Ch 16: Traveling Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 16: Traveling Waves

Problem 45c

Chapter 16, Problem 45c

FIGURE P16.45 is a snapshot graph at t = 0 s of a 5.0 Hz wave traveling to the left. Write the displacement equation for this wave.

Verified step by step guidance

Step 1: Analyze the graph to determine the amplitude (A), wavelength (λ), and frequency (f) of the wave. From the graph, the amplitude is 2 mm (maximum displacement from equilibrium), the wavelength is 2 m (distance between two consecutive peaks), and the frequency is given as 5.0 Hz.

Step 2: Use the relationship between frequency (f) and angular frequency (ω) to calculate ω. The formula is ω = 2πf. Substituting f = 5.0 Hz, ω = 2π × 5.0 rad/s.

Step 3: Use the relationship between wavelength (λ) and wave number (k) to calculate k. The formula is k = 2π/λ. Substituting λ = 2 m, k = 2π/2 rad/m.

Step 4: Write the general displacement equation for a traveling wave moving to the left. The equation is y(x, t) = A sin(kx + ωt). Since the wave is traveling to the left, the phase term includes a positive sign for ωt.

Step 5: Substitute the values of A, k, and ω into the displacement equation. A = 2 mm, k = π rad/m, and ω = 10π rad/s. The displacement equation becomes y(x, t) = 2 sin(πx + 10πt) mm.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Wave Equation

The wave equation describes the displacement of a wave as a function of position and time. For a sinusoidal wave, it is typically expressed as y(x, t) = A sin(kx - ωt + φ), where A is the amplitude, k is the wave number, ω is the angular frequency, and φ is the phase constant. This equation allows us to model how the wave propagates through space and time.

Frequency and Angular Frequency

Frequency (f) is the number of cycles a wave completes in one second, measured in Hertz (Hz). Angular frequency (ω) is related to frequency by the equation ω = 2πf. In this case, with a frequency of 5.0 Hz, the angular frequency can be calculated as ω = 2π(5.0) rad/s, which is essential for determining the wave's behavior over time.

Wave Number

The wave number (k) is defined as the number of wavelengths per unit distance and is given by the formula k = 2π/λ, where λ is the wavelength. The wavelength can be determined from the graph by measuring the distance between successive peaks. Understanding the wave number is crucial for writing the displacement equation accurately, as it relates to the spatial characteristics of the wave.

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