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The displacement of the medium in a sound wave is given by the equation \[{y_1} = Acos\left( {ax + bt} \right)\] where ,a and b are positive constants. The wave is reflected by an obstacle situated at x=0. The intensity of the reflected wave is 0.64 times that of the incident wave. Then, find out the wavelength and frequency of the incident wave.

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Last updated date: 13th Jun 2024
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Answer
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Hint: In an equation, wavelength is represented by the Greek letter lambda (λ). Depending on the type of wave, wavelength can be measured in meters, centimeters, or nanometers (1 m = 109 nm). The frequency, represented by the Greek letter nu (ν), is the number of waves that pass a certain point in a specified amount of time.

Solution step by step we know the displacement equation of wave is
\[y = Acos\left( {kx + \omega t} \right)\] ....(1)
 Now comparing equation 1 with the given equation \[{y_1} = Acos\left( {ax + bt} \right)\]
So, a = K and ω = b
Now, we know , $k = \dfrac{{2\pi }}{\lambda }$ , where $\lambda $ is the wavelength of the wave,
So, $a = \dfrac{{2\pi }}{\lambda }$
$\therefore \lambda = \dfrac{{2\pi }}{a}$
Also $\omega = 2\pi v$ , where v is the frequency of wave,
So, $v = \dfrac{b}{{2\pi }}$

So the wavelength and the frequency of the incident wave is $\dfrac{{2\pi }}{a},\dfrac{b}{{2\pi }}$ .

Note: Reflected and Transmitted waves at a boundary - definition. If a pulse is introduced at the left end of the rope, it will travel through the rope towards the right end of the medium. This pulse is called the incident pulse since it is incident towards (i.e., approaching) the boundary with the pole.