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A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x,t)=2.30mmcos[(6.98rad/m)x +1742rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

Short Answer

Expert verified
(a) 2.30 mm, (b) 277.3 Hz, (c) 0.9003 m, (d) 249.78 m/s, (e) negative x-direction, (f) 156.1 N, (g) 14.0 W.

Step by step solution

01

Identify the Amplitude

The amplitude of a wave is the coefficient in front of the cosine function in the wave equation. In the given wave function, the amplitude is 2.30 mm. Therefore, the amplitude is A=2.30mm.
02

Calculate the Frequency

The angular frequency ω is given by the term 1742rad/s in the wave function. The frequency f can be found using the formula f=ω2π. Therefore, f=17422π277.3Hz.
03

Determine the Wavelength

The wave number k is given by 6.98rad/m. The wavelength λ is related to the wave number by λ=2πk. Thus, λ=2π6.980.9003m.
04

Calculate the Wave Speed

The wave speed v can be calculated using the formula v=fλ. Given f277.3Hz and λ0.9003m, the wave speed is v=277.3×0.9003249.78m/s.
05

Identify the Direction of the Wave

The phase factor in the wave function includes +kx+ωt; the presence of +ωt indicates the wave is traveling in the negative x-direction.
06

Calculate the Tension in the Rope

The wave speed on a rope is given by v=Tμ, where μ is the linear mass density mL. With m=0.00338kg and L=1.35m, μ=0.003381.350.002504kg/m. Then, solving v=249.78=T0.002504 gives Tμv2156.1N.
07

Determine the Average Power Transmitted by the Wave

The average power P transmitted by a wave traveling along a string is given by P=12μω2A2v. Using μ0.002504kg/m, ω=1742rad/s, A=0.0023m, and v249.78m/s, we compute P=12(0.002504)(17422)(0.0023)2(249.78)14.0W.

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Key Concepts

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

Wave Function
A wave function is a mathematical representation of a wave. It describes how the wave travels through space and time. In this case, the wave function provided is y(x,t)=2.30mmcos[(6.98rad/m)x+1742rad/st)].
The wave function typically includes parameters like amplitude, angular wave number k, and angular frequency ω.
Understanding each component in the wave function helps in analyzing the wave's various properties, such as its direction and speed. For instance, the cosine function indicates this is a periodic wave, which means it repeats itself at regular intervals.
  • The coefficient 2.30mm is the amplitude, representing the maximum displacement from rest.
  • The term 6.98rad/m is the wave number, which relates to the wavelength.
  • The 1742rad/s term is the angular frequency, relating to the frequency.
Amplitude
Amplitude is a fundamental characteristic of waves. It represents the maximum extent of a wave's displacement from its equilibrium position.
In simple terms, it is how far the wave "reaches" or "grows" from its baseline up to the top-most point known as the wave crest.
In the wave equation y(x,t)=2.30mmcos[(6.98rad/m)x+1742rad/s)t], the amplitude is given by the coefficient in front, which is 2.30mm.
  • This parameter indicates the energy of the wave; a larger amplitude means the wave carries more energy.
  • Amplitude does not affect the speed of the wave; it solely impacts the energy level and intensity.
Wave Speed
Wave speed is a critical concept in wave motion, defining how fast a wave propagates through a medium.
You determine wave speed by multiplying the frequency of the wave f by its wavelength λ.
Mathematically expressed as v=fλ. For the given wave, with frequency 277.3Hz and wavelength 0.9003m, the wave speed is
  • 249.78m/s.
The speed of a wave depends on the medium through which it travels.
For instance, waves move faster in less dense materials and slower in denser ones. Understanding wave speed is vital for numerous practical applications, from engineering to communication.
Frequency
Frequency refers to how often the wave oscillates or completes a cycle in one second.
It is measured in Hertz (Hz), where one Hertz equals one cycle per second.
From the angular frequency ω provided in the wave equation, ω=1742rad/s, you can determine the frequency using the formula:
  • f=ω2π.
For this example, the frequency is approximately 277.3Hz.
Frequency is inversely related to the period of the wave, which is the time taken to complete one cycle.
  • Higher frequencies imply shorter periods and vice versa.
A higher frequency indicates more cycles per unit time, which means more energy in the wave.
Wavelength
Wavelength is the distance between successive crests (or any successive identical points) of a wave.
It is usually denoted by λ and is measured in meters (m).
Using the wave equation, the wave number k=6.98rad/m helps find the wavelength with the equation:
  • λ=2πk.
For our wave, this computes to about 0.9003m. Wavelength directly affects the wave's speed along with frequency.
It is an essential element in determining how waves interact with each other and their surrounding environment.
  • In optics, for instance, wavelengths determine the color of light.
  • In sound, they influence pitch.

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Most popular questions from this chapter

A musician tunes the C-string of her instrument to a fundamental frequency of 65.4 Hz. The vibrating portion of the string is 0.600 m long and has a mass of 14.4 g. (a) With what tension must the musician stretch it? (b) What percent increase in tension is needed to increase the frequency from 65.4 Hz to 73.4 Hz, corresponding to a rise in pitch from C to D?

By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m2 at a distance of 4.3 m from the source. (a) What is the intensity at a distance of 3.1 m from the source? (b) How much sound energy does the source emit in one hour if its power output remains constant?

A horizontal wire is stretched with a tension of 94.0 N, and the speed of transverse waves for the wire is 406 m/s. What must the amplitude of a traveling wave of frequency 69.0 Hz be for the average power carried by the wave to be 0.365 W?

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s. What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

A string with both ends held fixed is vibrating in its third harmonic. The waves have a speed of 192 m/s and a frequency of 240 Hz. The amplitude of the standing wave at an antinode is 0.400 cm. (a) Calculate the amplitude at points on the string a distance of (i) 40.0 cm; (ii) 20.0 cm; and (iii) 10.0 cm from the left end of the string. (b) At each point in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement? (c) Calculate the maximum transverse velocity and the maximum transverse acceleration of the string at each of the points in part (a).

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