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Chapter 15: Problem 18

The displacement of a wave traveling in the positive \(x\) -direction is \(y(x, t)=(3.5 \mathrm{cm}) \times \cos (2.7 x-92 t),\) where \(x\) is in \(\mathrm{m}\) and \(t\) is in \(\mathrm{s}\). What are the (a) frequency, (b) wavelength, and (c) speed of this wave?

Short Answer

Expert verified
The frequency is approximately \(14.63 \, \mathrm{Hz}\), the wavelength is approximately \(2.33 \, \mathrm{m}\), and the speed of the wave is approximately \(34.07 \, \mathrm{m/s}\)

Step by step solution

01

Identify Known Variables

From the given wave equation \(y(x, t)=(3.5 \, \mathrm{cm}) \times \cos (2.7 \, x-92 \, t)\), we can identify the amplitude \(A = 3.5 \, \mathrm{cm}\), the wave number \(k = 2.7 \, \mathrm{m^{-1}}\) and the angular frequency \(蠅 = 92 \, \mathrm{s^{-1}}\)
02

Calculate Frequency

The frequency \(f\) can be calculated as \(f = \frac{蠅}{2蟺}\). Substituting \(蠅 = 92 \, \mathrm{s^{-1}}\) gives the frequency \(f = \frac{92 \, \mathrm{s^{-1}}}{2蟺} \approx 14.63 \, \mathrm{Hz}\)
03

Calculate Wavelength

The wavelength \(位\) is given by the formula \(位 = \frac{2蟺}{k}\). Substituting \(k = 2.7 \, \mathrm{m^{-1}}\) into the formula gives the wavelength \(位 = \frac{2蟺}{2.7 \, \mathrm{m^{-1}}} \approx 2.33 \, \mathrm{m}\)
04

Calculate Speed of the Wave

The speed of the wave \(v\) is calculated by the formula \(v = \frac{蠅}{k}\). Substituting \(蠅 = 92 \, \mathrm{s^{-1}}\) and \(k = 2.7 \, \mathrm{m^{-1}}\) into the formula gives \(v = \frac{92 \, \mathrm{s^{-1}}}{2.7 \, \mathrm{m^{-1}}} \approx 34.07 \, \mathrm{m/s}\)

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Key Concepts

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

Frequency Calculation
To find the frequency of a wave, we need to understand its relation to angular frequency. Angular frequency (\(\omega\)) represents how fast something oscillates in radians per second. In the given wave equation, \(\omega\) is 92 s\(^{-1}\). Frequency (\(f\)) shows the number of cycles per second and is calculated by dividing the angular frequency by \(2\pi\).
  • Formula: \(f = \frac{\omega}{2\pi}\)
  • Substitute \(\omega = 92 \, \mathrm{s^{-1}}\)
  • Frequency: \(f \approx 14.63 \, \mathrm{Hz}\)
This means the wave oscillates around 14.63 times every second. Understanding frequency helps us know how fast the wave repeats its pattern over time.
Wavelength Determination
The wavelength of a wave is the distance over which the wave's shape repeats. From crest to crest or trough to trough, it signifies the length of one complete cycle of the wave. Wavelength (\(\lambda\)) can be found using the wave number (\(k\)). The wave number is like a spatial version of angular frequency. It indicates how many radians fit into a unit of distance. Given \(k = 2.7 \, \mathrm{m^{-1}}\), use the following formula:
  • Formula: \(\lambda = \frac{2\pi}{k}\)
  • Substitute \(k = 2.7 \, \mathrm{m^{-1}}\)
  • Wavelength: \(\lambda \approx 2.33 \, \mathrm{m}\)
This result informs us that every 2.33 meters, the wave pattern repeats. Understanding wavelength is crucial for knowing the spatial size of wave cycles.
Wave Speed
Wave speed (\(v\)) is how fast the wave travels through space. It is a combination of how frequently the wave oscillates and the distance over which it repeats. Wave speed links the frequency, wavelength, and angular frequency together. You can calculate it using this handy formula:
  • Formula: \(v = \frac{\omega}{k}\)
  • Use given values \(\omega = 92 \, \mathrm{s^{-1}}\) and \(k = 2.7 \, \mathrm{m^{-1}}\)
  • Wave Speed: \(v \approx 34.07 \, \mathrm{m/s}\)
This means the wave moves at approximately 34.07 meters per second in the medium. Understanding wave speed helps us predict how quickly a wave moves from one point to another, influencing applications in communication and transportation.

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