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Chapter 15: Problem 24

A wave on a string has a wave function given by $$ y(x, t)=(0.0200 \mathrm{~m}) \sin \left[\left(6.35 \mathrm{~m}^{-1}\right) x+\left(2.63 \mathrm{~s}^{-1}\right) t\right] $$ a) What is the amplitude of the wave? b) What is the period of the wave? c) What is the wavelength of the wave? d) What is the speed of the wave? e) In which direction does the wave travel?

Short Answer

Expert verified
Question: Determine the properties of the wave given by the wave equation as follows: $$ y(x, t) = (0.0200~m) \sin \left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right] $$ Properties to find: a) Amplitude b) Period c) Wavelength d) Speed of the wave e) Direction of the wave Answer: a) Amplitude: \(A = 0.0200~m\) b) Period: \(T \approx 2.39~s\) c) Wavelength: \(\lambda \approx 0.990~m\) d) Speed of wave: \(v \approx 0.414~m/s\) e) Direction of wave: negative x-direction

Step by step solution

01

Identify the amplitude

From the given wave equation, $$ y(x, t) = (0.0200~m) \sin \left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right] $$ Compare this equation with the general wave equation, we can immediately determine the amplitude as: $$ A = 0.0200~m $$
02

Identify the angular frequency and period

From the given wave equation, $$ y(x, t) = (0.0200~m) \sin \left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right] $$ We can determine the angular frequency (ω) as: $$ \omega = 2.63~s^{-1} $$ To calculate the period (T) of the wave, we can use the following relation between angular frequency and period: $$ \omega = \frac{2\pi}{T} $$ Now solve for the period T: $$ T = \frac{2\pi}{\omega} = \frac{2\pi}{2.63~s^{-1}} \approx 2.39~s $$
03

Identify the wave number and wavelength

From the given wave equation, $$ y(x, t) = (0.0200~m) \sin \left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right] $$ We can determine the wave number (k) as: $$ k = 6.35~m^{-1} $$ To calculate the wavelength (λ) of the wave, we can use the following relation between wave number and wavelength: $$ k = \frac{2\pi}{\lambda} $$ Now solve for the wavelength λ: $$ \lambda = \frac{2\pi}{k} = \frac{2\pi}{6.35~m^{-1}} \approx 0.990~m $$
04

Calculate the speed of the wave

The speed of the wave (v) can be calculated using the following relation between wavelength and period: $$ v = \frac{\lambda}{T} $$ Now substitute the previously calculated values of λ and T, $$ v = \frac{0.990~m}{2.39~s} \approx 0.414~m/s $$
05

Determine the direction of the wave

From the given wave equation, $$ y(x, t) = (0.0200~m) \sin \left[ (6.35~m^{-1}) x + (2.63~s^{-1}) t \right] $$ The wave function has a positive sign between the \(kx\) and \(ωt\) terms, indicating a wave traveling in the negative x-direction. The general form with a positive sign is: $$ y(x,t) = A \sin(kx - \omega t) $$ So, the wave travels in the negative x-direction.
06

Summary

Here are the answers to the problem: a) Amplitude: \(A = 0.0200~m\) b) Period: \(T \approx 2.39~s\) c) Wavelength: \(\lambda \approx 0.990~m\) d) Speed of wave: \(v \approx 0.414~m/s\) e) Direction of wave: negative x-direction

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Most popular questions from this chapter

A \(3.00-\mathrm{m}\) -long string, fixed at both ends, has a mass of \(6.00 \mathrm{~g}\). If you want to set up a standing wave in this string having a frequency of \(300 . \mathrm{Hz}\) and three antinodes, what tension should you put the string under?

A sinusoidal wave traveling in the positive \(x\) -direction has a wavelength of \(12 \mathrm{~cm},\) a frequency of \(10.0 \mathrm{~Hz},\) and an amplitude of \(10.0 \mathrm{~cm}\). The part of the wave that is at the origin at \(t=0\) has a vertical displacement of \(5.00 \mathrm{~cm} .\) For this wave, determine the a) wave number, d) speed, b) period, e) phase angle, and c) angular frequency, f) equation of motion.

The displacement from equilibrium caused by a wave on a string is given by \(y(x, t)=(-0.00200 \mathrm{~m}) \sin \left[\left(40.0 \mathrm{~m}^{-1}\right) x-\right.\) \(\left.\left(800 . \mathrm{s}^{-1}\right) t\right] .\) For this wave, what are the (a) amplitude, (b) number of waves in \(1.00 \mathrm{~m},\) (c) number of complete cycles in \(1.00 \mathrm{~s},\) (d) wavelength, and (e) speed?

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