/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 A wave traveling along the \(x\)... [FREE SOLUTION] | 91影视

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Chapter 16: Problem 24

A wave traveling along the \(x\) axis is described mathematically by the equation \(y=0.17 \sin (8.2 \pi t+0.54 \pi x),\) where \(y\) is the displacement (in meters), \(t\) is in seconds, and \(x\) is in meters. What is the speed of the wave?

Short Answer

Expert verified
The wave speed is approximately 15.19 m/s.

Step by step solution

01

Identify the Wave Equation Components

The equation given is \( y = 0.17 \sin(8.2 \pi t + 0.54 \pi x) \). Here, the general form for a traveling wave is \( y = A \sin(\omega t + kx) \). We can identify \( \omega = 8.2 \pi \) and \( k = 0.54 \pi \).
02

Relate \(\omega\) and \(k\) to Wave Speed

The wave speed \( v \) is related to the angular frequency \( \omega \) and the wave number \( k \) by the formula \( v = \frac{\omega}{k} \).
03

Substitute Known Values into the Speed Formula

Substitute \( \omega = 8.2 \pi \) and \( k = 0.54 \pi \) into the wave speed formula: \[ v = \frac{8.2 \pi}{0.54 \pi}. \] Simplify the expression by canceling out \( \pi \): \[ v = \frac{8.2}{0.54}. \]
04

Calculate the Wave Speed

Perform the division: \[ v = \frac{8.2}{0.54} \approx 15.19. \] Thus, the wave speed is approximately 15.19 meters per second.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Equation
A wave equation is a mathematical way to describe how a wave behaves along a medium. In our exercise problem, the wave equation given is: \[y = 0.17 \sin(8.2 \pi t + 0.54 \pi x) \]Here's what each part signifies:
  • Amplitude \((A)\): This is the maximum displacement the wave achieves, given by the coefficient in front of the sine function, 0.17 meters in this case.
  • Angular Frequency \((\omega)\): Represented by \(8.2 \pi\), it shows how many oscillations occur per second.
  • Wave Number \((k)\): Identified as \(0.54 \pi\), it represents how many wave cycles are in one meter.
When we're asked to determine properties like wave speed, understanding these components within the context of the wave equation is crucial.
Angular Frequency
Angular frequency helps us understand the speed of oscillation within a wave. Expressed in radians per second, it informs about the rapidity of wave cycles. In the equation:\[\omega = 8.2 \pi\]The angular frequency tells us how quickly the wave completes a cycle over time.
  • Relationship to Frequency: You can convert angular frequency to normal frequency \((f)\), measured in hertz (Hz), using the formula \( f = \frac{\omega}{2\pi} \).
  • Practical Example: For \(\omega = 8.2 \pi\), substituting into the formula gives \( f \approx 4.1 \text{ Hz}\).
Understanding angular frequency is essential to grasp how components of a wave contribute to its oscillatory behavior.
Wave Number
The wave number \((k)\) is a measure of the spatial frequency of a wave, indicating how many wave cycles exist in a unit distance. In our exercise:\[k = 0.54 \pi\]Higher wave numbers indicate more cycles per meter, meaning the wave is denser.
  • Relation to Wavelength: It connects directly to the wavelength \((\lambda)\) by the expression \( k = \frac{2\pi}{\lambda} \). This shows wavelengths inversely relate to wave numbers.
  • Example Calculation: For \(k = 0.54 \pi\), calculate the wavelength using \( \lambda = \frac{2\pi}{k} \), resulting in a longer wavelength for a lower wave number.
Wave number is pivotal to understanding how stretched or compact a wave appears.
Traveling Waves
Traveling waves are waves that move through a medium, carrying energy from one point to another. The wave function, shown in our exercise, is a typical representation of traveling waves:\[y = 0.17 \sin(8.2 \pi t + 0.54 \pi x)\]Key characteristics include:
  • Direction of Movement: A positive sign before the \(kx\) term indicates it is moving along the positive x-axis direction.
  • Wave Speed: Calculated using the formula \(v = \frac{\omega}{k}\). For our exercise, the calculated speed is approximately 15.19 meters per second.
Traveling waves matter because they provide insight into the transmission of energy in environments like air or water, showing how disturbances propagate.

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