Question

A circular wave travels outward from the origin. At one instant of time, the phase at r1 = 20 cm is 0 rad and the phase at r2 = 80 cm is 3p rad. What is the wavelength of the wave?

Short answer

The wavelength of the wave is$40cm$

Step-by-step solution

Given information

The position of the first point is$r_1=20cm=0.20m$

The phase at the first point is${\varphi }_1=0rad$

The position of the second point is$r_2=80cm=0.80m$

The phase at the second point is${\varphi }_2=3\mathrm{\pi }\mathrm{rad}$

The wavelength of the wave

The phase difference is given by ${\varphi }_2-{\varphi }_1=\frac{2\mathrm{\pi }}{\lambda }\left(r_2-r_1\right)$.

Substitute the given values

$$\begin{gathered} 3\mathrm{\pi }-0=\frac{2\mathrm{\pi }}{\mathrm{\lambda }}\left(0.80-0.20\right) \\ \mathrm{\lambda }=\frac{2}{3}\left(0.60\right) \\ \mathrm{\lambda }=0.40\mathrm{m} \\ \mathrm{\lambda }=40\mathrm{cm} \end{gathered}$$

Therefore, the wavelength of the wave is $40cm$.