Question
Express the equation of a simple harmonic progressive wave in different forms.
✓
Answer
A simple progressive wave travelling along the positive $x$-direction is given by $y=A \sin (\omega t-$ kx) ... (1)
where $A$ is the amplitude of the wave, $k$ is the wave number and co is the angular frequency.
Wave number, $k=\frac{2 \pi}{\lambda}$
$
\therefore y=A \sin \left(\omega t-\frac{2 \pi}{\lambda} x\right)
$
Angular frequency, $\omega=2 \pi n$, Eq. (2) can be written as
$
\begin{aligned}
\therefore y & =A \sin \left(2 \pi n t-\frac{2 \pi}{\lambda} x\right)^{} \\
y & =A \sin 2 \pi\left(n t-\frac{x}{\lambda}\right) \\
y & =A \sin 2 \pi n\left(t-\frac{x}{n \lambda}\right)
\end{aligned}
$
But $n \lambda=v$, the velocity of the wave.
$
y=A \sin 2 \pi n\left(t-\frac{x}{v}\right)
$
Also, writing $n=\frac{v}{\lambda}$ in Eq. (3), we get,
$
y=A \sin \frac{2 \pi}{\lambda}(v t-x)
$
Frequency of vibrations, $n=\frac{1}{T}$, Eq. (2) can be written as $y = A \sin 2 \pi\left(\overline{ T }-\frac{x}{\lambda}\right)$
Equations (1), (2), (3), (4), (5) and (6) are the different forms of the equation of a simple harmonic progressive wave.
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