The equation of a progressive wave is given by $y=5 \sin (100 \pi t-0.4 \pi x)$ where $y$ and $x$ are in…
The equation of a progressive wave is given by $y=5 \sin (100 \pi t-0.4 \pi x)$ where $y$ and $x$ are in $\mathrm{m}$ and $t$ is in $\mathrm{s}$.
(1) The amplitude of the wave is $5 \mathrm{~m}$.
(2) The wavelength of the wave is $5 \mathrm{~m}$.
(3) The frequency of the wave is $50 \mathrm{~Hz}$.
(4) The velocity of the wave is $250 \mathrm{~m} \mathrm{~s}^{-1}$.
Which of the following statements are correct?
(1), (2) and (3)
(2) and (3)
(1) and (4)
All are correct
Solution
The equation of a given progressive wave is
$y=5 \sin (100 \pi t-0.4 \pi x)$ ...(i)
The standard equation of a progressive wave is
$y=a \sin (\omega t-k x)$ ...(ii)
Comparing (i) and (ii), we get
$a=5 \mathrm{~m}, \omega=100 \pi \mathrm{rad} \mathrm{s}^{-1}, k=0.4 \pi \mathrm{m}^{-1}$
(1) Amplitude of the wave, $a=5 \mathrm{~m}$
(2) Wavelength of the wave,
$\lambda=\frac{2 \pi}{k}=\frac{2 \pi}{0.4 \pi}=5 \mathrm{~m}$
(3) Frequency of the wave,
$v=\frac{\omega}{2 \pi}=\frac{100 \pi}{2 \pi}=50 \mathrm{~Hz}$
(4) Velocity of the wave,
$v=v \lambda=\left(50 \mathrm{~s}^{-1}\right)(5 \mathrm{~m})=250 \mathrm{~m} \mathrm{~s}^{-1}$