E2.4 Relate wave speed, wavelength, and frequency

Ontario Grade 11 Physics

Waves and Sound

Ontario Grade 11 Physics — E2.4

A wave transfers a repeating disturbance from place to place. To describe how quickly it travels and how its pattern repeats, we use wave speed, wavelength, and frequency. These quantities are linked by one relationship. Before using it, we will define each quantity, its units, and the direction in which the wave travels.

What you will learn

Define wave speed, wavelength, and frequency using appropriate units.
Use the relationship between wave speed, wavelength, and frequency to solve problems.
Explain how changing frequency affects wavelength when wave speed stays the same.

1. The wave and its direction

Begin by choosing the physical system: the wave pattern travelling through a medium or through space. The medium is the material through which a wave travels, such as water or a stretched string. The wave pattern can move even though the material itself does not travel along with the pattern over long distances.

Choose a reference frame: the point of view from which positions and motion are described. For this lesson, imagine watching the wave from a stationary position. Choose the direction the wave travels as positive. For example, if the wave moves to the right, right is positive and left is negative. Wave speed describes how fast the pattern moves; when direction matters, state the direction in words.

A scalar has magnitude only. Frequency and wavelength are scalars. A vector has both magnitude and direction. Wave velocity is a vector, while wave speed is its magnitude. The relationship in this lesson uses wave speed, so a calculation gives a non-negative value. Add the direction of travel in words when it is known.

Wave speed tells how fast the wave pattern travels.
Wavelength is the distance between matching points on successive cycles, such as crest to crest.
Frequency is the number of complete cycles passing a point each second.

2. The quantities and their units

A cycle is one complete repeat of a wave pattern. A crest is a high point in the pattern, and a trough is a low point. Wavelength, represented by the Greek letter lambda, is the length of one cycle along the direction of travel. Its SI unit is the metre, written as m.

Frequency, represented by the letter f, counts how many cycles pass a fixed point in one second. Its SI unit is the hertz, written as Hz. One hertz means one cycle per second. Frequency is not the time for one cycle; it counts cycles in a given time.

Wave speed, represented by v, is the distance the wave pattern travels per unit of time. Its SI unit is metres per second, written as m/s. The symbol v here means wave speed, not a direction by itself. Keep units attached to measured or calculated values so that the quantities remain clear.

Wavelength: $\lambda$ , measured in metres (m).
Frequency: $f$ , measured in hertz (Hz), or cycles per second.
Wave speed: $v$ , measured in metres per second (m/s).

3. The relationship

In one second, a wave pattern completes f cycles. Each cycle has length lambda. The total distance covered by those cycles in one second is therefore f multiplied by lambda. That distance per second is the wave speed.

Use the governing relationship

v=f\lambda

. It relates wave speed to frequency and wavelength. If any two quantities are known, rearrange the relationship with ordinary algebra to find the third. For example, divide both sides by frequency to find wavelength, or divide by wavelength to find frequency.

Check the units as well as the arithmetic. Frequency in hertz is cycles per second, and wavelength is metres per cycle. Their product is metres per second because the cycle units cancel. For a wave travelling right, report the speed as a positive magnitude and state that it travels to the right. If it travels left, the speed remains a positive magnitude; the direction is left.

v=f\lambda

Use $v=f\lambda$ when speed, frequency, and wavelength are related.
At a fixed wave speed, a higher frequency means a shorter wavelength.
At a fixed wave speed, a lower frequency means a longer wavelength.

4. Using the relationship carefully

First identify the known quantities and the unknown. Convert values to SI units when necessary. Then select the form of the relationship that places the unknown by itself. Substitute values with their units, calculate, and round to a sensible number of significant figures based on the given data.

Finally, state what the result means for the wave and its direction. Check that the units match the quantity being found. Ask whether the size makes sense: for example, a wave with many cycles passing each second and a substantial wavelength should have a relatively large speed. This check can reveal a misplaced decimal or an incorrect rearrangement.

\lambda=\frac{v}{f}\quad\quad f=\frac{v}{\lambda}

Keep the units in the substitution.
Round the final value to match the precision of the supplied information.
Check the unit, magnitude, and direction in words.

Worked example

Find wave speed

A wave travels to the right. Its wavelength is

0.75\,\mathrm{m}

and its frequency is

8.0\,\mathrm{Hz}

. Find its speed.

Identify the system and direction
The system is the travelling wave pattern, viewed from a stationary position. Take right as positive. The known values are wavelength and frequency; the unknown is wave speed.
Choose the relationship
Wave speed equals frequency multiplied by wavelength. This applies because each cycle advances by one wavelength.
$v=f\lambda$
Substitute and calculate
Insert the values with their SI units. The frequency has two significant figures, so report the result to two significant figures.
$v=(8.0\,\mathrm{Hz})(0.75\,\mathrm{m})=6.0\,\mathrm{m/s}$

Answer: The wave speed is

6.0\,\mathrm{m/s}

to the right.

