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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 \(50.0-\mathrm{cm}\) -long wire with a mass of \(10.0 \mathrm{~g}\) is under a tension of \(50.0 \mathrm{~N}\). Both ends of the wire are held rigidly while it is plucked. a) What is the speed of the waves on the wire? b) What is the fundamental frequency of the standing wave? c) What is the frequency of the third harmonic?

The speed of light waves in air is greater than the speed of sound in air by about a factor of a million. Given a sound wave and a light wave of the same wavelength, both traveling through air, which statement about their frequencies is true? a) The frequency of the sound wave will be about a million times greater than that of the light wave. b) The frequency of the sound wave will be about a thousand times greater than that of the light wave. c) The frequency of the light wave will be about a thousand times greater than that of the sound wave. d) The frequency of the light wave will be about a million times greater than that of the sound wave. e) There is insufficient information to determine the relationship between the two frequencies.

Consider a guitar string stretching \(80.0 \mathrm{~cm}\) between its anchored ends. The string is tuned to play middle \(\mathrm{C},\) with a frequency of \(256 \mathrm{~Hz}\), when oscillating in its fundamental mode, that is, with one antinode between the ends. If the string is displaced \(2.00 \mathrm{~mm}\) at its midpoint and released to produce this note, what are the wave speed, \(v\), and the maximum speed, \(V_{\text {max }}\), of the midpoint of the string?

If transverse waves on a string travel with a velocity of \(50 \mathrm{~m} / \mathrm{s}\) when the string is under a tension of \(20 \mathrm{~N},\) what tension on the string is required for the waves to travel with a velocity of \(30 \mathrm{~m} / \mathrm{s} ?\) a) \(7.2 \mathrm{~N}\) c) \(33 \mathrm{~N}\) e) \(45 \mathrm{~N}\) b) \(12 \mathrm{~N}\) d) \(40 \mathrm{~N}\) f) \(56 \mathrm{~N}\)

Which of the following transverse waves has the greatest power? a) a wave with velocity \(v\), amplitude \(A\), and frequency \(f\) b) a wave of velocity \(v\), amplitude \(2 A\), and frequency \(f / 2\) c) a wave of velocity \(2 v\), amplitude \(A / 2\), and frequency \(f\) d) a wave of velocity \(2 v\), amplitude \(A\), and frequency \(f / 2\) e) a wave of velocity \(v\), amplitude \(A / 2\), and frequency \(2 f\)
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