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E3.4 · Explain properties and formation of standing waves

Learn to explain properties and formation of standing waves through clear examples and targeted practice.

Ontario Grade 11 Physics

Waves and Sound

How overlapping waves form fixed patterns on a string

A wave travels, but the material through which it moves usually does not travel along with it. On a string, for example, each small part moves while a disturbance passes. When two waves travelling in opposite directions overlap, their combined pattern can appear to stay in place. This is a standing wave. Its points of greatest and least motion stay at fixed positions. We will use a string to explain how the pattern forms and what its parts do.

What you will learn

Explain how two waves travelling in opposite directions can form a standing wave.
Describe nodes, antinodes, and the motion of the medium in a standing wave.
Explain why a string fixed at both ends can support only patterns with nodes at both ends.

1. Prerequisite bridge: motion, waves, and signs

A wave is a travelling disturbance. A transverse wave is one in which the medium moves at right angles to the direction the wave travels. For a wave on a horizontal string, the disturbance travels along the string while each part of the string moves up and down.

The system in this lesson is the string and the waves on it. Set the horizontal coordinate along the string, from the left end toward the right end. Choose upward displacement as positive and downward displacement as negative. Displacement describes how far and in which direction a point is from its resting position. It is a vector quantity because it includes direction. Amplitude is the greatest size of that displacement; it is a scalar and has no direction.

Waves can overlap. At any one point, the combined displacement is the sum of the individual displacements, including their signs. If one wave moves a point upward and another moves it downward by the same amount, their displacements cancel at that instant. If both displacements point the same way, they add.

A travelling wave has a pattern that moves along the medium. A standing wave has a pattern whose important positions remain in place. The medium still moves at many points, but the pattern does not travel along the string.

y_{\mathrm{total}}=y_1+y_2

A string’s parts move up and down while a transverse wave travels along it.
Displacement has direction; amplitude is a non-negative size.
Overlapping displacements combine by addition.

2. How a standing wave forms

Imagine a wave travelling along a stretched string. When it reaches a fixed end, it reflects and travels back along the string. The original wave and the reflected wave now travel in opposite directions. When waves with matching patterns repeatedly overlap, they can form a standing wave.

The two waves do not stop travelling. Instead, their overlap creates a pattern that stays in the same places. At some positions, the waves’ displacements cancel throughout the motion. At other positions, their displacements combine to produce the largest motion. Between these positions, points move up and down by different amounts.

A node is a point that always has zero displacement in the standing-wave pattern. An antinode is a point that reaches the greatest displacement. Nodes and antinodes are fixed positions; they do not travel along the string. Points between a node and an antinode move, but by less than the antinode.

A standing wave is not a row of string sections moving together in one direction. On opposite sides of a node, sections move in opposite directions at the same stage of the pattern. All points reach zero displacement together at certain instants, then move away from the resting line.

Reflection can make waves travel in opposite directions on the same string.
Repeated overlap creates fixed nodes and antinodes.
The string moves, even though the standing-wave pattern stays in place.

3. Reading the pattern

Picture the string’s resting position as a horizontal line. Mark two nodes on that line, with an antinode halfway between them. At one instant, the string may curve upward at the antinode. Later, it may curve downward there. The nodes remain on the resting line in both snapshots.

In a standing wave, neighbouring nodes are separated by half of a wavelength. A wavelength is the distance between matching points on a travelling wave’s repeating pattern. The distance from a node to the nearest antinode is one-quarter of a wavelength. These distances describe the spacing of the pattern, not the distance the string itself travels.

The amplitude of a standing wave is not the same at every position. It is zero at a node and greatest at an antinode. Thus, when describing the motion, identify where the point is on the string. A point at a node does not move; a point at an antinode moves the most.

A fixed end cannot move away from its support, so it is a node. If both ends of a string are fixed, both ends must be nodes. A pattern on that string must fit nodes at both ends. This is why some patterns fit the string and persist, while patterns that do not meet the end condition do not form the same stable standing-wave pattern.

d_{node\ to\ node}=

\frac{\lambda}{2}

Nodes have zero displacement; antinodes have the greatest displacement.
Neighbouring nodes are half a wavelength apart.
A fixed end is a node, so both ends of a fixed string are nodes.

4. Use the ideas to explain a pattern

When explaining a standing-wave diagram, begin by identifying the medium and its resting position. Then mark the nodes, including any fixed ends. Find the antinodes between them. Describe the motion using the sign convention: upward displacement is positive and downward displacement is negative.

Next, explain how the pattern could form. A wave reflects from an end and travels back. It overlaps with the wave travelling toward the end. The displacements add at each point. Repeated cancellation at some positions and repeated addition at others produce nodes and antinodes.

Finally, check whether the pattern fits the physical supports. A string fixed at both ends must have a node at each end. This check is about the shape and constraints of the system, not a calculation of speed or frequency.

A diagram can show two moments without implying that the wave pattern moves along the string. The nodes must line up at the same positions in both moments. An antinode can be above the resting line in one moment and below it later.

Use fixed positions to identify nodes and antinodes.
Explain formation through reflection and overlapping displacements.
Check that a fixed-end pattern has nodes at both supports.

