E2.3 Measure wave speed and compare theoretical and experimental values

Ontario Grade 11 Physics

Waves and Sound

Ontario Grade 11 Physics — E2.3

A wave can carry energy from one place to another. Its speed tells us how quickly a point on the wave pattern travels through a medium, or how quickly the pattern travels across a space. In this lesson, the physical system is a wave moving through a medium such as a stretched cord. We choose the direction the wave travels as positive. Wave speed is a scalar: it has magnitude but no direction. Wave velocity includes direction. The expectation here is to measure wave speed and compare an experimental value with a theoretical value.

What you will learn

Describe how to measure a wave's speed using distance and time.
Use frequency and wavelength to calculate a theoretical wave speed.
Compare experimental and theoretical values using units and percent difference.

1. Prerequisite bridge: distance, time, and wave features

Speed is distance travelled divided by the time taken. Distance is measured in metres, written as m. Time is measured in seconds, written as s. Speed in SI units is measured in metres per second, written as m/s. This relationship applies to a wave pattern as it moves.

A repeating wave has crests, the highest points, and troughs, the lowest points. Wavelength, symbolized by

\lambda

, is the distance between matching points on neighbouring cycles, such as crest to crest. It is measured in metres. Frequency, symbolized by

f

, is the number of complete cycles passing a point each second. It is measured in hertz, or cycles per second.

Frequency and period are connected. Period,

T

, is the time for one complete cycle and is measured in seconds. Frequency is the number of cycles per second, so a shorter period means a higher frequency. For example, a frequency of

4.0\ \mathrm{Hz}

means four cycles pass a point each second.

v=\frac{d}{t}

Distance divided by time gives speed.
Wavelength is a distance; frequency is a count of cycles per second.
Use SI units: metres, seconds, hertz, and metres per second.

2. Theoretical and experimental wave speed

The theoretical wave speed is calculated from the wave's frequency and wavelength. It is called theoretical because it comes from a model relating wave features, rather than from directly timing a wave over a measured distance. In this lesson, the model is that each cycle advances one wavelength in one period. The resulting speed is frequency multiplied by wavelength.

An experimental value is calculated from observations or measurements. For example, you could mark a distance along a cord and measure the time taken for a pulse to travel between the marks. A pulse is one short disturbance that moves along the medium. The measured distance divided by the measured time gives the experimental speed.

In a real investigation, record what was actually measured, including units and the method used. A proposed procedure is not evidence that measurements were taken. A simulation can help practise reading or predicting wave behaviour, but it is not a physical measurement. Example numbers in this lesson are practice data only; they are not claimed laboratory results.

A simple diagram can help set directions: mark the cord horizontally, choose right as positive, and show a crest moving right. If the wave travels left instead, its direction is left. Speed remains a positive scalar; velocity would need a direction. We compare speed magnitudes here.

The experimental and theoretical values may not match exactly. Timing and distance readings have limited precision. To compare them, calculate the absolute difference and express it relative to the theoretical value as a percentage. A smaller percent difference indicates closer agreement. This comparison does not by itself identify the cause of a difference.

v_{\mathrm{theory}}=f\lambda,\quad v_{\mathrm{exp}}=\frac{d}{t}

Theoretical speed uses frequency and wavelength.
Experimental speed uses a measured travel distance and measured travel time.
State the direction of travel when describing velocity; speed itself has no direction.
Keep the comparison focused on measured evidence and the chosen theoretical model.

3. Comparing values carefully

Before calculating, name the system, direction, known values, and unknown. The system is the travelling wave in its medium. Set the travel direction as positive. For a speed calculation, the answer is a positive magnitude; include a direction in words when the question asks which way the wave travels.

Keep units in the calculation. For the theoretical model, hertz is equivalent to inverse seconds, so multiplying frequency by wavelength gives metres per second. For the experimental model, metres divided by seconds also gives metres per second. If the units do not reduce to m/s, check the equation and substitutions.

