91Ó°ÊÓ

The given wave equation,$y=0.15\sin (0.79x-13t)$

We can find the displacementy at$\mathit{x}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{3}\mathbf{ }\mathit{m}\mathbf{,}\mathbf{}\mathit{t}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{16}\mathbf{ }\mathit{s}$by inserting these values in the given wave equation.

Then we can find the values of${\mathit{y}}_{\mathbf{m}}\mathbf{,}\mathit{k}\mathbf{,}\mathit{Ó\neg }$by comparing the given equation with the general equation of the wave.

The displacement of the resultant standing wavecan be calculated using the principle of superposition.

Formula:

The wavenumber of the wave,

$k=\frac{2\mathrm{π}}{\mathrm{λ}}$ (i)

The angular frequency of the wave,

$\mathrm{Ó\neg }=2\mathrm{Ï€}f$ (ii)

The transverse speed of the wave,

$v_m=\mathrm{Ó\neg }{y}_m$ (iii)

Here Here f is the frequency of the wave $\mathrm{λ}$is the wavelength of the wave and$y_m$ is the amplitude of the wave.

The general equation of a translational wave on a string is given as:

$y=y_{m}sin(kx-\mathrm{Ó\neg }t)$ (1)

The equation of the wave is given as,

$y=0.15\sin (0.79x-13t)$ (2)

Putting values of x =2.3 m and t =0.16 s in equation (2), we can calculate the displacement y as:

$$\begin{gathered} y=0.15\sin \left(0.79\left(2.3\right)-13\left(0.16\right)\right) \\ =-0.0389 \\ ~-0.039\mathrm{m} \end{gathered}$$

Therefore, the displacement y at $x=2.3 m,t=0.16 s$is -0.039 m.

Comparing equations (1) and (2), we can find the amplitude of the wave as:

$y_m=0.15\mathrm{m}$

Therefore, the value of $y_m$is 0.15 m.

Comparing equations (1) and (2), we can find the value of wavenumber as:

$k=0.79 m^{-1}$

Therefore, the value of k is$0.79{\mathrm{m}}^{-1}$

Comparing equations (1) and (2), we can find the value of the angular frequency as:

$\mathrm{Ó\neg }=13\frac{\mathrm{rad}}{\mathrm{s}}$

Therefore, the value of$\mathrm{Ó\neg }$ is$13\frac{\mathrm{rad}}{\mathrm{s}}$

From the given wave equation, we can conclude that the wave is traveling in the –x direction, so the sign of angular frequency ($\mathrm{Ó\neg }$)must be positive.

Thus, the sign of$\mathrm{Ó\neg }$ is positive.

The displacement y at$x=2.3 m,t=0.16 s$is -0.039 m.

Let's call it $y_1$.

The wave equation for the second wave can be written as,

$y_2=y_{m}sin(kx+\mathrm{Ó\neg }t)$

Substitute the values in the above expression, and we get,

$$\begin{gathered} y_2=0.15\sin \left(0.79\left(2.3\right)+13\left(0.16\right)\right) \\ =-0.101 \end{gathered}$$

Hence, displacement of the resulting standing wave can be calculated as,

$y(x,t)=y_1+y_2$

Substitute the values in the above expression, and we get,

$$\begin{gathered} y\left(x,t\right)=0.039-0.101 \\ =0.14\mathrm{m} \end{gathered}$$.

Hence, the value of the displacement of the resulting wave is -0.14 m.