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$ y = \left( {1.5m} \right)\sin \left( {0.400x} \right)\cos \left( {200t} \right) $

Where, $ x $ is in meters and $ t $ is in seconds. Determine the wavelength, frequency and speed of the interfering waves.

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From the question, we know that the equation of the standing waves is given as $ y = \left( {1.5m} \right)\sin \left( {0.400x} \right)\cos \left( {200t} \right) $ .

We know that the standard equation of stationary waves.

$ y = 2A\sin kx\cos wt $ where $ k = \dfrac{{2\pi }}{\lambda } $ and $ \omega = 2\pi f $ .

Here, $ \lambda $ is the wavelength, $ \omega $ is the angular wavelength, $ f $ is the frequency and $ k $ is angular wave number.

Now we compare the standard equation of standing wave with the given equation, we get,

$ k = 0.4 $ and $ \omega = 200\;{\rm{rad/s}} $ .

Now we substitute the values in the angular wavenumber formula, we get,

$0.4 = \dfrac{{2\pi }}{\lambda }\\

\lambda = \dfrac{{2\pi }}{{0.4}}\\

= 15.7\;{\rm{m}}$

Thus, the wavelength of the inferring wave is 15.17 m.

Now we substitute the values in the angular velocity formula, we get,

$200 = 2\pi f\\

f = \dfrac{{2\pi }}{{200}}\\

= 31.8\;{\rm{Hz}}$

Thus, the frequency of the inferring wave is 31.8 Hz.

The speed of the interfering eaves is expressed as,

$ v = \dfrac{\omega }{k} $

Now we substitute the values of $ \omega $ as 200 rad/s and $ k $ as 0.4 in the above expression. We get,

$v = \dfrac{{200}}{{0.4}}\\

= 500\;{\rm{m/s}}$

Thus, the velocity of the inferring wave is 500 m/s.

Stationary (standing) waves are produced when two waves having equal speed, frequency, and amplitude are travelling in opposite directions. One of the common examples of stationary waves is plucking of a string of guitar or violin. The general equation of stationary wave is given as $ y = 2A\cos kx\sin \omega t $ .