Question

Use the wave equation to find the speed of a wave given in terms of the general function: $\mathit{h}\left(x,t\right)$

localid="1660990709658" $\mathit{y}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{m}\mathbf{m}\mathbf{)}\mathit{h}\mathbf{[}\left(30m^{-1}\right)\mathbf{x}\mathbf{+}\left(6.0s^{-1}\right)\mathbf{t}\mathbf{]}$.

Short answer

The speed ofthe given wave is 0.2 m/s

Step-by-step solution

The given data

The given wave equation, $y(\mathrm{x},\mathrm{t})=(4.00)h[\left(30\right)\mathrm{x}+(6.0)\mathrm{t}]$

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.

Calculation of the speed of the wave

By comparingthegiven equation withthesolution of the wave equation (i)

$y(\mathrm{x},\mathrm{t})=(4.00)h[\left(30\right)\mathrm{x}+(6.0)\mathrm{t}]$

We can get,

1. Angular frequency of the wave is$\omega =6.0\mathrm{rad}/\mathrm{s}$
2. Angular wave number$k=30.0{\mathrm{m}}^{-1}$

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

$$\begin{gathered} v=\frac{6.0}{30.0} \\ =0.2\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of the speed is 0.2 m/s .