13.2 Wave Properties: Speed, Amplitude, Frequency, and Period

Wave Properties

Waves transfer energy from one place to another without transferring matter. The properties covered here describe how a wave looks, how fast it moves, and how much energy it carries. These same properties apply whether you're dealing with sound waves, light waves, or waves on a string.

Understanding wave properties also explains everyday observations: why a bass guitar sounds lower than a violin, why light bends when it enters water, and how radio stations broadcast at specific frequencies. The math connecting these properties is straightforward, but you need to be comfortable moving between the equations fluidly.

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Properties of Waves

Amplitude is the maximum displacement of a wave from its equilibrium (rest) position, measured in meters. It tells you how much energy the wave carries. A wave with twice the amplitude carries four times the energy, since energy is proportional to the square of amplitude. On a diagram, amplitude is the vertical distance from the midline to a crest (or to a trough).

Frequency ($f$) is the number of complete wave cycles that pass a fixed point per second, measured in Hertz (Hz), where $1 \text{ Hz} = 1 \text{ cycle/second}$. Higher frequency means more cycles packed into each second. For sound, higher frequency means higher pitch. For electromagnetic waves, higher frequency means higher energy.

Period ($T$) is the time it takes for one complete wave cycle to pass, measured in seconds. Period and frequency are reciprocals of each other:

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

A wave with a frequency of 50 Hz has a period of $\frac{1}{50} = 0.02$ seconds. If you know one, you always know the other.

Wavelength ($\lambda$) is the distance between two consecutive points that are in phase, such as crest to crest or trough to trough, measured in meters. Wavelength and frequency are inversely related when wave speed is constant: as one goes up, the other goes down.

Wave speed ($v$) is how fast the wave propagates through a medium, measured in meters per second. The medium determines the speed. A sound wave in air travels at about 343 m/s, but in steel it travels at roughly 5,960 m/s. For a given medium under constant conditions, wave speed stays the same regardless of frequency or amplitude.

Relationships Between Wave Characteristics

The central equation connecting speed, frequency, and wavelength is:

$v = f\lambda$

This says that wave speed equals frequency times wavelength. Since the medium fixes the speed, increasing frequency forces wavelength to decrease, and vice versa. You can rearrange this equation two ways depending on what you're solving for:

• $\lambda = \frac{v}{f}$ (solve for wavelength)
• $f = \frac{v}{\lambda}$ (solve for frequency)

You can also express wavelength in terms of period and speed by substituting $f = \frac{1}{T}$:

$\lambda = vT$

This is useful when a problem gives you the period instead of the frequency. For a constant wave speed, a longer period means a longer wavelength.

The wave equation describes the displacement $y$ of any point on a sinusoidal wave as a function of position $x$ and time $t$:

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

• $A$ is the amplitude
• $k$ is the wave number, defined as $k = \frac{2\pi}{\lambda}$, with units of rad/m
• $\omega$ is the angular frequency, defined as $\omega = \frac{2\pi}{T} = 2\pi f$, with units of rad/s

The wave number tells you how many radians of phase fit per meter of space, while angular frequency tells you how many radians of phase pass per second. These connect the spatial and temporal parts of wave behavior into a single expression.

Wave Interactions

Superposition states that when two or more waves overlap in the same region, the resulting displacement at any point is the algebraic sum of the individual displacements. This is sometimes called the principle of superposition, and it applies to all linear waves.

Interference is what results from superposition:

• Constructive interference occurs when waves are in phase (crests align with crests). The amplitudes add, producing a larger combined wave.
• Destructive interference occurs when waves are out of phase (crests align with troughs). The amplitudes partially or fully cancel.

Standing waves form when two waves with the same frequency and amplitude travel in opposite directions through the same medium. Instead of traveling, the wave pattern appears to vibrate in place, with fixed points called nodes (zero displacement) and points of maximum displacement called antinodes. You'll see these on vibrating strings and in air columns.

Dispersion occurs when wave speed in a medium depends on frequency. Different frequency components of a complex wave then travel at different speeds, causing the wave to spread out. A prism separating white light into colors is a classic example of dispersion.

Applications of Wave Concepts

1. Determining wave properties in different media

Example: Find the wavelength of a 440 Hz sound wave (the note A4) traveling through air at 343 m/s.

1. Write the relevant equation: $\lambda = \frac{v}{f}$
2. Substitute: $\lambda = \frac{343 \text{ m/s}}{440 \text{ Hz}}$
3. Solve: $\lambda = 0.78 \text{ m}$

That same 440 Hz tone in water (where sound travels at about 1,480 m/s) would have a wavelength of $\frac{1480}{440} = 3.36 \text{ m}$. The frequency stays the same, but the wavelength stretches because the wave moves faster. In general, sound travels fastest in solids, then liquids, then gases, because of differences in how tightly molecules are coupled.

2. Analyzing wave behavior at boundaries

When a wave crosses from one medium into another, its frequency does not change. This is a critical point. The wave speed changes (determined by the new medium), so the wavelength must adjust to keep $v = f\lambda$ true.

• Moving into a slower medium: wavelength decreases
• Moving into a faster medium: wavelength increases

Example: A wave on a thick rope travels at 4 m/s with a wavelength of 0.5 m. It enters a thinner rope where the speed is 6 m/s. The frequency is $f = \frac{4}{0.5} = 8 \text{ Hz}$, which stays constant. The new wavelength is $\lambda = \frac{6}{8} = 0.75 \text{ m}$.

3. Applying wave properties to electromagnetic waves

All electromagnetic waves travel at the speed of light in a vacuum: $c = 3 \times 10^8 \text{ m/s}$. The same wave equation applies, just with $c$ replacing $v$:

$c = f\lambda$

Example: Find the frequency of a radio wave with a wavelength of 3 m.

1. Write the equation: $f = \frac{c}{\lambda}$
2. Substitute: $f = \frac{3 \times 10^8 \text{ m/s}}{3 \text{ m}}$
3. Solve: $f = 1 \times 10^8 \text{ Hz} = 100 \text{ MHz}$

This falls in the FM radio range. The electromagnetic spectrum spans from radio waves (long wavelength, low frequency) through microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays (short wavelength, high frequency). All travel at the same speed in vacuum; only their frequencies and wavelengths differ.