Use a sensible number of significant figures. Significant figures are the meaningful digits in a measured value. Do not report a result with far more precision than the measurements support. When comparing two speeds, use the same units and sensible rounding before interpreting the difference.

A percent difference is a comparison of the size of the gap with the theoretical value. The absolute value bars mean to use the positive size of the difference, whether the experimental result is higher or lower. The theoretical value is the reference in this lesson, so it belongs in the denominator.

\%\,\mathrm{difference}=\frac{|v_{\mathrm{exp}}-v_{\mathrm{theory}}|}{v_{\mathrm{theory}}}\times100\%

Write known values with units before substitution.
Check that both values being compared use the same units.
A percent difference is non-negative and is measured in percent.

4. Planning a measurement and checking quality

A possible physical procedure is to make a pulse on a cord, mark two points a known distance apart, and time the pulse as it travels between them. This is a proposed procedure, not a completed experiment. Use a metre ruler or tape for distance and a stopwatch or suitable timer for time. Record the measured values, their units, and the observed travel direction.

Timing one pass can make reaction time a large part of the result. If appropriate to the setup, measure a longer travel distance or repeat the timing and use the same method consistently. Do not change the recorded evidence to make it match the theoretical value. Report what the measurements support.

A separate theoretical calculation needs a frequency and wavelength for the same wave. These can be measured or supplied in a question. Make sure they refer to the wave being compared with the travel-time measurement. A mismatch could reflect measurement limits, inconsistent conditions, or an incorrect reading; further investigation would be needed to decide which.

Finish with three checks. The units should be m/s. The speed should be positive, with the direction stated separately if needed. The value should be reasonable for the given distance and time: a longer distance in the same time means a greater speed, and a longer time for the same distance means a smaller speed.

Record observations; do not treat a proposed method as measured evidence.
Compare values for the same wave and use consistent units.
Check units, direction, significant figures, and reasonableness.

Worked example

1. Calculate theoretical wave speed

Practice data: a wave has frequency

3.5\ \mathrm{Hz}

and wavelength

0.80\ \mathrm{m}

. It travels to the right along a cord. Find its theoretical speed.

Set the system and direction
The system is the wave travelling along the cord. Take right as positive. The requested quantity is speed, so report a positive magnitude and state the direction separately.
Choose the model
Use the relationship between frequency, wavelength, and theoretical wave speed. Frequency counts cycles per second, and each cycle spans one wavelength.
$v_{\mathrm{theory}}=f\lambda$
Substitute with units
Insert the given frequency and wavelength. The units combine to metres per second.
$v_{\mathrm{theory}}=(3.5\ \mathrm{Hz})(0.80\ \mathrm{m})=2.8\ \mathrm{m/s}$

Answer: The theoretical wave speed is

2.8\ \mathrm{m/s}

to the right.

Check: The units are m/s because

\mathrm{Hz}=1/\mathrm{s}

. The positive speed and stated rightward direction match the situation. A speed of a few metres per second is consistent with the given frequency and wavelength.

Worked example

2. Calculate experimental wave speed

Practice data, not reported laboratory results: a pulse travels

1.20\ \mathrm{m}

to the left along a cord in

0.75\ \mathrm{s}

. Find the experimental speed.

Set the system and direction
The system is the pulse travelling along the cord. Take left as the direction of travel. The unknown is its experimental speed.
Use measured distance and time
The measured distance is divided by the measured time. Distance and time are both positive measurements, so the calculated speed is a positive magnitude.
$v_{\mathrm{exp}}=\frac{d}{t}$
Substitute and round
Use the practice measurements with their units. Both values have two significant figures, so report the speed to two significant figures.
$v_{\mathrm{exp}}=\frac{1.20\ \mathrm{m}}{0.75\ \mathrm{s}}=1.6\ \mathrm{m/s}$

Answer: The experimental speed is

1.6\ \mathrm{m/s}

to the left.