Check: The units are hertz times metres, equivalent to metres per second. A wave covering

0.75\,\mathrm{m}

per cycle for

8.0

cycles each second travels

6.0\,\mathrm{m}

each second, so the result is reasonable.

Worked example

Find wavelength

A wave travels left at

12\,\mathrm{m/s}

. Its frequency is

3.0\,\mathrm{Hz}

. Find its wavelength.

Identify the system and direction
The system is the wave pattern, viewed from a stationary position. Take right as positive, so the stated direction of travel is left. The unknown wavelength is a distance and is not negative.
Rearrange the relationship
Starting from wave speed equals frequency times wavelength, divide by frequency to isolate wavelength.
$\lambda=\frac{v}{f}$
Substitute and calculate
Use the speed magnitude and the given frequency. The two significant figures in the supplied values support a two-significant-figure result.
$\lambda=\frac{12\,\mathrm{m/s}}{3.0\,\mathrm{Hz}}=4.0\,\mathrm{m}$

Answer: The wavelength is

4.0\,\mathrm{m}

. The wave travels left.

Check: Dividing metres per second by cycles per second gives metres per cycle, reported as metres. A wave travelling

12\,\mathrm{m}

each second at three cycles each second has a wavelength of

4.0\,\mathrm{m}

, which is reasonable.

Worked example

Find frequency

A wave travelling to the right has a wavelength of

0.40\,\mathrm{m}

and a speed of

2.4\,\mathrm{m/s}

. Find its frequency.

Identify the system and direction
The system is the travelling wave pattern, viewed from a stationary position. Take right as positive. The known quantities are speed and wavelength; the unknown is frequency.
Rearrange the relationship
Divide wave speed by wavelength to isolate frequency.
$f=\frac{v}{\lambda}$
Substitute and calculate
Substitute the SI values. The given values have two significant figures, so give the result to two significant figures.
$f=\frac{2.4\,\mathrm{m/s}}{0.40\,\mathrm{m}}=6.0\,\mathrm{Hz}$

Answer: The frequency is

6.0\,\mathrm{Hz}

. The wave travels to the right.

Check: Metres per second divided by metres gives cycles per second, or hertz. Six cycles per second, each

0.40\,\mathrm{m}

long, account for

2.4\,\mathrm{m}

of travel per second.

Common mistakes and how to avoid them

Treating wavelength as the distance from a crest to the nearest trough.

Correction: A wavelength is one complete cycle. Measure between matching points, such as crest to the next crest.

Using frequency as though it were the time for one cycle.

Correction: Frequency counts cycles per second and is measured in hertz. Use the frequency value in

v=f\lambda

.

Reporting a negative wave speed because the wave travels left.

Correction: Wave speed is a magnitude and is non-negative. State the direction separately; wave velocity includes direction.

Assuming that a larger frequency always means a larger wave speed.

Correction: The relationship connects speed, frequency, and wavelength. If speed stays the same, increasing frequency makes wavelength shorter.

Lesson summary

Wavelength is the distance for one complete cycle; frequency is cycles per second.
Wave speed is the distance travelled by the wave pattern per second.
Use $v=f\lambda$ and keep SI units with each value.
Report speed as a magnitude and give the direction separately when it is known.
Check units, significant figures, and whether the answer is physically reasonable.

Check your understanding

Question 1

A wave travels at

10\,\mathrm{m/s}

and has a wavelength of

2.0\,\mathrm{m}

. What is its frequency?

$5.0\,\mathrm{Hz}$
$20\,\mathrm{Hz}$
$0.20\,\mathrm{Hz}$
correctIndex

Show answer and explanation

5.0\,\mathrm{Hz}

Use

f=v/\lambda

. Dividing

10\,\mathrm{m/s}

by

2.0\,\mathrm{m}

gives

5.0\,\mathrm{Hz}

.

Question 2

For a wave travelling at the same speed, what happens to its wavelength if its frequency increases?

The wavelength decreases.
The wavelength increases.
The wavelength stays the same.
correctIndex is integer not comments

Show answer and explanation

The wavelength decreases.

Since

v=f\lambda

and speed is fixed, a greater frequency must be paired with a smaller wavelength.

Key terms

Cycle: One complete repeat of a wave pattern.
Frequency: The number of complete cycles passing a point each second, measured in hertz.
Wavelength: The distance between matching points on successive cycles, measured in metres.
Wave speed: The distance the wave pattern travels per unit of time, measured in metres per second.
Scalar: A quantity with magnitude only.
Vector: A quantity with magnitude and direction.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation E2.4. 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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E2.4 · Relate wave speed, wavelength, and frequency

Ontario Grade 11 Physics

Ontario Grade 11 Physics — E2.4

What you will learn

1. The wave and its direction

2. The quantities and their units

3. The relationship

4. Using the relationship carefully

Find wave speed

Find wavelength

Find frequency

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