Standing Waves

The video lesson for this topic is coming soon.

Essential Question

How can a wave stand still?

Pluck a guitar string and it vibrates in a pattern that seems frozen in place, even though waves are supposed to travel. By the end of this lesson, you should be able to identify nodes and antinodes, name harmonics, and use λ = 2L/n to find wavelengths on a string.

What Is a Standing Wave?

A wave reflects off a fixed end
The two waves interfere
The pattern stays in place

A string fixed between two posts vibrates in three loops. The solid blue curve shows the string at one moment and the dashed blue curve shows it half a cycle later. Above the string, a blue arrow labeled incoming points right and a red arrow labeled reflected points left. Caption: the loops grow and shrink in place.

Send a wave down a string tied to a wall and it reflects back. The incoming wave and the reflected wave overlap and interfere. At just the right frequencies, they combine into a standing wave: a pattern of loops that grow and shrink in place instead of traveling down the string. The solid and dashed curves show the string half a cycle apart.

Nodes and Antinodes

Node: never moves
Antinode: moves the most
Nodes: destructive
Antinodes: constructive

A string fixed between two posts vibrating in three loops. Four black dots mark the nodes, labeled N, at both ends and at two points in between. The middle of each loop is labeled A for antinode. Caption: N is a node, A is an antinode.

A node is a point on a standing wave that never moves. At a node, the incoming and reflected waves always cancel (destructive interference). An antinode is a point halfway between nodes that swings up and down the farthest. At an antinode, the two waves always add (constructive interference). The fixed ends of a string are always nodes.

One Loop = Half a Wavelength

Each loop is half a wavelength
Node to node: λ/2
Two loops make one λ

A string fixed between two posts vibrating in two loops. A double arrow above the first loop is labeled lambda over 2. A double arrow below both loops is labeled lambda. Caption: one loop is half a wavelength.

One full wavelength of a wave has a crest and a trough. A single loop of a standing wave is only the crest part (or only the trough part), so each loop is half a wavelength. The distance from one node to the next node is λ/2, and two loops together make one full wavelength.

Harmonics

Only certain patterns fit
Each pattern is a harmonic
n = number of loops

Three strings of the same length, stacked. The top string vibrates in one loop, labeled 1st harmonic. The middle string vibrates in two loops, labeled 2nd. The bottom string vibrates in three loops, labeled 3rd.

A string fixed at both ends must have a node at each end, so only patterns with a whole number of loops can form. Each allowed pattern is called a harmonic. The harmonic number n equals the number of loops: the 1st harmonic (also called the fundamental) has 1 loop, the 2nd harmonic has 2 loops, and so on.

Wavelength from String Length

λ = 2L / n

λ = wavelength, in meters (m)
L = length of the string, in meters (m)
n = harmonic number (the number of loops)

The equation lambda = 2L / n in a yellow box. Below it, three cards: lambda is wavelength in m, L is string length in m, and n is the harmonic number, the number of loops.

Each loop is half a wavelength, and n loops fill the whole string. So L = n(λ/2). Solving for wavelength gives λ = 2L/n. For the 1st harmonic, the wavelength is twice the length of the string.

Resonance

Each harmonic: its own frequency
Match it: huge waves
Like pushing a swing

A graph of the string's amplitude versus driving frequency. The amplitude stays small except for three tall, narrow peaks at f1, 2 f1, and 3 f1.

Each harmonic happens at its own frequency, called a natural frequency. If f₁ is the frequency of the 1st harmonic, the others are 2f₁, 3f₁, and so on. When you shake the string at exactly one of these frequencies, each push adds to the wave already there and the amplitude grows huge. This is resonance. It’s the same reason a swing goes higher when you push it in rhythm. Shake at any other frequency and the string barely moves.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Counting Nodes

A string fixed at both ends vibrates in its 3rd harmonic. How many nodes and how many antinodes does it have?

Loops: 3rd harmonic means n = 3, so there are 3 loops
Antinodes: one antinode in the middle of each loop: 3 antinodes
Nodes: a node at each end plus one between each pair of loops: 2 + 2
Answer: 4 nodes and 3 antinodes. Nodes always outnumber antinodes by one on a string fixed at both ends.

Example 2: Finding Wavelength

A 1.20 m string vibrates with 3 loops. What is the wavelength of the waves on the string?

A string fixed between two posts vibrating in three loops, with a double arrow under the whole string labeled 1.20 m.

Knowns: L = 1.20 m, n = 3 (3 loops)
Equation: λ = 2L/n
Substitute: λ = 2(1.20 m) / 3 = 2.40 m / 3
Answer: λ = 0.800 m

Check: each loop is 1.20 m ÷ 3 = 0.400 m, and two loops make one wavelength, 0.800 m.

Example 3: Guitar String

A 0.650 m guitar string vibrates in its 1st harmonic at 196 Hz. What is the speed of the waves on the string?

A string fixed between two posts vibrating in one loop, with a double arrow under it labeled 0.650 m and a label f = 196 Hz.

Knowns: L = 0.650 m, n = 1, f = 196 Hz
Wavelength: λ = 2L/n = 2(0.650 m)/1 = 1.30 m
Wave equation: v = fλ = (196 Hz)(1.30 m) = 254.8 m/s
Answer: v = 255 m/s (three significant figures)

Example 4: Which Harmonic?

A 2.0 m rope fixed at both ends has a standing wave with a wavelength of 0.80 m. Which harmonic is it?

Knowns: L = 2.0 m, λ = 0.80 m
Rearrange: λ = 2L/n, so n = 2L/λ
Substitute: n = 2(2.0 m) / 0.80 m = 4.0 m / 0.80 m. The meters cancel.
Answer: n = 5, so it is the 5th harmonic (5 loops)

Big Ideas

Standing wave: a wave + its reflection
Nodes never move; antinodes move most
Each loop is half a wavelength
λ = 2L/n, where n = number of loops

Back to the essential question: a wave can look like it is standing still when it interferes with its own reflection. Some points, the nodes, always cancel, and others, the antinodes, swing the most, so the pattern stays in place. Each harmonic fits a whole number of half wavelengths on the string, so λ = 2L/n. Next up, we’ll meet a whole family of waves that can travel through empty space in The Electromagnetic Spectrum.