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)=(2.30mm)⁡cos⁡[(16.98 rad/m)x+(742 rad/s)t]y(x,t)=\left(2.30\operatorname{mm)}\cos[\left(16.98\text{ }rad/m\right)x+(742\text{ }rad/s\right)t]y(x,t)=(2.30mm)cos[(16.98 rad/m)x+(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m1.35\text{ m}1.35 m and a mass of 0.00338kg⁡0.00338\operatorname{kg}0.00338kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength.

Ch 15: Mechanical Waves

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 15: Mechanical Waves

Problem 28abc

Chapter 15, Problem 28abc

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y(x,t)=$\left$(2.30$\operatorname{mm)}\]\cos$[$\left$(16.98$\text{ }$rad/m$\right$)x+(742$\text{ }$rad/s$\right$)t]$ . Being more practical, you measure the rope to have a length of $1.35$\text{ m}$$ and a mass of $0.00338$\operatorname{kg}$$ . You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength.

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To find the amplitude (a), identify the coefficient of the cosine function in the wave equation y(x, t) = 2.30 mm cos[(16.98 rad/m)x + (742 rad/s)t]. The amplitude is the maximum displacement from the equilibrium position, which is 2.30 mm.

To determine the frequency (b), use the angular frequency ω given in the wave function, which is 742 rad/s. The frequency f can be found using the relation f = ω / (2π).

For the wavelength (c), use the wave number k, which is 16.98 rad/m. The wavelength λ is related to the wave number by the formula λ = 2π / k.

To find the wave speed (d), use the relationship between wave speed v, frequency f, and wavelength λ: v = f * λ. Calculate f and λ from the previous steps and use them to find v.

To determine the direction the wave is traveling (e), observe the sign of the terms in the wave function. The positive sign in the term (16.98 rad/m)x + (742 rad/s)t indicates the wave is traveling in the negative x-direction.

To find the tension in the rope (f), use the formula for wave speed v = sqrt(T/μ), where T is the tension and μ is the linear mass density. First, calculate μ = mass/length = 0.00338 kg / 1.35 m. Then rearrange the formula to solve for T: T = μ * v^2.

To calculate the average power transmitted by the wave (g), use the formula P_avg = (1/2) * μ * ω^2 * A^2 * v, where A is the amplitude, ω is the angular frequency, and v is the wave speed. Substitute the values obtained from previous steps into this formula.

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Key Concepts

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

Wave Function

The wave function y(x, t) = A cos(kx + ωt) describes the displacement of a wave at any position x and time t. Here, A is the amplitude, k is the wave number, and ω is the angular frequency. The wave function provides information about the wave's characteristics, such as amplitude, frequency, and direction of travel.

Wave Speed

Wave speed (v) is the rate at which a wave propagates through a medium. It is calculated using the formula v = ω/k, where ω is the angular frequency and k is the wave number. Wave speed is crucial for understanding how quickly energy is transmitted along the medium, such as a rope in this context.

Tension in the Rope

Tension in the rope affects the speed of wave propagation. For a wave on a string, the wave speed v is related to the tension T and the linear mass density μ (mass per unit length) by the formula v = sqrt(T/μ). Understanding tension is essential for calculating wave speed and analyzing wave behavior on the rope.

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Related Practice

Textbook Question

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y (x, t) = (2.30 mm) \cos [(16.98 r a d / m) x + (742 r a d / s) t]$ . 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: (f) tension in the rope; (g) average power transmitted by the wave.

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Textbook Question

Threshold of Pain. You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are 7.5 m from it, you measure its intensity to be 0.11 W/m². An intensity of 1.0 W/m² is often used as the 'threshold of pain.' How much closer to the source can you move before the sound intensity reaches this threshold?

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Textbook Question

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y(x,t)=$\left$(2.30$\operatorname{mm)}\]\cos$[$\left$(16.98$\text{ }$rad/m$\right$)x+(742$\text{ }$rad/s$\right$)t]$ . Being more practical, you measure the rope to have a length of $1.35$\text{ m}$$ and a mass of $0.00338$\operatorname{kg}$$ . You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;

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Textbook Question

At a distance of 7.00 x 10¹² m from a star, the intensity of the radiation from the star is 15.4 W/m². Assuming that the star radiates uniformly in all directions, what is the total power output of the star?

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Textbook Question

Energy Output. 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. What is the intensity at a distance of 3.1 m from the source?

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Textbook Question

Energy Output. 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. How much sound energy does the source emit in one hour if its power output remains constant?

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