Snell's law describes refraction quantitatively:

$\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$

where $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, $v_1$ and $v_2$ are the wave speeds in each medium, and $n_1$, $n_2$ are the refractive indices.

Wave interactions and phenomena

The superposition principle says that when two or more waves overlap, the net displacement at any point is the algebraic sum of the individual displacements. This leads to:

• Constructive interference: Waves in phase add up, producing a larger amplitude.
• Destructive interference: Waves out of phase cancel, reducing amplitude.

Standing waves form when two identical waves travel in opposite directions (typically a wave and its reflection). The result is a pattern of fixed nodes (zero displacement) and antinodes (maximum displacement). You'll see these in vibrating strings and resonant tubes.

Beats occur when two waves of slightly different frequencies overlap. The resulting amplitude oscillates at the beat frequency:

$f_{\text{beat}} = |f_1 - f_2|$

Dispersion is when different frequency components of a wave travel at different speeds in a medium. A prism separating white light into colors is a familiar example. In dispersive media, the group velocity $v_g$ (speed of the wave packet's envelope) differs from the phase velocity $v_p$ (speed of individual crests).

Wave attenuation and energy transfer

As a wave propagates, its amplitude typically decreases due to energy dissipation in the medium. This attenuation often follows an exponential decay:

$A(x) = A_0 e^{-\alpha x}$

where $A_0$ is the initial amplitude, $\alpha$ is the attenuation coefficient (units of $\text{m}^{-1}$), and $x$ is the distance traveled.

For energy and intensity:

• Energy density is proportional to $A^2$. This is why amplitude matters so much for wave energy.
• Intensity (power per unit area) for a point source spreading in three dimensions falls off with the inverse square law:

$I \propto \frac{1}{r^2}$

This isn't because energy is lost; it's because the same energy spreads over a larger spherical surface as $r$ increases. Attenuation and geometric spreading are separate effects, and in real situations both contribute to a wave weakening with distance.