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E2.1 · Use terminology for waves, interference, standing waves, and resonance

Learn to use terminology for waves, interference, standing waves, and resonance through clear examples and targeted practice.

Ontario Grade 11 Physics

Waves and Sound

Ontario Grade 11 Physics · Study topic E2.1

A scalar has size only. Frequency is a scalar. A vector has size and direction, such as a displacement that is upward or downward. In the rope examples, the system is the rope and its wave pattern. The rope lies along a horizontal line. Right is the positive direction along the rope, and upward displacement is positive. These directions describe different things: the wave pattern can travel right while parts of the rope move up and down. A wave diagram should show the rest position, which is the rope’s undisturbed position.

What you will learn

1. Prerequisite bridge: describing waves

A wave is a travelling disturbance that transfers energy from place to place. A mechanical wave travels through a material called a medium. A rope, air, and water can each be a medium. In a rope wave, the pattern travels along the rope while sections of rope move around their rest positions.
A transverse wave has a disturbance at right angles to the direction the wave travels. A wave travelling right along a rope while the rope moves up and down is transverse. A longitudinal wave has a disturbance in the same direction as the wave travels. In a sound wave in air, compressions and spread-out regions move along the air. A compression is a region where particles are closer together than usual.
A wave diagram can show displacement against position at one instant. Displacement tells how far and in which direction a point is from its rest position. A crest is a high point above the rest position; a trough is a low point below it. Amplitude, AA, is the greatest displacement from the rest position. It is a distance measured in metres, m\mathrm{m}.
Wavelength, λ\lambda, is the distance between matching points on consecutive cycles, such as crest to crest. It is measured in metres. Frequency, ff, is the number of complete cycles per second. Its SI unit is the hertz, Hz\mathrm{Hz}. Period, TT, is the time for one complete cycle, measured in seconds. Frequency and period describe how often a source repeats its motion; they are not distances.
Read a diagram’s axes before describing it. On a displacement-versus-position diagram, the horizontal spacing between repeated crests is a wavelength. On a displacement-versus-time graph for one location, the time for a complete cycle is a period. The diagram’s horizontal axis determines which quantity can be read.

2. Interference: adding displacements

Interference is the result of two or more waves overlapping in the same region. During the overlap, displacements combine at each location. This combining is called superposition. For a rope, use the sign convention already set: upward displacement is positive and downward displacement is negative.
When overlapping waves displace the medium in the same direction, their signed displacements add to make a larger displacement. This is constructive interference. When their displacements point in opposite directions, one signed displacement is added to the other. This is destructive interference. The result depends on both displacement sizes and directions. Opposing displacements do not always cancel completely, and the combined displacement is not always smaller than each individual displacement.
For example, displacements of +0.030 m+0.030\ \mathrm{m} and −0.010 m-0.010\ \mathrm{m} combine to +0.020 m+0.020\ \mathrm{m}. The result is upward. It is smaller than the first displacement but larger than the second. Equal displacements in opposite directions combine to zero at that location while the waves overlap.
Interference describes what happens during overlap. It does not mean a wave is permanently destroyed. After pulses pass through one another, they continue along the rope. Keep the displacement direction separate from the wave’s travel direction: an upward displacement does not by itself mean that the wave travels upward.
ycombined=y1+y2y_{\mathrm{combined}}=y_1+y_2

3. Standing waves and resonance

A standing wave is a pattern that appears to stay in place rather than travel along the medium. It can form when waves travelling in opposite directions overlap repeatedly. The medium still moves in parts of the pattern, even though the pattern as a whole appears stationary.
A node is a location on a standing wave that remains at zero displacement. An antinode is a location with the greatest repeated displacement. A simple diagram can show a sequence of node, antinode, node. The section between neighbouring nodes is called a loop, and it contains one antinode. Nodes and antinodes describe locations in the pattern, not separate waves.
Resonance is a large response when a system is driven at a frequency matching one of its natural frequencies. A natural frequency is a frequency at which a system readily vibrates. The driving frequency is the frequency of the repeated input. Both are measured in hertz. For example, a rope driven at a matching frequency can develop a standing-wave pattern with a large amplitude.
Interference, standing waves, and resonance are connected but not interchangeable terms. Interference describes overlapping waves. A standing wave describes a pattern. Resonance describes a large response under a matching-frequency condition. Interference can help form a standing wave, and a standing-wave pattern may be part of a resonant response.
fdriving=fnaturalf_{\mathrm{driving}}=f_{\mathrm{natural}}

4. Choose precise terms

When describing a wave, state what is moving and what the diagram shows. A transverse wave can travel horizontally while the medium moves vertically. Amplitude describes maximum displacement from rest, while wavelength describes a distance between repeated points. A node is a fixed location in a standing-wave pattern, not a point where the entire medium is motionless.
When describing overlap, state whether displacements have the same or opposite directions. For a numerical displacement, include its sign, unit, and direction. A positive answer means upward under the rope convention used here; a negative answer means downward. Amplitude itself is a non-negative distance, not a signed direction.
When describing resonance, name the driving frequency and the natural frequency, then say whether they match and whether the response is large. A larger response alone does not establish the matching-frequency condition. Precise terminology makes clear which wave feature or process is being described.

