Question 463: Waves

Question

A simple harmonic progressive wave is represented as $y=0.03 \sin \pi(2 t-0.01 x) \mathrm{m}$. At a given instant of time, the phase difference between two particles 25 m apart is

Accepted Answer

The equation of the simple harmonic wave is given as:
$ y = 0.03 \sin \pi(2 t - 0.01 x) \, 	ext{m} $
This can be rewritten to match the standard wave equation format:
$ y = 0.03 \sin (2 \pi t - 0.01 \pi x) \, 	ext{m} $
Comparing this with the general equation for a progressive wave:
$ y = a \sin (\omega t - k x) $
we identify the wave number $ k $ as:
$ k = 0.01\pi $
The wavelength $ \lambda $ can be found using the relationship:
$ k = \frac{2 \pi}{\lambda} \quad \Rightarrow \quad \lambda = \frac{2 \pi}{0.01 \pi} = 200 \, 	ext{m} $
To find the phase difference ($\Delta \phi$) between two particles 25 meters apart, we use the formula:
$ \Delta \phi = k x = \frac{2 \pi}{\lambda} 	imes x $
Plugging in the values, we have:
$ \Delta \phi = \frac{2 \pi}{200} 	imes 25 = \frac{\pi}{4} \, 	ext{rad} $
Thus, the phase difference between the two particles separated by 25 meters is $\frac{\pi}{4} \, 	ext{rad}$.

Answer

$\pi \mathrm{~rad}$

Answer

$\frac{\pi}{2} \mathrm{~rad}$

Answer

$\frac{\pi}{4} \mathrm{~rad}$

Answer

$\frac{\pi}{8} \mathrm{~rad}$

Waves

A simple harmonic progressive wave is represented as y=0.03 \sin \pi(2 t-0.01 x) \mathrm{m}. At a given instant of time, the phase difference between two particles 25 m apart is

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