Illustrate superposition, standing waves, and beat frequencies

Ontario Grade 11 Physics

Waves and Sound

How waves combine, form fixed patterns, and produce changing loudness

A wave transfers energy while points in its medium move around their usual positions. Displacement describes how far and in which direction a point is from its rest position. It is measured in metres. Frequency counts cycles per second and is measured in hertz. Displacement has a direction; frequency does not. In this lesson, the system is the wave medium, such as a string, or the pair of sounds reaching a listener. For a string, choose upward displacement as positive and downward displacement as negative. For sound, beat frequency is a non-negative rate, so it has no positive or negative direction.

What you will learn

Describe superposition by adding wave displacements at the same place and time.
Explain how equal-frequency waves of equal amplitude travelling in opposite directions can form standing waves with nodes and antinodes.
Calculate and interpret beat frequency from two sound frequencies.

1. Superposition: combine signed displacements

When waves overlap, each contributes to the displacement at the same place and time. The principle of superposition says that the total displacement equals the sum of the individual displacements. Displacement is signed: its sign tells whether it points along or opposite to the chosen positive direction.

If two displacements point in the same direction, they reinforce each other. This is constructive interference. If they point in opposite directions, they partly or fully cancel. This is destructive interference. Interference describes the result of overlap; neither wave is permanently removed.

For example, if upward is positive, a wave moving a point up by

0.03\ \mathrm{m}

contributes

+0.03\ \mathrm{m}

. A wave moving it down by

0.01\ \mathrm{m}

contributes

-0.01\ \mathrm{m}

. Their sum is upward by

0.02\ \mathrm{m}

. The signs are essential.

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

Add signed displacements at the same place and time.
Same-direction contributions reinforce; opposite-direction contributions cancel partly or fully.
Displacement is measured in metres.

2. Standing waves: fixed nodes and antinodes

A standing wave is a repeating pattern that appears to remain in place instead of travelling along the medium. In the ideal pattern described here, it forms when two waves of equal frequency and equal amplitude travel in opposite directions along the same medium. Their displacements combine continuously by superposition.

A node is a point that remains at zero displacement in the ideal standing-wave pattern. The equal, opposite contributions cancel there. An antinode is a point where the displacement reaches its greatest size. Points between nodes move back and forth. The nodes and antinodes stay in fixed positions, but this does not mean that the whole medium is still.

A string fixed at both ends has nodes at its fixed ends. Along the string, nodes and antinodes alternate. This description identifies the pattern without requiring a separate wave equation.

Equal-frequency, equal-amplitude waves travelling in opposite directions can form an ideal standing wave.
Nodes remain at zero displacement; antinodes reach the greatest displacement.
The medium moves between nodes.

3. Beats: a repeating change in loudness

When two sound waves with slightly different frequencies reach a listener together, their combined sound can grow louder and softer repeatedly. One rise and fall in loudness is a beat. Beat frequency is the number of beats per second. It is a scalar measured in hertz.

The beat frequency is the absolute difference between the two sound frequencies. Absolute difference means subtract the smaller frequency from the larger, so the answer is never negative. When the source frequencies are close, the changing loudness can be heard as separate beats.

This model explains why two nearly matched musical notes can sound as though their loudness is pulsing. As the waves shift between stronger and weaker overlap, the combined sound changes in loudness. The relationship below describes the beat rate; it does not claim that any particular sound has been measured.

f_{\mathrm{beat}}=|f_1-f_2|

Beats are repeated increases and decreases in loudness from two nearby sound frequencies.
Beat frequency is a non-negative rate measured in hertz.
A larger frequency difference means more beats per second.

4. Choose the model and check the result

For overlapping waves at one point, add the signed displacements. For a fixed pattern with nodes and antinodes, identify a standing wave. For two nearby sound frequencies that create changing loudness, find the beat frequency.

Keep the quantities distinct. Displacement is a signed length in metres. Frequency and beat frequency are scalar rates in hertz. Do not add frequencies to find the displacement at a point, or treat a node as a point where the entire medium has stopped moving.

For a numerical result, check that its unit matches the quantity. State the direction when a displacement has a sign. A beat frequency cannot be negative. Finally, ask whether the size of the result fits the situation.

Match the relationship to the physical situation.
Use signs and directions for displacement; use a non-negative value for beat frequency.
Check units, direction where needed, and physical reasonableness.

Worked example

Adding overlapping displacements

At the same point on a string, wave A produces an upward displacement of

0.024\ \mathrm{m}

and wave B produces a downward displacement of

0.009\ \mathrm{m}

. Find the total displacement.

Set direction and identify the unknown
The system is the point on the string where the waves overlap. Choose upward as positive and downward as negative. The unknown is the combined displacement.
Apply superposition
Superposition requires adding the displacements at the same place and time. The downward contribution is negative with this sign convention.
$y_{\mathrm{total}}=(+0.024\ \mathrm{m})+(-0.009\ \mathrm{m})$
Calculate and interpret
The difference is positive, so the combined displacement is upward. Since both values are given to the nearest $0.001\ \mathrm{m}$ , report the difference to that decimal place.
$y_{\mathrm{total}}=+0.015\ \mathrm{m}$

Answer: The total displacement is

0.015\ \mathrm{m}

upward.

