14.2 Periodic Waves

Periodic waves repeat at regular intervals, and you describe them using period, frequency, wavelength, amplitude, and wave speed. The two equations that do most of the work are $T = \frac{1}{f}$ and $\lambda = \frac{v}{f}$. These relationships help you connect wave graphs, equations, and physical descriptions throughout Unit 14.

Why This Matters for the AP Physics 2 Exam

Periodic waves give you the vocabulary and equations you need for the rest of Unit 14, including the Doppler effect, interference, standing waves, and diffraction. On the exam you will often analyze wave graphs and equations, so you need to read amplitude, wavelength, period, and frequency directly from representations. Creating and interpreting models such as graphs, equations, and verbal descriptions shows up across multiple-choice questions and in free-response work that asks you to translate between different representations of the same wave.

Key Takeaways

• Period $T$ is the time for one full cycle (seconds); frequency $f$ is cycles per second (Hz), and they are inverses: $T = \frac{1}{f}$.
• Wavelength $\lambda$ is the distance between matching points on a wave, like crest to crest, and connects to speed and frequency by $\lambda = \frac{v}{f}$.
• Amplitude is the maximum displacement from equilibrium and is independent of period and frequency.
• A wave's energy increases as its frequency increases.
• For sound, higher frequency means higher pitch.
• You can describe a sinusoidal wave as displacement versus time, $x(t) = A\cos(\omega t) = A\cos(2\pi f t)$, or displacement versus position, $y(x) = A\cos\left(2\pi \frac{x}{\lambda}\right)$.

Physical Properties of Periodic Waves

Period and Frequency

Periodic waves are oscillations that repeat at regular intervals as they travel through a medium or space. The time for one complete cycle to pass a fixed point is the period ($T$), measured in seconds.

• Period ($T$) is how long it takes for the wave pattern to repeat.
• Frequency ($f$) is how many complete cycles happen per second, measured in hertz (Hz).
• They are inverses: $f = \frac{1}{T} \quad \text{and} \quad T = \frac{1}{f}$

Amplitude is the maximum displacement from the equilibrium position. It relates to the wave's energy but stays independent of period and frequency.

• A larger amplitude carries more energy.
• Changing frequency does not automatically change amplitude. These are separate properties.
• A wave's energy increases with increasing frequency, and amplitude is a separate property that does not depend on period or frequency.

Frequency shows up in familiar ways:

• Sound waves: higher frequency means higher pitch (like a flute), and lower frequency means lower pitch (like a bass drum).
• Electromagnetic waves: higher-frequency light carries more energy than lower-frequency light.

Wavelength ($\lambda$) is the distance between successive corresponding positions on a wave, such as crest to crest, trough to trough, or any point on one cycle to the same point on the next cycle.

Sinusoidal Wave Equations

Sinusoidal waves follow a pattern you can write with trig functions. These equations let you predict the wave's displacement at a given time or position.

Displacement at a fixed location as a function of time:

$x(t) = A \cos(\omega t) = A \cos(2\pi ft)$

Where:

• $x(t)$ is the displacement at time $t$
• $A$ is the amplitude (maximum displacement)
• $\omega$ is the angular frequency in radians per second
• $f$ is the frequency in hertz

Displacement at a fixed time as a function of position:

$y(x) = A \cos\left(2\pi\frac{x}{\lambda}\right)$

Where:

• $y(x)$ is the displacement at position $x$
• $\lambda$ is the wavelength

These two forms describe the same wave in two ways: displacement versus time at one location, and displacement versus position at one instant.

Wavelength, Speed, and Frequency

Wavelength, speed, and frequency are tied together by one relationship:

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

Wavelength is proportional to wave speed and inversely proportional to frequency. That leads to a few useful conclusions:

• If frequency increases while speed stays constant, wavelength decreases.
• If speed increases while frequency stays constant, wavelength increases.
• If wavelength increases while speed stays constant, frequency decreases.

For example, sound waves in air travel at about 343 m/s:

• A 100 Hz tone has a wavelength of 3.43 meters.
• A 10,000 Hz tone has a wavelength of about 3.43 centimeters.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Identify which quantities you are given and which the question wants. Most periodic wave problems reduce to $T = \frac{1}{f}$ or $\lambda = \frac{v}{f}$.
• Watch your units. Frequency in Hz, period in seconds, speed in m/s, and wavelength in meters keep your answers consistent.
• Use functional dependence. If a question doubles the frequency at constant speed, predict that wavelength is cut in half without plugging in numbers.

Free Response

• Be ready to translate between representations: a graph of displacement versus time, a graph of displacement versus position, an equation, and a sentence description should all describe the same wave.
• From a displacement versus time graph, read the period off the time axis, then get frequency from $f = \frac{1}{T}$.
• From a displacement versus position graph, read the wavelength off the position axis, then combine with speed using $\lambda = \frac{v}{f}$.
• Read amplitude from the height of the curve in either graph. It does not change when frequency or period changes.

Common Trap

• A displacement versus time graph and a displacement versus position graph look almost identical, but one axis is time and the other is position. The repeat length is the period in one and the wavelength in the other. Always check the axis label first.

Practice Problem 1: Wave Period and Frequency

Solution

(a) First find the frequency from the number of cycles per second: $f=\frac{240\ \text{cycles}}{30\ \text{s}}=8\ \text{Hz}$ Then find the period using $T=\frac{1}{f}$: $T=\frac{1}{8\ \text{Hz}}=0.125\ \text{s}$

(b) The frequency is $f=8\ \text{Hz}$.

Therefore, the wave has a period of 0.125 s and a frequency of 8 Hz.

Practice Problem 2: Wave Speed Calculation

Solution

To find the wave speed, rearrange the wave relationship $\lambda = \frac{v}{f}$ into:

$v = f \times \lambda$

Where:

• $v$ is the wave speed (what we are solving for)
• $f$ is the frequency (50 Hz)
• $\lambda$ is the wavelength (0.4 meters)

Substituting the values: $v = 50\ \text{Hz} \times 0.4\ \text{m} = 20\ \text{m/s}$

Therefore, the wave travels through the medium at 20 meters per second.

Practice Problem 3: Wavelength Determination

Solution

To find the wavelength, use:

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

Where:

• $\lambda$ is the wavelength (what we are solving for)
• $v$ is the wave speed (343 m/s)
• $f$ is the frequency (440 Hz)

Substituting the values: $\lambda = \frac{343 \text{ m/s}}{440 \text{ Hz}}$ $\lambda = 0.78 \text{ m}$

Therefore, the wavelength of the 440 Hz sound wave is 0.78 meters, or about 78 centimeters.

Common Misconceptions

• Amplitude and frequency are not the same thing. A louder sound has a larger amplitude, but a higher-pitched sound has a higher frequency. You can change one without changing the other.
• Higher amplitude does not mean higher frequency. Energy increases with amplitude and also increases with frequency, but those are two separate ways energy depends on a wave.
• Wave speed is set by the medium, not by frequency. When you change frequency in a given medium, the wavelength changes to keep $\lambda = \frac{v}{f}$ true; the speed stays the same.
• Wavelength and period are not interchangeable. Wavelength is a distance (meters) and period is a time (seconds). They show up on different axes when you graph a wave.
• $T = \frac{1}{f}$ only works when frequency is in hertz and period is in seconds. Convert before plugging in.

Related AP Physics 2 Guides

• 14.3 Boundary Behavior of Waves and Polarization
• 14.6 Wave Interference and Standing Waves
• 14.7 Diffraction
• 14.1 Properties of Wave Pulses and Waves
• 14.4 Electromagnetic Waves
• 14.5 The Doppler Effect