Course Alignment

This Algebra-Based Full Unit Review covers Mechanical Waves and Wave Speed and complements OpenStax University Physics Volume 1, Chapter 16.

BEST USED

✓ After studying the unit

✓ Before starting homework

✓ Before a quiz or exam

Your Review Plan

Complete these six stages in order. Each stage builds on the previous one and prepares you for the final readiness check.

6 stages • Approximately 45–60 minutes

Warm-Up Check

Before you begin, take a moment to see what you already remember. Do not worry about getting everything right. This is only a starting point.

ACTIVITY 1

Recall Activity 1

Identify the relevant wave quantities and connect them using the simplest wave relation. A wave travels at 24 m/s with wavelength 3.0 m. Find its frequency.

WARM-UP • ACTIVITY 1 Recall Activity 1 f = v/λ = 24/3.0 = 8.0 Hz. WHY IT WORKS Use v = fλ and solve for the unknown wave quantity.

ACTIVITY 2

Recall Activity 2

Use the defining relation and keep units consistent. A 5.0 Hz oscillator drives a wave. What is the period?

WARM-UP • ACTIVITY 2 Recall Activity 2 T = 1/f = 0.200 s. WHY IT WORKS Frequency and period are reciprocals.

ACTIVITY 3

Recall Activity 3

Connect a qualitative change in the medium to the wave speed. If string tension increases by a factor of 4 while μ is unchanged, how does the wave speed change?

WARM-UP • ACTIVITY 3 Recall Activity 3 The speed doubles. WHY IT WORKS For a string v = √(F_T/μ), so v scales with the square root of tension.

Ready to strengthen your understanding?

You've refreshed what you already know. Next, you'll reinforce the essential concepts that will help you solve problems with confidence. Need to see the learning path again?

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Core Concepts

Core Concepts

KEY CONCEPT 1

Wave Variables and Traveling-Wave Speed

A periodic mechanical wave is described by amplitude A, wavelength λ, frequency f, and period T. Frequency is set by the source. The medium determines the propagation speed. Once v and f are known, λ follows from v = fλ. EXAMPLE A 15 Hz wave traveling at 45 m/s has λ = 3.0 m. SENSEI NOTE When a wave enters a new medium, frequency remains tied to the source while speed and wavelength can change.

Review the physical meaning before applying the equation.

KEY CONCEPT 2

Wave Speed on a Stretched String

For a stretched string, the wave speed increases with tension F_T and decreases with linear mass density μ = m/L. This gives a direct algebraic model for how the physical properties of the medium control propagation. EXAMPLE For F_T = 100 N and μ = 0.040 kg/m, v = √(2500) = 50 m/s. SENSEI NOTE Use tension in newtons and μ in kilograms per meter before applying the square root.

Review the physical meaning before applying the equation.

KEY CONCEPT 3

Energy and Power in Sinusoidal String Waves

A sinusoidal wave on a string carries energy because each string element has kinetic and elastic potential energy. The average power increases with string density, angular frequency, wave-speed, and the square of amplitude. EXAMPLE Doubling amplitude on the same string at the same frequency multiplies the average power by four. SENSEI NOTE Power depends on A², so amplitude changes have a strong effect on energy transport.

Review the physical meaning before applying the equation.

Ready to apply these ideas?

You've reinforced the essential concepts. Now it's time to put them into practice by working through guided examples and building your problem-solving confidence. Need a quick reminder?

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Guided Practice

Now combine the wave concepts in representative college-physics problems.

PRACTICE 1

Worked Example

Use the central wave relations and show each algebraic step. A wave has f = 12 Hz and travels at 36 m/s. Find λ and T.

PRACTICE 1 Worked Example λ = 36/12 = 3.0 m; T = 1/12 = 0.0833 s. WHY IT WORKS Use v = fλ for space and T = 1/f for time.

PRACTICE 2

Guided Problem

Identify the medium quantities before applying the propagation-speed relation. A 0.80 kg string is 20 m long and held at 72 N tension. Find the wave speed.

PRACTICE 2 Guided Problem μ = 0.80/20 = 0.040 kg/m; v = √(72/0.040) = 42.4 m/s. WHY IT WORKS Compute the linear mass density first, then use the string-speed relation.

PRACTICE 3

Independent Problem

Connect the wave model to energy transport or a change in conditions. A string has μ = 0.020 kg/m, tension 50 N, and supports a 10 Hz wave of amplitude 0.030 m. Find v and Pavg.

PRACTICE 3 Independent Problem v = 50 m/s; ω = 62.8 rad/s; Pavg ≈ 1.78 W. WHY IT WORKS First use v = √(F_T/μ), then substitute into Pavg = (1/2) μω²A²v.

Ready to check your understanding?

You've practiced the essential skills with guidance. Now it's time to solve a few short problems on your own and confirm you're ready to move forward. Need a quick reminder?

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Confidence Check

Try these short questions without looking back. Your goal is reliable reasoning, not speed.

QUICK CHECK 1

Read a Wave Description

State the relationship first, then evaluate. A string-wave speed is 30 m/s. If the tension is multiplied by 9 at the same μ, what is the new speed?

QUICK CHECK 1 Read a Wave Description 90 m/s. WHY IT WORKS v ∝ √F_T, so multiplying tension by 9 multiplies speed by 3.

QUICK CHECK 2

Check the Medium–Source Distinction

Choose the correct statement and justify it physically. A 20 Hz source sends waves into two strings. The second string supports twice the wave speed. Compare their wavelengths.

QUICK CHECK 2 Check the Medium–Source Distinction The second wavelength is twice as large. WHY IT WORKS The source frequency stays 20 Hz, so λ = v/f is proportional to wave speed.

QUICK CHECK 3

Connect Propagation and Local Motion

State the relevant physical distinction clearly. If amplitude triples while all other string-wave quantities stay fixed, by what factor does average power change?

QUICK CHECK 3 Connect Propagation and Local Motion By a factor of 9. WHY IT WORKS Pavg is proportional to A².

How did it go?

You've checked your understanding. Take one final look at the essential ideas before deciding what to do next. Need a quick reminder?

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Summary

Before moving on, take one final look at the wave ideas that should accompany every later calculation.

KEY TAKEAWAY 1

Use the Source–Medium Separation

The source fixes frequency; the medium fixes wave speed; wavelength adjusts through v = fλ.

KEY TAKEAWAY 2

Connect String Properties to Speed

On a stretched string, v = √(F_T/μ): more tension raises speed, while greater linear density lowers it.

KEY TAKEAWAY 3

Amplitude Controls Energy Transport Strongly

For sinusoidal string waves, average power scales as A² and also depends on frequency and medium properties.

Ready for your next step?

You've reviewed the essential ideas one last time. Now choose the resource that best matches how confident you feel. Need a quick reminder?

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Next Step

FR-07 • CONFIDENCE CHECK CONTINUED

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