Check: The result is in metres, the correct unit for displacement. It is smaller than the upward contribution because the downward wave partly cancels it. The result has two significant figures, consistent with the subtraction of values given to the nearest

0.001\ \mathrm{m}

.

Worked example

Recognizing a node in an ideal standing wave

Two equal-amplitude waves of the same frequency travel in opposite directions along a string. At a marked point, their displacements always cancel. What is the point called, and what happens there?

Identify the system and pattern
The system is the string. The waves have equal frequency and equal amplitude, and travel in opposite directions, so their repeated superposition can form an ideal standing-wave pattern.
Use superposition at the marked point
The question states that the two contributions cancel at this point. Their total displacement is therefore zero, which matches the definition of a node.
$y_{\mathrm{total}}=0\ \mathrm{m}$
Describe the nearby motion
The marked point stays at zero displacement in the pattern. Other points between nodes move back and forth, so the whole string is not still.

Answer: The marked point is a node. Its displacement remains zero in the ideal standing-wave pattern.

Check: Zero displacement is consistent with cancellation of equal contributions. The unit is metres. The conclusion applies to the marked node, not to every point on the string.

Worked example

Finding the beat frequency

Two sound sources produce frequencies of

440\ \mathrm{Hz}

and

446\ \mathrm{Hz}

. Find the beat frequency and state what it means.

Identify the system and unknown
The system is the two sounds reaching the same listener. Their frequencies are scalars. The unknown is the number of loudness beats per second.
Apply the beat-frequency relationship
Beat frequency is the absolute difference between the two source frequencies, so it cannot be negative.
$f_{\mathrm{beat}}=|446\ \mathrm{Hz}-440\ \mathrm{Hz}|$
Calculate and interpret
The frequency difference is six hertz. This means six loudness beats occur each second.
$f_{\mathrm{beat}}=6\ \mathrm{Hz}$

Answer: The beat frequency is

6\ \mathrm{Hz}

, meaning the loudness rises and falls six times per second.

Check: Hertz is the correct unit for a frequency. Six beats per second matches the six-hertz difference between the source frequencies. The result is positive and has one significant figure.

Common mistakes and how to avoid them

Adding displacement magnitudes without considering direction.

Correction: Choose a positive direction and add signed displacements. With upward positive, a downward displacement is negative.

Assuming a node means the entire string does not move.

Correction: A node remains at zero displacement in the ideal standing-wave pattern. Other points between nodes move.

Calling one of the source frequencies the beat frequency.

Correction: Beat frequency is the absolute difference between the two source frequencies.

Treating frequency as a vector with a direction.

Correction: Frequency is a scalar. Displacement has a direction, represented here by a positive or negative sign.

Lesson summary

Superposition adds signed displacements at the same place and time.
An ideal standing wave can form from equal-amplitude waves of the same frequency travelling in opposite directions. Nodes remain at zero displacement; antinodes have the greatest displacement.
Two nearby sound frequencies can produce beats. Beat frequency is the absolute difference between the frequencies.
Displacement is measured in metres; frequency is measured in hertz. Check signs, units, and physical meaning.

Check your understanding

Question 1

At one point, one wave produces a displacement of

-0.012\ \mathrm{m}

and another produces

+0.012\ \mathrm{m}

. What is the combined displacement?

$0\ \mathrm{m}$
$+0.024\ \mathrm{m}$
$-0.012\ \mathrm{m}$
$+0.012\ \mathrm{m}$

Show answer and explanation

0\ \mathrm{m}

The equal displacements have opposite signs, so their sum is zero.

Question 2

What is the name for a point that remains at zero displacement in an ideal standing-wave pattern?

Antinode
Node
Beat
Frequency

Show answer and explanation

Node

A node is a point that stays at zero displacement in the pattern.

Question 3

Two sound frequencies are

512\ \mathrm{Hz}

and

518\ \mathrm{Hz}

. What is their beat frequency?

$1030\ \mathrm{Hz}$
$6\ \mathrm{Hz}$
$-6\ \mathrm{Hz}$
$518\ \mathrm{Hz}$

Show answer and explanation

6\ \mathrm{Hz}

The absolute difference is

|518\ \mathrm{Hz}-512\ \mathrm{Hz}|=6\ \mathrm{Hz}

. Beat frequency is non-negative.

Key terms

Displacement: The distance and direction of a point from its rest position.
Superposition: The rule that overlapping waves produce a total displacement equal to the sum of their individual displacements.
Interference: The result of overlapping waves, in which displacements reinforce or cancel.
Standing wave: A repeating pattern that appears fixed in place, with nodes and antinodes.
Node: A point that remains at zero displacement in an ideal standing-wave pattern.
Antinode: A point in a standing-wave pattern where displacement reaches its greatest size.
Beat: One repeated increase and decrease in loudness caused by two nearby sound frequencies.
Hertz: The SI unit of frequency, equal to one cycle per second.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Physics (SPH3U), expectation E3.3. 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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