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 14th Edition

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Young & Freedman Calc 14th Edition

Ch 15: Mechanical Waves

Problema 28de

Capitolo 15, Problema 28de

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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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.

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Percorso guidato

08:30

Intro to Wave Functions

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.

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Percorso guidato

07:19

Intro to Waves and Wave Speed

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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Percorso guidato

07:19

Intro to Waves and Wave Speed

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