Check: Metres divided by seconds gives m/s. The positive speed is paired with the leftward direction. The result is reasonable: travelling about

1.2\ \mathrm{m}

in less than a second requires a speed greater than

1\ \mathrm{m/s}

.

Worked example

3. Compare theoretical and experimental values

Practice comparison data: a wave has theoretical speed

2.4\ \mathrm{m/s}

, and the measured experimental speed is

2.1\ \mathrm{m/s}

. Both values describe motion to the right. Find the percent difference, using the theoretical value as the reference.

Identify the comparison
The system is the same travelling wave. Both speeds have the same direction and units, so compare their magnitudes. The theoretical value is the reference.
Find the positive difference
Subtract the speeds and take the absolute value so the difference is positive whether the experimental value is above or below the theoretical one.
$|2.1\ \mathrm{m/s}-2.4\ \mathrm{m/s}|=0.3\ \mathrm{m/s}$
Calculate percent difference
Divide the difference by the theoretical speed and multiply by one hundred percent. Round to one significant figure, consistent with the difference shown.
$\frac{0.3\ \mathrm{m/s}}{2.4\ \mathrm{m/s}}\times100\%=10\%$

Answer: The experimental value is about 10% different from the theoretical value, and it is lower.

Check: The speed units cancel in the ratio, leaving a percent. The difference is positive and smaller than the theoretical speed, so a result of 10% is reasonable. Both values describe rightward motion.

Common mistakes and how to avoid them

Calling frequency the time for one cycle.

Correction: Frequency is cycles per second. Period is the time for one cycle.

Reporting the experimental speed without units or travel direction.

Correction: Include m/s and state the direction in words when it is relevant. Speed is a scalar; direction belongs to velocity or to the description of travel.

Dividing by the experimental value when the comparison specifies theoretical speed as the reference.

Correction: Put the stated reference value in the denominator. Here, use theoretical speed.

Presenting a planned measurement as if it had already been performed.

Correction: Label a procedure as proposed. Only call values experimental measurements when they come from observations that were actually recorded.

Lesson summary

Theoretical wave speed is calculated from frequency and wavelength.
Experimental wave speed is calculated from measured distance and travel time.
Compare values using the same units, and state direction separately from speed.
Report appropriate significant figures and check units and reasonableness.

Check your understanding

Question 1

A wave has frequency

5.0\ \mathrm{Hz}

and wavelength

0.60\ \mathrm{m}

. What is its theoretical speed?

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

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 are m/s.

Question 2

A pulse moves

2.0\ \mathrm{m}

to the right in

0.50\ \mathrm{s}

. Which result correctly states its experimental speed?

$4.0\ \mathrm{m/s}$ to the right
$0.25\ \mathrm{m/s}$ to the right
$4.0\ \mathrm{m}$ to the right
$4.0\ \mathrm{m/s}$ to the left

Show answer and explanation

4.0\ \mathrm{m/s}

to the right

Distance divided by time gives

2.0\ \mathrm{m}/0.50\ \mathrm{s}=4.0\ \mathrm{m/s}

. The pulse travels to the right.

Key terms

Wavelength: The distance between matching points on neighbouring wave cycles.
Frequency: The number of complete wave cycles that pass a point each second, measured in hertz.
Theoretical value: A value calculated using a model or relationship.
Experimental value: A value calculated from observations or measurements that were actually made.
Significant figures: The meaningful digits in a measured value, used to guide how precisely a result is reported.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation E2.3. 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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E2.3 · Measure wave speed and compare theoretical and experimental values

Ontario Grade 11 Physics

Ontario Grade 11 Physics — E2.3

What you will learn

1. Prerequisite bridge: distance, time, and wave features

2. Theoretical and experimental wave speed

3. Comparing values carefully

4. Planning a measurement and checking quality

1. Calculate theoretical wave speed

2. Calculate experimental wave speed

3. Compare theoretical and experimental values

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