Respuesta :
We have a wave function: D(y,t) and we want to know some things about it. 1. The direction the wave is travelling is negative y direction or -y. 2. Since sound waves are longitudinal waves, this sound wave is oscillating along the y axis. 3. The wavelength we can get from k=2π/λ, k is the wave number, λ is the wavelength. So λ=2π/k=6.28/8.96=0.7 m. 4. Before i get the wave speed i will calculate the period of oscillation. It can be calculated from: ω=2πf where ω is angular frequency and f is wave frequency. So f=ω/2π=3140/6.28=500 Hz and the period is T=1/f=1/500=0.002 s. 5. Wave speed is v=λ*f= 0.7*500=350 m/s.
(a). The direction of propagation of the sound wave is the negative direction of Y-axis.
(b). Oscillation of sound waves is along the Y-axis.
(c). The wavelength of the sound wave is [tex]\boxed{0.70\text{ m}}[/tex].
(d). The wave speed is [tex]\boxed{350\text{ m/s}}[/tex].
(e). The time period of oscillation is [tex]\boxed{0.002\text{ s}}[/tex].
Further explanation:
Part (a)
Given:
A sound wave is described by [tex]D({y,t})=({0.0200\,{\text{mm}}}){\text{ sin}}\left[ {({8.96\,{\text{rad/m}}})y+({3140\,{\text{rad/s}}})t+\dfrac{\pi}{4}\,{\text{rad}}}\right][/tex]
Here, [tex]y[/tex] is the position (in meter) and [tex]t[/tex] is the time (in sec).
Concept:
The standard equation of wave travelling in a particular direction is given by:
[tex]D({y,t})=A\sin(ky+\omega t+\phi)[/tex]
Compare the given equation with the standard expression of a travelling wave:
[tex]\begin{aligned}A&=0.0200\text{ mm}\\k&=8.96\text{ rad/m}}\\\omega&=3140\text{ rad/s}}\\\phi&=\dfrac{\pi}{4}\\\end{aligned}[/tex]
In the displacement equation if [tex]\omega t[/tex] and [tex]ky[/tex] have the same sign then the direction of wave propagation will be negative direction of Y-axis.
And if [tex]\omega t[/tex] and [tex]ky[/tex] have different sign, then the direction of propagation of wave will be in the positive direction of Y-axis.
Here the coefficient [tex]\omega t[/tex] and [tex]ky[/tex] have the same sign so the direction of propagation of sound wave is in the negative direction of Y axis.
Part (b)
The wave expressed in the equation is a sound wave and thesound waves are the longitudional waves.
The longitudional waves are the waves in which the oscillation of the waves particles occur in the direction of the propagation of the wave.
On comparison of the give wave equation, it states that the direction of propagation of the wave is along the directon of Y-axis. So, the sound wave is oscillating along the Y-axis.
Thus, the direction of oscillation of the wave is along the Y-axis.
Part (c)
The wavelength can be obtained from the expression given below:
[tex]k=\dfrac{{2\pi}}{\lambda}[/tex]
Here, [tex]k[/tex] represents wave number and [tex]\lambda[/tex] represents the wavelength.
Rearrange the above expression for [tex]\lambda[/tex].
[tex]\lambda=\dfrac{2\pi}{k}[/tex]
Substitute [tex]8.96\text{ rad/m}[/tex] for [tex]k[/tex] in the above expression.
[tex]\begin{aligned}\lambda&=\dfrac{2\pi}{8.96}\text{ m}\\&=0.70\text{ m}\end{aligned}[/tex]
Thus, the wavelength of the wave is [tex]\boxed{0.70\text{ m}}[/tex].
Part (d)
In order to calculate the speed of the wave we have to find the wave frequency.
The wave frequency of the wave can be determined by:
[tex]f=\dfrac{\omega}{2\pi}[/tex]
Substitute [tex]3140\text{ rad/s}[/tex] for [tex]\omega[/tex] in the above expression.
[tex]\begin{aligned}f&=\dfrac{3140}{2\pi}\\&= 500\,{\text{Hz}}\end{aligned}[/tex]
Now speed of wave is calculated by using the expression given below.
[tex]V=f\cdot \lambda[/tex]
Substitute [tex]500\,{\text{Hz}}[/tex] for [tex]f[/tex] and [tex]0.7\,{\text{m}}[/tex] for [tex]\lambda[/tex] in the above expression.
[tex]\begin{aligned}V&=500\text{ Hz}\times0.70\text{ m}\\&=350\,{\text{m/s}}\end{aligned}[/tex]
Thus, the wave speed for the given wave is [tex]\boxed{350\text{ m/s}}[/tex].
Part (e)
The time period of oscillation of wave is the reciprocal of the wave frequency.
The time period of oscillation of wave is expressed as:
[tex]T=\dfrac{1}{f}[/tex]
Substitute [tex]500\,{\text{Hz}}[/tex] for [tex]f[/tex] in the above expression.
[tex]T = 0.002\,{\text{s}}[/tex]
Thus, the time period of oscillation of the wave is [tex]\boxed{0.002\text{ s}}[/tex]
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Answer details:
Grade: College
Subject: Physics
Chapter: Wave motion
Keywords:
Sound wave, D(y,t), 0.0200 mm, sin, (8.96 rad/m)y, (3140 rad/s)t, pi/4 rad, y in m, t in s, direction of wave travelling, axis of oscillation, wavelength, wave speed, period, negative y direction, -y, y axis, 0.7m, 500 hz, 0.002s, 350 m/s.
