Question

Two waves, each having a frequency of $100Hz$ and a wavelength of $2.0cm$, are travelling in the same direction on a string.

1. When the second wave was produced $0.015s$, later than the first one at the same place, the phase difference is $3\mathrm{\pi }$.
2. When the two waves were produced at the same instant but the first one was produced a distance $4.0cm$ behind the second one, the phase difference is $4\mathrm{\pi }$.

If each of the waves in above cases has an amplitude of $2.0mm$, what would be the amplitudes of the resultant waves.

Answer 1

Step 1: Given

Amplitude of the waves: $A=2mm=2\times {10}^{-3}m$

Phase difference in first case: ${\varphi }_1=3\mathrm{\pi }$

Phase difference in first case: ${\varphi }_2=4\mathrm{\pi }$

Step 2: Formula Given

$R=2A\cos \frac{\varphi }{2}$

Step 3: Find the resultant amplitude

Calculate the resultant amplitude when the phase difference is $3\mathrm{\pi }$, using the formula.

$$\begin{gathered} R=2A\cos \frac{\varphi }{2} \\ =2\times 2\times {10}^{-3}\times \cos \frac{3\mathrm{\pi }}{2} \\ =4\times {10}^{-3}\times 0 \\ =0mm \end{gathered}$$

Calculate the resultant amplitude when the phase difference is $4\mathrm{\pi }$, using the formula.

$$\begin{gathered} R=2A\cos \frac{\varphi }{2} \\ =2\times 2\times {10}^{-3}\times \cos \frac{4\mathrm{\pi }}{2} \\ =4\times {10}^{-3}\times 1 \\ =4\times {10}^{-3} \\ =4mm \end{gathered}$$

Hence, the resultant amplitude when phase difference is $3\mathrm{\pi }$ is $0mm$ and when phase difference is $4\mathrm{\pi }$ is $4mm$.