E2.2 Investigate mechanical waves and interference

Ontario Grade 11 Physics

Waves and Sound

SPH3U study topic E2.2: describing waves and investigating how they overlap

A mechanical wave is a travelling disturbance that needs a material medium, such as a rope, spring, or air. The wave pattern travels through the medium while parts of the medium move around their resting positions. Before describing or calculating, define the system and directions. For a rope viewed from the room, the system can be the rope and its wave. Choose right as positive along the rope and upward as positive for displacement. Displacement is a vector: it has both size and direction. Frequency, wavelength, and wave speed are scalar quantities with size but no direction. A direction can still be given for the wave's travel.

What you will learn

Describe mechanical waves and distinguish transverse from longitudinal motion.
Relate wave speed, frequency, wavelength, and period.
Plan a wave investigation and distinguish measured observations from simulated output.
Use superposition to predict the displacement when waves overlap.

1. Describe a mechanical wave

A transverse wave has medium particles moving at right angles to the wave's direction of travel. For example, a wave can travel right along a rope while points on the rope move up and down. A longitudinal wave has particles moving parallel to the direction of travel. Sound travelling through air is an example. A compression is a region where the particles are closer together. A rarefaction is a region where they are farther apart.

A wave diagram shows displacement at different positions at one instant. Its vertical axis represents displacement, not the distance travelled by the wave. A graph of displacement versus time at one fixed point shows how that point moves over time. Label the axes and indicate the direction of wave travel so that particle motion is not confused with the motion of the pattern.

Amplitude,

A

, is the greatest displacement from the resting position. Wavelength,

\lambda

, is the distance between matching points on neighbouring cycles, such as crest to crest. Frequency,

f

, is the number of complete cycles passing a point each second. Its SI unit is the hertz,

\mathrm{Hz}

, equivalent to cycles per second. Period,

T

, is the time for one complete cycle, measured in seconds. Wave speed,

v

, is the distance the wave pattern travels per second, measured in metres per second.

For a repeating wave, period and frequency are reciprocals. In one second,

f

cycles pass a point. Each cycle spans one wavelength, so the wave travels a distance of

f\lambda

metres in that second. This gives the wave-speed relationship. Use values that describe the same repeating wave.

v=f\lambda,\qquad f=\frac{1}{T}

A mechanical wave needs a material medium.
Amplitude and wavelength are measured in metres; period is measured in seconds.
Frequency is measured in hertz, and wave speed is measured in metres per second.
State the direction of travel and distinguish it from the direction of particle motion.

2. Investigate waves and record evidence

A wave investigation can use a rope or long spring. Define the system as the rope or spring and its wave, viewed from the room. Choose right as positive for direction along the rope and upward as positive for displacement. Keep the equipment clear of people and objects, and use only a small motion that the equipment can safely support.

One possible procedure is to make a small pulse by moving one end briefly upward and returning it to rest. Observe the direction in which the pulse pattern travels and the direction in which the material moves. Repeat with a downward pulse. Then move the end repeatedly to create a pattern of cycles. This is a proposed procedure; it does not claim that any measurements have already been made.

If suitable timing and measuring tools are available, record the time for several cycles and the distance across several wavelengths. Estimate the period by dividing the total time by the number of cycles. Estimate wavelength by dividing the measured distance by the number of wavelengths. Frequency can be found from the number of cycles divided by elapsed time. Record units, the direction convention, what was changed, and what was observed. A sketch can show the resting position, a crest or pulse, and the direction of travel.

Evidence from a real setup consists of observations or readings actually collected from that setup. Evidence from a simulation is simulated output, not a physical measurement. Keep those descriptions separate. A useful conclusion answers the investigation question, refers to recorded evidence, and notes limits such as difficulty timing cycles or locating a crest precisely.

T=\frac{t}{N}

State the system, direction convention, procedure, and units.
Record only observations and measurements that were actually obtained.
Identify simulation output as simulated evidence, not as a physical measurement.
Note measurement limits when explaining what the evidence supports.

3. Explain interference with superposition

When waves overlap in the same medium, their displacements add at each point and instant. This rule is called superposition. Use one resting position and one sign convention for both waves. With upward chosen as positive, an upward displacement is positive and a downward displacement is negative.

Interference is the combined effect of waves while they overlap. Constructive interference occurs when displacements point in the same direction. The resultant displacement is then larger than either individual displacement. Destructive interference occurs when displacements point in opposite directions. The resultant is smaller than the larger individual displacement. Equal and opposite displacements produce zero resultant displacement at that point and instant.

Zero displacement at one point does not mean that the waves have disappeared. It means that their signed displacements cancel there at that instant. To predict an overlap, draw both displacements from the same resting line, assign signs using the chosen direction, and add them. The sign of the result gives its direction; the magnitude gives its distance from rest.

When investigating overlap, distinguish a direct observation on a real rope or spring from output viewed in a simulation and from a prediction made from a diagram. Each can help answer a question, but they are not the same kind of evidence.

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

Add signed displacements at the same place and instant.
Displacements in the same direction reinforce; opposite displacements reduce one another.
Zero resultant displacement is cancellation at a point and instant, not the disappearance of the waves.

Worked example

Find the speed of a repeating wave

A repeating wave travels right along a rope. Its frequency is

6.0\,\mathrm{Hz}

and its wavelength is

0.80\,\mathrm{m}

. Find its speed and direction.