Two snapshots of one standing-wave pattern

Position	Left fixed end	Between the ends	Right fixed end
Instant A	Node: zero displacement	Antinode: positive, greatest displacement	Node: zero displacement
Instant B	Node: zero displacement	Antinode: negative, greatest displacement	Node: zero displacement

Worked example

Explaining a fixed string’s pattern

A stretched string is fixed at its left and right ends. A standing-wave drawing shows both ends still, with one point between them moving the most. Explain the features and how the pattern forms.

Identify the system and direction
The system is the string and the waves on it. Take the positive horizontal direction from the left support to the right support, and take upward displacement as positive. The string’s fixed supports cannot move.
Name the marked positions
The two fixed ends are nodes because their displacement stays zero. The point between them that moves the most is an antinode. It moves upward and downward as the pattern changes.
Explain the formation
A wave reflects from an end and travels back along the string. It overlaps with the wave travelling the other way. Their displacements combine, repeatedly cancelling at the nodes and producing the greatest motion at the antinode.

Answer: The fixed ends are nodes, the moving point with the greatest motion is an antinode, and the pattern forms through the overlap of waves travelling in opposite directions.

Check: The explanation fits the supports: neither end moves. It also distinguishes the stationary pattern from the moving string.

Worked example

Interpreting a point between a node and an antinode

A point on a standing-wave string lies between a node and an antinode. Describe how it moves and compare its motion with the motion at those two special positions.

Set the sign convention
Use upward displacement as positive and downward displacement as negative. The system is the vibrating string; the point’s motion is measured relative to the string’s resting position.
Compare the positions
The point between the node and antinode moves up and down. Its greatest displacement is less than the antinode’s. Unlike the node, it does not stay at zero displacement throughout the pattern.
Describe what remains fixed
The node and antinode remain at fixed positions along the string. The point between them moves vertically, so its location along the string does not travel with the standing-wave pattern.

Answer: The point moves up and down with an intermediate amount of motion. It moves more than a node, which remains still, and less than an antinode, which moves the most.

Check: The comparison is based on displacement from the resting position. It does not confuse the point’s up-and-down motion with motion of the pattern along the string.

Worked example

Checking a proposed string pattern

A drawing of a string fixed at both ends shows one end as a node and the other end moving as an antinode. Can this be the standing-wave pattern of that string? Explain.

Identify the physical constraint
The system is a string attached to supports at both ends. Choose upward as positive displacement. A fixed support prevents the string end from moving up or down.
Apply the node definition
A point that always has zero displacement is a node. Therefore, each fixed end must be a node, not an antinode.
Judge the drawing
The proposed pattern makes one fixed end an antinode. That would require the end to move, which conflicts with the support. The drawing cannot represent a standing-wave pattern on this fixed-end string.

Answer: No. Both fixed ends must be nodes, so a pattern with an antinode at one fixed end is not possible for this string.

Check: The conclusion follows directly from the physical constraint that the attached ends cannot move.

Common mistakes and how to avoid them

Thinking that nodes travel along the string.

Correction: The waves travel in opposite directions, but the nodes and antinodes of the standing pattern stay at fixed positions.

Calling an antinode a point that remains still.

Correction: A node remains at zero displacement. An antinode reaches the greatest displacement.

Assuming the string itself travels along with the standing pattern.

Correction: The string’s parts move up and down. The pattern of nodes and antinodes stays in place.

Treating a fixed end as an antinode.

Correction: A fixed end cannot move, so it is a node.

Lesson summary

Standing waves form when waves travelling in opposite directions overlap and their displacements combine.
Nodes remain at zero displacement. Antinodes reach the greatest displacement.
The standing-wave pattern stays in place even though parts of the medium move.
A fixed end is a node. A string fixed at both ends must have nodes at both ends.

Check your understanding

Question 1

Which statement describes a node?

It always has zero displacement.
It always has the greatest displacement.
It moves along the string with the pattern.
It is the point where the string moves fastest along its length.

Show answer and explanation

It always has zero displacement.

A node is a fixed position that remains at zero displacement in the standing-wave pattern.

Question 2

What forms a standing wave on a string?

Two waves travelling in opposite directions overlap.
One wave stops moving at the middle of the string.
The string moves steadily in one direction.
A wave travels only toward a fixed end and does not reflect.

Show answer and explanation

Two waves travelling in opposite directions overlap.

A reflected wave can travel opposite to the incoming wave. Their repeated overlap creates the standing pattern.

Question 3

A string is fixed at both ends. Which feature must be present at each end?

A node
An antinode
The greatest displacement
A travelling crest that remains there

Show answer and explanation

A node

A fixed end cannot move, so its displacement stays zero. That makes it a node.

Key terms

Amplitude: The greatest size of a point’s displacement from its resting position.
Antinode: A fixed position in a standing wave where the medium reaches its greatest displacement.
Displacement: How far and in which direction a point is from its resting position.
Node: A fixed position in a standing wave that always has zero displacement.
Standing wave: A wave pattern formed by overlapping waves travelling in opposite directions, with nodes and antinodes at fixed positions.
Transverse wave: A wave in which the medium moves at right angles to the direction the wave travels.
Wavelength: The distance between matching points on successive repeats of a wave pattern.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Physics (SPH3U), expectation E3.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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