Question

Use the wave equation to find the speed of a wave given by -

$\mathit{y}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{m}\mathbf{)}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{[}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{00}{\mathbf{m}}^{\mathbf{-}\mathbf{1}}\mathbf{)}\mathbf{x}\mathbf{‐}\mathbf{(}\mathbf{7}\mathbf{.}\mathbf{00}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}\mathbf{)}\mathit{t}\mathbf{]}$.

Short answer

The speed of the wave is 1.75 m/s

Step-by-step solution

The given data

The given equation for the wave,$y(\mathrm{x},\mathrm{t})=(3.0\mathrm{mm})sin[(4.00{\mathrm{m}}^{-1})\mathrm{x}‐(7.00{\mathrm{s}}^{-1})t]$

Understanding the concept of the wave equation

The solution of wave equation is given by,

$\mathit{y}\left(x,t\right)\mathbf{=}{\mathit{y}}_{\mathbf{m}}\mathit{s}\mathit{i}\mathit{n}\left(kx-\omega t\right)$

By comparing this equation with the given equation, we can calculate the speed of the given wave.

Formula:

The speed of the wave,$v=\omega /k$ (i)

Calculation of the speed of wave

By comparing the given equation with the solution of the wave equation

$y(\mathrm{x},\mathrm{t})=(3.0\mathrm{mm})sin[(4.00{\mathrm{m}}^{-1})\mathrm{x}‐(7.00{\mathrm{s}}^{-1})t]$

Comparing the given equation with general that is,$y\left(x,t\right)=y_m\mathrm{sinsin}\left(kx-wt\right)$ we get

We can get, angular frequency,$\omega =7.00\mathrm{rad}/\mathrm{s}$

Wave number,$k=4.00/\mathrm{rad}$

Using equation (i), we get the speed of the wave as

$$\begin{gathered} v=\frac{7.00{\mathrm{s}}^{-1}}{4.00{\mathrm{m}}^{-1}} \\ =1.75\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of the speed of the wave is$1.75\mathrm{m}/\mathrm{s}$