Unit 14 · Topic 14.2

14.2 Periodic Waves

A periodic wave repeats itself, so you can describe it with a period T, a frequency $f = \frac{1}{T}$ and a wavelength λ. These link to the speed by v = fλ. Two graphs describe a wave: displacement against position is a snapshot of the whole wave at one instant, and displacement against time follows one point of the medium.

Key terms

period
frequency
wavelength
wave speed
pitch
sinusoidal wave

Describing a periodic wave

Period T: the time for one complete cycle, in seconds.
Frequency f: the number of cycles per second, in hertz (Hz). It's the reciprocal of the period, $T = \frac{1}{f}$ .
Wavelength λ: the distance from one point on the wave to the next matching point, such as crest to crest or trough to trough, in meters.
Amplitude A: the maximum displacement from equilibrium. It has nothing to do with the period or frequency; you can have a big slow wave or a small fast one.

The wave equation v = fλ

In one period, a wave moves forward exactly one wavelength. So its speed is $v = \frac{\lambda}{T} = f\lambda$ . The equation sheet writes it as $\lambda = \frac{v}{f}$ .

In a given medium, the speed is fixed, so frequency and wavelength trade off: double the frequency and the wavelength halves. When a wave moves into a new medium, the frequency stays the same (it's set by the source) and the wavelength changes with the speed.

For sound, frequency is heard as pitch: a higher frequency sounds higher. Humans hear roughly 20 Hz to 20,000 Hz. Two notes an octave apart have frequencies in a 2 : 1 ratio, which is part of why they sound so alike.

For waves of the same amplitude in the same medium, a higher-frequency wave carries more energy, because each bit of the medium is moved back and forth more quickly.

Two kinds of graph

A displacement-versus-position graph is like a photograph of a rope at one instant. The distance between neighboring crests is the wavelength. The equation sheet describes this shape as $y(x) = A\cos\left(2\pi \frac{x}{\lambda}\right)$ .

A displacement-versus-time graph tracks one single point of the medium as it bobs up and down. The time between neighboring peaks is the period. The sheet describes this as $x(t) = A\cos(2\pi f t)$ .

The two graphs can look identical, so always read the horizontal axis label before you measure anything. From the pair together you can get both λ and T, and then $v = \frac{\lambda}{T}$ .

Sinusoidal waves

A wave made by a source moving in simple harmonic motion has a smooth sine or cosine shape, called sinusoidal. The cosine forms on the sheet describe a wave whose displacement is at its maximum, A, when x = 0 or t = 0.

You won't need to manipulate these equations heavily. Mostly you'll read A, λ, T or f from them or from a graph, then use v = fλ.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Wavelength of a musical note
A tuning fork vibrates at 440 Hz in air, where sound travels at 343 m/s. Find the wavelength and the period of the sound.
Show the solutionHide the solution
1. Step 1: $\lambda = \dfrac{v}{f} = \dfrac{343\text{ m/s}}{440\text{ Hz}} = 0.780$ m.
2. Step 2: $T = \dfrac{1}{f} = \dfrac{1}{440} = 2.27 \times 10^{-3}$ s, or 2.27 ms.
Answer: λ ≈ 0.780 m; T ≈ 2.27 ms.
Example 2Calculator allowed
Reading two graphs
A displacement-versus-position graph of a wave on a rope shows crests every 0.60 m and a maximum displacement of 3.0 cm. A displacement-versus-time graph for one point on the rope shows peaks every 0.020 s. Find the amplitude, wavelength, frequency and speed.
Show the solutionHide the solution
1. Step 1: Amplitude: the maximum displacement, 3.0 cm, can be read from either graph.
2. Step 2: Wavelength: crest-to-crest distance on the position graph, 0.60 m.
3. Step 3: Period: peak-to-peak time on the time graph, 0.020 s, so $f = \dfrac{1}{0.020\text{ s}} = 50$ Hz.
4. Step 4: Speed: v = fλ = (50 Hz)(0.60 m) = 30 m/s.
Answer: A = 3.0 cm, λ = 0.60 m, f = 50 Hz, v = 30 m/s.
Example 3Calculator allowed
Changing frequency in the same medium (trap)
A speaker plays a 500 Hz tone with a wavelength of 0.686 m. The frequency is raised to 1000 Hz. What happens to the speed and wavelength of the sound in the same room?
Show the solutionHide the solution
1. Step 1: The speed of sound depends on the air, not the source, so it stays the same: v = (500 Hz)(0.686 m) = 343 m/s.
2. Step 2: With v fixed, $\lambda = \dfrac{v}{f} = \dfrac{343}{1000} = 0.343$ m. Doubling f halves λ.
3. Step 3: The trap is assuming that a higher frequency travels faster. It doesn't: the high and low notes from a band reach you at the same time.
Answer: Speed stays 343 m/s; the wavelength halves to 0.343 m.

Common mistakes

Reading a wavelength off a displacement-versus-time graph. The spacing on a time graph is the period; check the axis label first.
Thinking amplitude affects the frequency or speed. Amplitude is independent of both.
Assuming frequency changes when a wave enters a new medium. The frequency stays the same; the speed and wavelength change.
Measuring wavelength from a crest to the next trough. That's only half a wavelength.

On the exam

Graph questions are common: given one or two graphs of a wave, find A, λ, T, f or v, or sketch the graph for a point or a moment the question describes.
Expect to predict a new wavelength or frequency when something changes. State what stays fixed (the speed in a medium, or the frequency across a boundary) and reason from v = fλ.

Connected topics

Videos

Understanding Longitudinal and Transverse Waves, Wavelength, and Period using Graphs
Flipping PhysicsWatch on YouTube (opens in a new tab)
Amplitude, period, frequency and wavelength of periodic waves | Physics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
Wave Period and Frequency
Bozeman ScienceWatch on YouTube (opens in a new tab)
Wave Speed Equation Derivation and Demonstration
Flipping PhysicsWatch on YouTube (opens in a new tab)
Period, Frequency, Amplitude, & Wavelength - Waves
The Organic Chemistry TutorWatch on YouTube (opens in a new tab)
High School Physics - The Wave Equation
Dan Fullerton (APlusPhysics)Watch on YouTube (opens in a new tab)

Check yourself

4 questions on 14.2 Periodic Waves. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A tuning fork vibrates at 440 Hz in air, where sound travels at 343 m/s. What is the wavelength of the sound?

Time (s)	Displacement of point P (cm)
0	0
0.010	2.0
0.020	0
0.030	−2.0
0.040	0
0.050	2.0

Experimental data. A sinusoidal wave travels along a long rope. A motion sensor records the transverse displacement of one point P on the rope at several times.

Question 2 of 4Calculator allowed

What is the frequency of the wave?

Question 3 of 4Calculator allowed

The wave travels along the rope at 12 m/s. What is its wavelength?

Question 4 of 4Calculator allowed

What information can be found from this table alone, without knowing the wave speed?

0 of 4 answered