Question

A wave has an angular frequency of$\mathbf{110}\mathit{r}\mathit{a}\mathit{d}\mathbf{/}\mathit{s}$and a wavelength of 1.80m. (a)Calculate the angular wave number and (b)Calculate the speed of the wave.

Short answer

1. The angular wave number is$3.49{\mathrm{m}}^{‐1}$
2. The speed of the wave is$31.53\mathrm{m}/\mathrm{s}$

Step-by-step solution

The given data

• The angular frequency of a wave,$\omega =110\mathrm{rad}/\mathrm{s}$
• The wavelength of a wave,$\lambda =1.8\mathrm{m}$

Understanding the concept of the wave equation

Wave number is the inverse of wavelength. The speed of the wave is equal to the product of its wavelength and frequency. The angular frequency is equal to 2π times frequency.

We can use the equations of wavenumber and the speed of the wave to calculate them.

Formulae:

The angular wave number of a wave equation, $k=\frac{2\tau \tau }{\lambda }$(i)

The speed of the wave, $v=\lambda f$(ii)

The angular frequency of the wave, $\omega =2\tau \tau f$(iii)

(a) Calculation for finding the wave equation

The angular wave number using equation (i) is calculated by substituting the values in the equation (i).

$$\begin{gathered} k=\frac{2\mathrm{\tau \tau }}{1.8\mathrm{m}} \\ =3.49{\mathrm{m}}^{-1} \end{gathered}$$

The value of the angular wave number is role="math" localid="1657278936633" $3.49{\mathrm{m}}^{-1}$.

(b) Calculation of wave speed

The speed of the wave using equations (ii) and (iii) frequency function is given as:

$$\begin{gathered} v=\frac{\lambda \omega }{2\tau \tau } \\ =\frac{1.8\times 110}{2\tau \tau } \\ =31.53\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of wave speed is $31.53\mathrm{m}/\mathrm{s}$.