Use the wave equation to find the speed of a wave given by – $\mathit{y}\left(x,t\right)\mathbf{=}\left(2.00mm\right){\left[\left(20m^{-1}\right)x‐\left(4.0s^{-1}\right)t\right]}^{\mathbf{0}\mathbf{.}\mathbf{5}}$.

The given wave equation,$y(\mathrm{x},\mathrm{t})=(2.00\mathrm{mm}){[\left(20{\mathrm{m}}^{-1}\right)\mathrm{x}‐\left(4.0{\mathrm{s}}^{-1}\right)\mathrm{t}]}^{0.5}$

By comparing the given wave equation with the standard form, we can find the wavenumber and angular velocity. Using these values, we can find the speed of the wave.

Formula:

The general expression of the wave, $y\left(x,t\right)=y_m\sin \left(kx-\omega t\right)$ (i)

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

Here, $y$is displacement,$y_m$ is the amplitude of the wave, $k$is the angular wave number, $\omega$is the Angular frequency of the wave,$t$ is time.

By comparingthegiven equation with a solution of the wave equation, we get

1. Angular frequency of a wave is$\omega =4\mathrm{rad}/\mathrm{s}$
2. Angular wave number$k=20.00{\mathrm{m}}^{-1}$
3. Amplitude of wave $y_m=2.00$

Using equation (ii), the speed of the wave is given as:

$$\begin{gathered} v=\frac{4.00}{20.00} \\ =0.2\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of the speed is $0.2\mathrm{m}/\mathrm{s}$.