Course Alignment

This Algebra-Based Focused 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 focused stages in order. Each stage reinforces your understanding and prepares you for a final confidence check.

6 stages • Approximately 15–20 minutes

Warm-Up Check

Before you begin, take a minute to reactivate what you already know. This quick warm-up will help you focus on the most important ideas before moving on.

ACTIVITY 1

Key Ideas

Recall the component definitions of velocity and acceleration. A wave travels at 24 m/s with wavelength 3.0 m. Find its frequency.

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

ACTIVITY 2

Common Mistakes

Identify the relationship that controls the wave quantity. A 5.0 Hz oscillator drives a wave. What is the period?

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

ACTIVITY 3

Quick Application

Distinguish circular speed from circular velocity. If string tension increases by a factor of 4 while μ is unchanged, how does the wave speed change?

WARM-UP • ACTIVITY 3 Quick Application 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.

Key relation: Wave speed depends on the restoring and inertial properties of the medium.

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.

Key relation: Wave speed depends on the restoring and inertial properties of the medium.

KEY CONCEPT 3

How the Focused Ideas Connect

Describe the wave, identify what the medium fixes, then apply the appropriate relation.

Strategy: representation → relation → calculation → physical interpretation.

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

Apply the reinforced ideas to two representative wave situations.

PRACTICE 1

Guided Example

Use independent components connected by one elapsed time. A wave has f = 12 Hz and travels at 36 m/s. Find λ and T.

PRACTICE 1 Guided 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

Independent Check

Add the velocities in a common coordinate system. A 0.80 kg string is 20 m long and held at 72 N tension. Find the wave speed.

PRACTICE 2 Independent Check 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

Focused Setup Strategy

State the known wave quantities and identify whether the medium or source controls each one before substituting numbers.

Then check units and whether the result is physically consistent with the medium.

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

Now confirm the two highest-leverage ideas without looking back.

QUICK CHECK 1

Wave Relationship Check

Identify velocity and acceleration at the top of the path. A string-wave speed is 30 m/s. If the tension is multiplied by 9 at the same μ, what is the new speed?

CONFIDENCE CHECK 1 Wave Relationship Check 90 m/s. WHY IT WORKS v ∝ √F_T, so multiplying tension by 9 multiplies speed by 3.

QUICK CHECK 2

Circular and Relative Directions

State both directions explicitly. A 20 Hz source sends waves into two strings. The second string supports twice the wave speed. Compare their wavelengths.

CONFIDENCE CHECK 2 Circular and Relative Directions 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

Interpret Your Focused Check

Use the two results above to decide whether to continue or revisit the matching core concept.

READINESS GUIDE 2 correct: You’re ready to continue. • 1 correct: Review the missed idea, then continue. • 0 correct: Revisit Core Concepts or choose more practice.

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 most transferable wave rules.

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

Medium and source roles stay distinct

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

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

Before moving on, take one final look at the essential ideas you’ll want to remember.

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