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: (d) wave speed; (e) direction the wave is traveling;

Ch 15: Mechanical Waves

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 15: Mechanical Waves

Problem 26de

Chapter 15, Problem 26de

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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To find the wave speed, we need to use the formula for wave speed $ v $, which is given by $ v = \frac{\omega}{k} $, where $ \omega $ is the angular frequency and $ k $ is the wave number. From the wave function $ y(x, t) = 2.30 \text{ mm} \cos[(16.98 \text{ rad/m})x + (742 \text{ rad/s})t] $, we identify $ \omega = 742 \text{ rad/s} $ and $ k = 16.98 \text{ rad/m} $.

Substitute the values of $ \omega $ and $ k $ into the wave speed formula: $ v = \frac{742 \text{ rad/s}}{16.98 \text{ rad/m}} $. This will give you the wave speed in meters per second.

To determine the direction the wave is traveling, examine the sign of the terms in the wave function. The wave function is $ y(x, t) = 2.30 \text{ mm} \cos[(16.98 \text{ rad/m})x + (742 \text{ rad/s})t] $. The positive sign in front of $ t $ indicates that the wave is traveling in the negative x-direction.

The wave speed can also be verified using the physical properties of the rope. The speed of a wave on a string is given by $ v = \sqrt{\frac{T}{\mu}} $, where $ T $ is the tension in the rope and $ \mu $ is the linear mass density. Calculate $ \mu $ using $ \mu = \frac{\text{mass}}{\text{length}} = \frac{0.00338 \text{ kg}}{1.35 \text{ m}} $.

If the tension $ T $ is known or can be measured, substitute $ \mu $ and $ T $ into the formula $ v = \sqrt{\frac{T}{\mu}} $ to verify the wave speed calculated from the wave function.

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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) = 2.30mm cos[(16.98 rad/m)x + (742 rad/s)t] describes the displacement of the wave at any position x and time t. It is a mathematical representation of the wave's oscillation, where the cosine function indicates a harmonic wave, and the coefficients provide information about amplitude, wave number, and angular frequency.

Wave Speed

Wave speed is the rate at which a wave propagates through a medium. It can be calculated using the formula v = ω/k, where ω is the angular frequency (742 rad/s) and k is the wave number (16.98 rad/m). This relationship shows how the frequency and wavelength of the wave determine its speed, which is crucial for understanding wave dynamics.

Direction of Wave Travel

The direction of wave travel is determined by the sign of the terms in the wave function. In y(x, t) = 2.30mm cos[(16.98 rad/m)x + (742 rad/s)t], the positive sign between the wave number and angular frequency indicates the wave is traveling in the negative x-direction. This concept helps in visualizing the movement of the wave along the rope.

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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 $y (x, t) = (2.30 mm) \cos [(16.98 r a d / m) x + (742 r a d / s) t]$ . 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 jet plane at takeoff can produce sound of intensity 10.0 W/m² at 30.0 m away. But you prefer the tranquil sound of normal conversation, which is 1.0 μW/m². Assume that the plane behaves like a point source of sound. (a) What is the closest dis-tance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

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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: (a) amplitude; (b) frequency; (c) wavelength.

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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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