Worked example

Interference with opposite displacements

At one point on a rope, an upward pulse has displacement +0.030 m+0.030\ \mathrm{m} and a downward pulse has displacement −0.010 m-0.010\ \mathrm{m}. Find the combined displacement while they overlap and name the type of interference.
  1. Set the system and convention
    The system is the rope at the overlap point. Upward displacement is positive and downward displacement is negative. The unknown is the combined displacement at that point.
  2. Combine the displacements
    During overlap, the displacements add with their signs. Since they point in opposite directions, this is destructive interference.
    ycombined=(+0.030 m)+(−0.010 m)y_{\mathrm{combined}}=(+0.030\ \mathrm{m})+(-0.010\ \mathrm{m})
  3. Interpret the result
    The positive result means the point is displaced upward. Keep the direction in the answer because displacement is a vector.
    ycombined=+0.020 my_{\mathrm{combined}}=+0.020\ \mathrm{m}
Answer: The combined displacement is 0.020 m0.020\ \mathrm{m} upward. The overlap is destructive interference.
Check: The unit is metres, and the positive sign agrees with an upward result. The answer is smaller than the upward displacement but larger than the downward displacement, which is reasonable for these opposing pulse sizes. It is reported to two significant figures.

Worked example

Recognizing features of a standing wave

A rope’s pattern appears to stay in place. Two locations remain at zero displacement, and a location between them moves through the greatest displacement. Name these features and describe the overall pattern.
  1. Identify the pattern
    The system is the rope and its wave pattern. A pattern that appears to stay in place is a standing wave.
  2. Name the locations
    Locations that remain at zero displacement are nodes. The location with the greatest repeated displacement is an antinode.
  3. Describe the part between nodes
    The section between neighbouring nodes is a loop. In this description, the loop contains the antinode.
Answer: The rope has a standing wave. The two locations at zero displacement are nodes, and the location of greatest displacement is an antinode. The section between neighbouring nodes is a loop.
Check: The terms match the stated behaviour: nodes stay at zero displacement, while the antinode has the greatest repeated displacement. The description does not confuse a stationary-looking pattern with a motionless rope.

Worked example

Identifying resonance

A system has a natural frequency of 5.0 Hz5.0\ \mathrm{Hz}. It is driven repeatedly at 5.0 Hz5.0\ \mathrm{Hz} and shows a large vibration response. Identify the process and explain why.
  1. Identify the system information
    The system is the object being driven. Frequency is a scalar, so a direction is not needed. The relevant facts are the natural frequency, the driving frequency, and the large response.
  2. Compare the frequencies
    Resonance is identified when the driving frequency matches a natural frequency and the response is large. Both frequencies are measured in hertz.
    fdriving=fnatural=5.0 Hzf_{\mathrm{driving}}=f_{\mathrm{natural}}=5.0\ \mathrm{Hz}
  3. State the interpretation
    The matching frequencies and large response together describe resonance. Neither frequency is a direction or a displacement.
Answer: The system is undergoing resonance because its driving frequency matches its natural frequency and the response is large.
Check: Both frequencies are 5.0 Hz5.0\ \mathrm{Hz}, so their units and values match. The stated large response fits the meaning of resonance.

Common mistakes and how to avoid them

Calling amplitude the distance from crest to trough.
Correction: Amplitude is measured from the rest position to a crest or trough. For a symmetric wave, crest-to-trough distance is twice the amplitude.
Treating wavelength and period as interchangeable.
Correction: Wavelength is a distance; period is a time. Check whether the diagram’s horizontal axis shows position or time.
Saying destructive interference always makes both waves disappear or makes the result smaller than either displacement.
Correction: Add signed displacements. Opposing displacements can cancel, reduce, or leave a result that is larger than the smaller displacement.
Calling every point on a standing wave a node.
Correction: Only locations that remain at zero displacement are nodes. Antinodes are locations with the greatest repeated displacement.
Using resonance as another name for a standing wave.
Correction: A standing wave names a pattern. Resonance names a large response when the driving frequency matches a natural frequency.

Lesson summary

Check your understanding

Question 1

A transverse wave travels right along a rope. Which description is correct?
  1. The wave travels right while the rope moves up and down.
  2. The wave and every part of the rope must travel right together.
  3. The rope moves along its length, so the wave is longitudinal.
  4. The wave has no direction because its displacement is vertical.
Show answer and explanation
The wave travels right while the rope moves up and down.
In a transverse wave, the disturbance is at right angles to the travel direction. The rope can move up and down while the pattern travels right.

Question 2

Two equal-size pulses overlap at one point. One displacement is upward and the other is downward. What is the combined displacement there?
  1. Twice the upward displacement
  2. Zero, due to destructive interference
  3. The size of one pulse, downward
  4. A larger upward displacement because two pulses are present
Show answer and explanation
Zero, due to destructive interference
Equal displacements in opposite directions add to zero at that location while they overlap. This is destructive interference.

Question 3

Which situation describes resonance?
  1. A wave has a crest above its rest position.
  2. A point on a standing wave remains at zero displacement.
  3. A system has a large response when its driving frequency matches a natural frequency.
  4. Two overlapping waves always cancel completely.
Show answer and explanation
A system has a large response when its driving frequency matches a natural frequency.
A large response at a matching driving and natural frequency is resonance. The other options describe a wave feature, a node, or an incorrect claim about interference.

Key terms

Amplitude
The greatest displacement from the rest position.
Antinode
A location on a standing wave with the greatest repeated displacement.
Frequency
The number of complete cycles per second, measured in hertz.
Interference
The result when two or more waves overlap.
Node
A location on a standing wave that remains at zero displacement.
Resonance
A large response when a system’s driving frequency matches a natural frequency.
Standing wave
A wave pattern that appears to stay in place rather than travel along the medium.
Wavelength
The distance between matching points on consecutive cycles.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation E2.1. It is a study resource, not an official curriculum publication.

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