Define the system and direction
The system is the wave on the rope, viewed from the room. Right is positive for travel. The known values are frequency and wavelength; the unknown is wave speed.
Choose the relationship
For a repeating wave, wave speed equals frequency multiplied by wavelength. A hertz is a cycle per second, so the product has units of metres per second.
$v=f\lambda$
Substitute and report
Substitute the SI values. The stated travel direction is right, so include it in the result.
$v=(6.0\,\mathrm{Hz})(0.80\,\mathrm{m})=4.8\,\mathrm{m/s}$

Answer: The wave travels at

4.8\,\mathrm{m/s}

to the right.

Check: The units reduce to metres per second, as required for speed. The answer has two significant figures, matching the given values. Six cycles per second, each spanning

0.80\,\mathrm{m}

, means the pattern travels

4.8\,\mathrm{m}

each second.

Worked example

Find frequency from period

At a fixed point on a spring, one complete cycle takes

0.25\,\mathrm{s}

. Find the frequency.

Identify the known and unknown
The system is the spring. The observation point is fixed relative to the room. The given period is the time for one cycle; frequency is the number of cycles per second.
Use the reciprocal relationship
Frequency and period are reciprocals. Taking the reciprocal of the period gives the number of cycles in one second.
$f=\frac{1}{T}$
Substitute and check units
Substitute the period in seconds. Inverse seconds are expressed as hertz.
$f=\frac{1}{0.25\,\mathrm{s}}=4.0\,\mathrm{Hz}$

Answer: The frequency is

4.0\,\mathrm{Hz}

.

Check: The units are inverse seconds, or hertz. Four cycles per second means each cycle takes

0.25\,\mathrm{s}

, which matches the given period. The result has two significant figures.

Worked example

Predict displacement during overlap

At one point on a rope, pulse A would displace the rope

+0.04\,\mathrm{m}

and pulse B would displace it

-0.03\,\mathrm{m}

at the same instant. Up is positive. Find the resultant displacement.

Define the system and sign
The system is the rope at the point where the pulses overlap. Upward displacement is positive and downward displacement is negative. The unknown is the combined displacement.
Apply superposition
The displacements have opposite signs, so they partly cancel. Add the signed values rather than adding their sizes alone.
$y_{\mathrm{resultant}}=y_A+y_B$
Substitute and interpret
The result is positive, so the point is displaced upward under the chosen convention.
$y_{\mathrm{resultant}}=(+0.04\,\mathrm{m})+(-0.03\,\mathrm{m})=+0.01\,\mathrm{m}$

Answer: The resultant displacement is

0.01\,\mathrm{m}

upward.

Check: All displacements are in metres. The result is smaller than either starting displacement because they oppose each other. Its positive sign agrees with the larger upward displacement, so the result is reasonable.

Common mistakes and how to avoid them

Treating the height of a wave diagram as the distance the wave has travelled.

Correction: The vertical axis shows displacement. Label the axes and use an arrow to show the direction in which the wave pattern travels.

Adding the sizes of opposing displacements as if both were positive.

Correction: Choose a positive direction and add signed displacements. Opposite directions have opposite signs.

Saying destructive interference destroys the waves.

Correction: Destructive interference reduces the resultant displacement while waves overlap. Zero displacement at a point is not the disappearance of the waves.

Reporting simulation output as a physical measurement.

Correction: Label simulation output as simulated evidence. Call a value a measurement only when it was measured in a real setup.

Lesson summary

A mechanical wave is a travelling disturbance that needs a medium.
For repeating waves, $v=f\lambda$ and $f=1/T$ .
A useful investigation states its setup, procedure, observations, units, and limitations.
Superposition adds signed displacements at the same point and instant.
Constructive interference increases resultant displacement; destructive interference reduces it.

Check your understanding

Question 1

A wave has frequency

5.0\,\mathrm{Hz}

and wavelength

0.60\,\mathrm{m}

. What is its speed?

$3.0\,\mathrm{m/s}$
$8.3\,\mathrm{m/s}$
$0.12\,\mathrm{m/s}$
$5.6\,\mathrm{m/s}$

Show answer and explanation

3.0\,\mathrm{m/s}

Use

v=f\lambda

. The product is

(5.0\,\mathrm{Hz})(0.60\,\mathrm{m})=3.0\,\mathrm{m/s}

. The units and two significant figures are appropriate.

Question 2

At one instant, two pulses displace a rope point by

+0.02\,\mathrm{m}

and

-0.02\,\mathrm{m}

. What is the resultant displacement?

$+0.04\,\mathrm{m}$
$-0.02\,\mathrm{m}$
$0\,\mathrm{m}$
$+0.02\,\mathrm{m}$

Show answer and explanation

0\,\mathrm{m}

The equal displacements have opposite signs. Their signed sum is zero, so the point is at its resting position at that instant.

Key terms

Mechanical wave: A travelling disturbance that needs a material medium.
Medium: The material through which a wave travels.
Amplitude: The greatest displacement from the resting position.
Wavelength: The distance between matching points on neighbouring cycles.
Frequency: The number of complete cycles passing a point each second.
Period: The time taken for one complete cycle.
Superposition: The rule that overlapping wave displacements add at a point.
Interference: The combined effect of waves while they overlap.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation E2.2. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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E2.2 · Investigate mechanical waves and interference

Ontario Grade 11 Physics

SPH3U study topic E2.2: describing waves and investigating how they overlap

What you will learn

1. Describe a mechanical wave

2. Investigate waves and record evidence

3. Explain interference with superposition

Find the speed of a repeating wave

Find frequency from period

Predict displacement during overlap

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through SPH3U

About this lesson and its review