Answer:
[tex]\phi=2.43\ rad=139.2^{\circ}[/tex]
Explanation:
The resultant wave of two identical traveling waves of amplitude [tex]y_M[/tex] with a phase difference [tex]\phi[/tex] between them is:
[tex]Y(x,t)=y_Msin(kx-\omega t+\phi)+y_Msin(kx-\omega t)[/tex]
Using the trigonometric formula
[tex]sin(a)+sin(b)=2sin(\frac{a+b}{2})cos(\frac{a-b}{2})[/tex]
we have:
[tex]Y(x,t)=2y_Msin(\frac{2kx-2\omega t+\phi}{2})cos(\frac{\phi}{2})=2y_Mcos(\frac{\phi}{2})sin(kx-\omega t+\frac{\phi}{2})[/tex]
where we can see that the amplitude of the combined wave is [tex]Y_M=2y_Mcos(\frac{\phi}{2})[/tex], which we want to be equal to [tex]0.7y_M[/tex], so we do:
[tex]2y_Mcos(\frac{\phi}{2})=0.7y_M[/tex]
[tex]cos(\frac{\phi}{2})=0.35[/tex]
[tex]\phi=2Arccos(0.35)=2.43\ rad[/tex]
