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A transverse harmonic wave on a string is described by $y(x, t)=3 \sin (36 t+0.018 x+\pi / 4)$, where $x, y$ are in $\mathrm{cm}$ and $t$ in sec. The positive direction of $x$-axis is from left to right.
(i) Is this a travelling or stationary wave? If it is travelling, what are the speed and direction of its propagation?
(ii) What are its amplitude and frequency?
(iii) What is the initial phase at the origin?
(iv) What is the least distance between two successive crests in the wave?
(i) Is this a travelling or stationary wave? If it is travelling, what are the speed and direction of its propagation?
(ii) What are its amplitude and frequency?
(iii) What is the initial phase at the origin?
(iv) What is the least distance between two successive crests in the wave?
Solution:
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Verified Answer
Wave equation given,
$y(x, t)=3 \sin (36 t+0.018 x+\pi / 4)$
The general equation of a plane progressive wave is $y(x, t)=a \sin \left[\frac{2 \pi}{\lambda}(v t+x)+\phi_0\right]$
Comparing we get,
(i) The given wave is a travelling wave moving from right to left.
Equating coefficient of $t$
Velocity, $v=\frac{36}{0.018}=2000 \mathrm{~cm} / \mathrm{s}$
(ii) Amplitude, $a=3 \mathrm{~cm}$
$$
\begin{aligned}
&\frac{2 \pi}{\lambda}=0.018 \\
&\Rightarrow \lambda=\text { wavelength }=\frac{2 \pi}{0.018}=349 \mathrm{~cm}
\end{aligned}
$$
Frequency, $v=\frac{v}{\lambda}=\frac{2000}{349}=5.73 \mathrm{~Hz}$.
(iii) Initial phase $=\phi_0=\frac{\pi}{4}$ radian
(iv) Least distance between two successive crests = wavelength $=349 \mathrm{~cm} .=3.5 \mathrm{~m}$
$y(x, t)=3 \sin (36 t+0.018 x+\pi / 4)$
The general equation of a plane progressive wave is $y(x, t)=a \sin \left[\frac{2 \pi}{\lambda}(v t+x)+\phi_0\right]$
Comparing we get,
(i) The given wave is a travelling wave moving from right to left.
Equating coefficient of $t$
Velocity, $v=\frac{36}{0.018}=2000 \mathrm{~cm} / \mathrm{s}$
(ii) Amplitude, $a=3 \mathrm{~cm}$
$$
\begin{aligned}
&\frac{2 \pi}{\lambda}=0.018 \\
&\Rightarrow \lambda=\text { wavelength }=\frac{2 \pi}{0.018}=349 \mathrm{~cm}
\end{aligned}
$$
Frequency, $v=\frac{v}{\lambda}=\frac{2000}{349}=5.73 \mathrm{~Hz}$.
(iii) Initial phase $=\phi_0=\frac{\pi}{4}$ radian
(iv) Least distance between two successive crests = wavelength $=349 \mathrm{~cm} .=3.5 \mathrm{~m}$
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