Question

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

Short answer

The speed of the given wave is 0.2 m/s .

Step-by-step solution

The given data

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

Understanding the concept of the wave equation

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.

Step 3: Calculation of the speed of the wave

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 $0.2\mathrm{m}/\mathrm{s}$.