91Ó°ÊÓ

$$\begin{gathered} {\mathrm{y}}_1\left(\mathrm{x},\mathrm{t}\right)=\left(4.60\mathrm{mm}\right)\sin \left(2\mathrm{Ï€³\ae }-400\mathrm{Ï€³Ù}\right) \\ {\mathrm{y}}_1\left(\mathrm{x},\mathrm{t}\right)=\left(5.6\mathrm{mm}\right)\sin \left(2\mathrm{Ï€³\ae }-400\mathrm{Ï€³Ù}+0.80\mathrm{Ï€}\mathrm{rad}\right) \end{gathered}$$

Amplitude of the third wave,$y_{3m}=5.00\mathrm{mm}$

When two waves travel alongthesame string, the amplitude of each wave can be found relative to the resultant wave. Usingthephasor technique, the phase angle can be calculated.

Formula:

The phase angle of two connecting lines, $\mathrm{θ}=\frac{\mathrm{Adjacent}\mathrm{Side}}{\mathrm{Resultant}}......\left(1\right)$

From the given equations of the waves, we can see that the amplitude of the first wave is 4.6 m and the amplitude of the second wave is 5.6 mm. From the wave equations, it is also clear that the phase difference between these two waves is$0.80\mathrm{π}\mathrm{rad}\mathrm{or}144°$. This means that the angle between the amplitude of these two waves is $0.80\mathrm{π}\mathrm{rad}\mathrm{or}144°$So, if we add these two vectors, we will get the resultant. It can be done as follows:

To find the amplitude relative to the first wave with phase difference $144°$, we can use the standard technique for addition of two vectors. The resultant vector will have only y component caused by $y_2\mathrm{as}{\mathrm{y}}_1$has only x component.

$$\begin{gathered} y_R=\left(y_2\right)\sin \left(144°\right)\left(\mathrm{where}{\mathrm{y}}_2=5.6\mathrm{mm}\right) \\ =3.29\mathrm{mm} \end{gathered}$$

Hence, the value of the amplitude is 3.29 mm

x component of the resultant amplitude would be

$$\begin{gathered} y_{Rx}=y_1-y_2\cos \left(180-144\right) \\ =4.6-5.6\cos 36 \\ =0.06950 \end{gathered}$$

To find the angle relative to wave 1, we can use equation (1) in the given figure to the phase of resultant wave,

$$\begin{gathered} \mathrm{θ}={\cos }^{-1}\left(\frac{0.06950}{3.29}\right) \\ =88.8° \end{gathered}$$

Therefore, the phase angle relative to wave 1 is$88.8°$

For the amplitude to be maximum,the third wave should be in phase with the resultant wave.

So the angle must be equal to $88.8°$relative to the first wave.