Course Alignment

This Calculus-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. For y(x,t) = 0.020 sin(4x − 12t), identify k and ω.

WARM-UP • ACTIVITY 1 Recall Activity 1 k = 4 rad/m and ω = 12 rad/s. WHY IT WORKS Compare the function with y = A sin(kx − ωt + φ).

ACTIVITY 2

Recall Activity 2

Use the defining relation and keep units consistent. Using k = 4 rad/m and ω = 12 rad/s, find the wave speed.

WARM-UP • ACTIVITY 2 Recall Activity 2 v = ω/k = 3.0 m/s. WHY IT WORKS A point of constant phase moves so that kx − ωt is constant.

ACTIVITY 3

Recall Activity 3

Connect a qualitative change in the medium to the wave speed. For y = A sin(kx − ωt), what derivative gives the transverse velocity of a string element?

WARM-UP • ACTIVITY 3 Recall Activity 3 ∂y/∂t. WHY IT WORKS The element’s position changes in time at fixed x.

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

Sinusoidal Traveling Waves and Phase

Write a traveling wave as y(x,t) = A sin(kx − ωt + φ). The wave number k measures spatial phase change and angular frequency ω measures temporal phase change. Constant phase gives the propagation speed. EXAMPLE For y = 0.030 sin(5x − 20t), λ = 2π/5 = 1.26 m, f = 20/(2π) = 3.18 Hz, and v = 4.0 m/s. SENSEI NOTE The sign of the kx and ωt terms determines propagation direction; kx − ωt moves in +x.

Review the physical meaning before applying the equation.

KEY CONCEPT 2

String Dynamics and Local Particle Motion

The wave speed on a stretched string follows from tension and linear density. The wave function also lets you calculate the transverse velocity and acceleration of a particular string element by partial differentiation at fixed position. EXAMPLE For y = A sin(kx − ωt), u_y = −Aω cos(kx − ωt) and a_y = −ω²y. SENSEI NOTE The local transverse particle speed is not the same quantity as the wave propagation speed ω/k.

Review the physical meaning before applying the equation.

KEY CONCEPT 3

Wave Equation and Energy Transport

A sinusoidal traveling wave satisfies the one-dimensional wave equation. Time derivatives give local particle motion, while spatial derivatives describe the shape. For a string, the average power carried by a sinusoidal wave follows from the local force and transverse velocity. EXAMPLE For y = A sin(kx − ωt), both second derivatives reproduce y, requiring v² = ω²/k². SENSEI NOTE The wave equation links the spatial curvature of the waveform to its temporal acceleration.

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. For y = 0.040 sin(2.5x − 15t), find λ, f, and wave speed.

PRACTICE 1 Worked Example λ = 2π/2.5 = 2.51 m; f = 15/(2π) = 2.39 Hz; v = 15/2.5 = 6.0 m/s. WHY IT WORKS Read k and ω directly, then use their definitions and v = ω/k.

PRACTICE 2

Guided Problem

Identify the medium quantities before applying the propagation-speed relation. For the same wave, find the maximum transverse speed of a string element.

PRACTICE 2 Guided Problem u_max = Aω = (0.040)(15) = 0.60 m/s. WHY IT WORKS Differentiate with respect to time and take the maximum magnitude of the cosine factor.

PRACTICE 3

Independent Problem

Connect the wave model to energy transport or a change in conditions. A string wave has μ = 0.015 kg/m, A = 0.020 m, f = 25 Hz, and v = 40 m/s. Find Pavg.

PRACTICE 3 Independent Problem ω = 157 rad/s; Pavg = (1/2)(0.015)(157²)(0.020²)(40) ≈ 2.96 W. WHY IT WORKS Convert frequency to angular frequency before using the sinusoidal-wave power expression.

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 wave is y = A sin(kx + ωt). Which direction does it propagate?

QUICK CHECK 1 Read a Wave Description In the −x direction. WHY IT WORKS Constant phase kx + ωt = constant gives x = −(ω/k)t + constant.

QUICK CHECK 2

Check the Medium–Source Distinction

Choose the correct statement and justify it physically. For a sinusoidal wave, how is local transverse acceleration related to displacement?

QUICK CHECK 2 Check the Medium–Source Distinction a_y = −ω²y. WHY IT WORKS Two time derivatives of a sinusoid reproduce the displacement with a factor −ω².

QUICK CHECK 3

Connect Propagation and Local Motion

State the relevant physical distinction clearly. For y = A sin(kx − ωt), show the ratio (∂²y/∂t²)/(∂²y/∂x²).

QUICK CHECK 3 Connect Propagation and Local Motion The ratio is ω²/k² = v². WHY IT WORKS Both derivatives are proportional to −y, leaving only the squared coefficients.

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

Read Physics Directly From Phase

From y = A sin(kx ∓ ωt + φ), identify k, ω, direction, λ, f, and v = ω/k.

KEY TAKEAWAY 2

Separate Propagation From Particle Motion

Use ω/k for wave speed, but use time derivatives of y(x,t) for the motion of individual medium elements.

KEY TAKEAWAY 3

The Wave Equation Encodes Propagation

A traveling sinusoid satisfies ∂²y/∂x² = (1/v²)∂²y/∂t², with v = ω/k.

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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