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1. A
harmonic wave function is given by: $E(x,t)=A\sin (kx-\omega t)$ with A, k, and ω are real constants.
a.
What
is the magnitude of the wavelength λ,
the periode T, the frequency f and the phase velocity vph?
b.
Define
the circular wavenumber k and the
angular frequency ω.
c.
If
$E(x,t)={{10}^{3}}\sin \left[ \pi \left\{ \left( 3\times {{10}^{6}}{{m}^{-1}}
\right)x-\left( 9\times {{10}^{14}}{{s}^{-1}} \right)t \right\} \right]V/m$ ,
calculate the phase velocity, the wavelength , the frequency, the periode, and
the amplitude of the wave.
Give also the units
of each quantity.
d.
What
are the amplitude and the complex representation of amplitude of the given wave
function (c)?
Solution:
a.
$k=\frac{2\pi
}{\lambda }\to \lambda =\frac{2\pi }{k}$ ; $T=\frac{2\pi }{\omega }$; $f=\frac{1}{T}=\frac{\omega
}{2\pi }$ ; and ${{v}_{ph}}=-\frac{{{\left( {}^{\partial \varphi }/{}_{\partial
t} \right)}_{x={{x}_{o}}}}}{{{\left( {}^{\partial \varphi }/{}_{\partial x}
\right)}_{t={{t}_{o}}}}}$ with $\varphi \left( x,t \right)=kx-\omega t$ , ${{v}_{ph}}=\frac{\omega
}{k}$ .
b.
k is the number of
waves per unit length; ω is the
number of waves per unit time.
c.
${{v}_{ph}}=-\frac{{{\left(
{}^{\partial \varphi }/{}_{\partial t} \right)}_{x}}}{{{\left( {}^{\partial
\varphi }/{}_{\partial x} \right)}_{t}}}=-\frac{{{9.10}^{14}}}{-{{3.10}^{6}}}={{3.10}^{8}}m/s$
; $\lambda =\frac{2\pi }{k}$ with $k={{3.10}^{6}}\pi $, $\lambda
={}^{2}/{}_{3}{{.10}^{-6}}m$ ; $f=\frac{\omega }{2\pi }$ with $\omega
={{9.10}^{14}}$, $f={{4,5.10}^{14}}Hz$ , $T={}^{1}/{}_{f}={}^{2}/{}_{9}{{.10}^{-14}}s$
; $A={{10}^{3}}{}^{V}/{}_{m}$ .
d. $E(x,t)={{10}^{3}}\sin
({{3.10}^{6}}\pi x-{{9.10}^{14}}\pi t)={{10}^{3}}\cos ({{3.10}^{6}}\pi
x-{{9.10}^{14}}\pi t{{-}^{\pi }}{{/}_{2}})$
$={{10}^{3}}\cos
({{3.10}^{6}}\pi x{{-}^{\pi }}{{/}_{2}}-{{9.10}^{14}}\pi t)$
Working this equation out, we obtain: $E(x,t)={{10}^{3}}\cos ({{3.10}^{6}}\pi x-{}^{\pi }/{}_{2})\cos
({{9.10}^{14}}\pi t)+{{10}^{3}}\sin ({{3.10}^{6}}\pi x-{}^{\pi }/{}_{2})\sin
({{9.10}^{14}}\pi t)$
$E(x,\omega )=\left[ {{10}^{3}}\cos
\left( {{3.10}^{6}}\pi x{{-}^{\pi }}{{/}_{2}} \right) \right]+i\left[
{{10}^{3}}\sin \left( {{3.10}^{6}}\pi x{{-}^{\pi }}{{/}_{2}} \right) \right]$
$={{10}^{3}}{{e}^{i({{3.10}^{6}}\pi
x{{-}^{\pi }}{{/}_{2}})}}={{10}^{3}}{{e}^{-{{i}^{\pi
}}{{/}_{2}}}}{{e}^{i({{3.10}^{6}}\pi x)}}$. Complex representation of amplitude : $={{10}^{3}}{{e}^{-i{}^{\pi }/{}_{2}}}$ .
2. Define the complex representation of amplitude of each of the following wave functions:
a.
${{\psi
}_{1}}(t)=A\cos (\omega t+{{\phi }_{o}})$
b.
${{\psi
}_{2}}(z,t)=A\cos (kz-\omega t)$
c.
${{\psi
}_{3}}(x,y,z,t)=A\cos (\vec{k}.\vec{r}+\omega t)$
Solution:
a.
${{\psi
}_{1}}(t)=A\cos (\omega t+{{\phi }_{o}})=A\cos \omega t\cos {{\phi }_{o}}-A\sin
\omega t\sin {{\phi }_{o}}$
Complex representation: ${{\Psi
}_{1}}(\omega )=A\cos {{\phi }_{o}}-iA\sin {{\phi }_{o}}=A{{e}^{-i{{\phi
}_{o}}}}$
Truly, we can also express ψ1(t) in term of complex
number: ${{\psi }_{1}}(t)=$ Re $\left[
{{\Psi }_{1}}(\omega ){{e}^{-i\omega t}} \right]=$ Re $\left[ A{{e}^{-i{{\phi }_{o}}}}{{e}^{-i\omega t}} \right]$ thus,
the complex representation of amplitude is ${{\psi }_{1}}(\omega
)=A{{e}^{-i{{\phi }_{o}}}}$ .
b.
${{\psi
}_{2}}(z,t)=A\cos (kz-\omega t)=A\cos kz\cos \omega t+A\sin kz\sin \omega t$
${{\Psi }_{2}}(z,\omega )=A\cos
kz+iA\sin kz=A{{e}^{ikz}}$
We can also write ${{\psi
}_{2}}(z,t)=$ Re $\left[ {{\Psi
}_{2}}(z,\omega ){{e}^{-i\omega t}} \right]=$Re $\left[ A{{e}^{ikz}}{{e}^{-i\omega t}} \right]$ thus the complex representation of amplitude =
A.
c.
${{\psi
}_{3}}(x,y,z,t)=A\cos (\vec{k}.\vec{r}+\omega t)=A\cos
({{k}_{x}}x+{{k}_{y}}y+{{k}_{z}}z+\omega t)$
$=A\cos
({{k}_{x}}x+{{k}_{y}}y+{{k}_{z}}z)\cos \omega t-A\sin
({{k}_{x}}x+{{k}_{y}}y+{{k}_{z}}z)\sin \omega t$
${{\Psi }_{3}}(x,y,z,\omega )=A\cos
({{k}_{x}}x+{{k}_{y}}y+{{k}_{z}}z)-iA\sin ({{k}_{x}}x+{{k}_{y}}y+{{k}_{z}}z)$
$=A{{e}^{-i\vec{k}.\vec{r}}}$
Thus, the complex representation of amplitude is A.
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