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Chapter 14: Problem 25

A simple harmonic wave of wavelength \(16 \mathrm{cm}\) and amplitude \(2.5 \mathrm{cm}\) is propagating along a string in the negative \(x\) -direction at \(35 \mathrm{cm} / \mathrm{s} .\) Find its (a) angular frequency and (b) wave number. (c) Write a mathematical expression describing the displacement y of this wave (in centimeters) as a function of position and time. Assume the displacement at \(x=0\) is a maximum when \(t=0\)

Short Answer

Expert verified
(a) The angular frequency is approximately \(13.746 \mathrm{s}^{-1}\). (b) The wave number is approximately \(0.3927 \mathrm{cm}^{-1}\). (c) The wave displacement can be represented by the equation \(y(x,t) = 2.5\sin(0.3927x + 13.746t )\) centimeters.

Step by step solution

01

Calculation of Frequency

First, find the frequency, \(f\), of the harmonic wave. The speed, \(v\), of a wave is given by the product of its wavelength, \(λ\), and its frequency, \(f\). Therefore, rearranging \(v = λf\) we get \(f = v/λ\). Substituting the given values: \(f = 35\mathrm{cm}/s)/(16\mathrm{cm}) = 2.1875 \mathrm{s}^{-1}\) (approximate to the third decimal place).
02

Calculation of Angular Frequency

Next, calculate the angular frequency, \(ω\), using the relationship \(ω = 2πf\). Substituting the calculated frequency: \(ω = 2π*2.1875 \mathrm{s}^{-1} = 13.746 \mathrm{s}^{-1}\) (approximate to the third decimal place). This is the angular frequency of the wave.
03

Calculation of Wave Number

Calculate the wave number, \(k\), which is given by \(k = 2π/λ\). By substituting the given wavelength: \( k = 2π/(16 \mathrm{cm}) = 0.3927 \mathrm{cm}^{-1}\) (approximate to the third decimal place). This is the wave number of the wave.
04

Write the Wave Equation

Now the wave equation that describes the displacement, \(y\), as a function of position, \(x\), and time, \(t\), can be written. For this wave traveling in the negative x direction, the equation is \(y(x,t) = A\sin(kx - ωt + φ)\). Given that the wave displacement is maximum at \(x=0\) and \(t=0\), this refers to a maximum in the sinusoidal function where the phase angle, \(φ\), is zero. Therefore, the equation becomes \(y(x,t) = 2.5\sin(0.3927x + 13.746t )\) centimeters.

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