91Ó°ÊÓ

1. Amplitude of the two waves, $y_m=12\text{ n³¾}$.
2. Wavelength of the two waves,$\mathrm{λ}=35\text{ c³¾}$.
3. Phase difference between them, $\mathrm{Ï•}=\mathrm{Ï€}/3$.

The amplitude and the phase difference for the two waves are given. Using this we can find the resultant wave. Once we get the resultant wave, we can find the required quantities.

Formula:

The expression of the displacement of the wave,

$\mathit{y}\mathbf{=}{\mathit{y}}_{\mathbf{m}}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{k}\mathbf{x}\mathbf{−}\mathbf{Ó\neg }\mathbf{t}\mathbf{)}$ …(¾±)

The resultant wave equation,

$\mathit{Y}\mathbf{=}\mathbf{2}{\mathit{y}}_{\mathbf{m}}\mathit{c}\mathit{o}\mathit{s}\left(\frac{{\displaystyle \mathrm{Ï•}}}{{\displaystyle 2}}\right)\mathit{s}\mathit{i}\mathit{n}\left(kx−\mathrm{Ó\neg }t+\frac{{\displaystyle \mathrm{Ï•}}}{{\displaystyle 2}}\right)$ …(¾±¾±)

The amplitude and the phase difference between the two waves is given, so, using equation (i), we can write the displacement equations as:

$y_1=y_m\sin (kx−\mathrm{Ó\neg }t)$

$y_2=y_m\sin (kx−\mathrm{Ó\neg }t+\mathrm{Ï•})$

From equation (ii), the resultant amplitude is given as:

$\begin{array}{c}{Y}_m=2×(12\text{ n³¾})×\cos \left(\frac{\mathrm{Ï€}}{6}\right)\\ =20.78\text{ n³¾}\\ Y_m≈21\text{ n³¾}\end{array}$

Hence, the amplitude of the resultant amplitude during interference is $21\text{ n³¾}$.

Wavelength of the superimposed waves do not change, hence, the resultant wavelength would also bethesame asthewavelength of the sound waves.

${\mathrm{λ}}_{resul\tan t}=35\text{ c³¾}$

Hence, the value of the resultant wavelength is $35\text{ c³¾}$.

As the waves are travelling in the opposite direction

The phase difference between them will be$\mathrm{Ï•}=\mathrm{Ï€}$

Using equation (ii) and the given values, we get the amplitude as:

$Y_m=2y_m\cos \left(\mathrm{Ï•}\right)$

Substitute all the value in the above equation.

$\begin{array}{l}{Y}_m=2×12\text{ n³¾}×\cos \left(\frac{\mathrm{Ï€}}{2}\right)\\ =|2×12\text{ n³¾}×\left(1\right)|\\ Y_m=24\text{ n³¾}\end{array}$

Hence, the value of net amplitude is$24\text{ n³¾}$ .

Change in direction would have no effect on the wavelength of the resultant wave

${\mathrm{λ}}_{resul\tan t}=35\text{ c³¾}$

Hence, the value of the wavelength is $35\text{ c³¾}$.