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

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552당신이 사용하는 게 아니라요?교과서 변경

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Young & Freedman Calc 15th Edition

Ch 15: Mechanical Waves

문제 26de

15장, 문제 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;

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

관련 실천

교과서 질문

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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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?

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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