WAVE_ADDITION2
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\[
\begin{array}{l}
 http://physicsmatayom.spaces.live.com \\
 http://www.vchakarn.com/vb\log /32546 \\
 wave\_addition2 \\
 1.Let\,y1 = R1*\sin (\theta  + \phi 1)\,and\,y1 = R2*\sin (\theta  + \phi 2)\, \\
 2.if\,y3 = y1 + y2 = R3*\sin (\theta  + \phi 3)\,then \\
 3.R3 = \sqrt {R3_x ^2  + R3_y ^2 } \,;vectorR3 = vectorR1 + vectorR2 \\
 3.1R3_x  = R1*\cos \phi 1 + R2*\cos \phi 2 \\
 3.2R3_y  = R1*\sin \phi 1 + R2*\sin \phi 2\,\,and \\
 4.\,1\tan \phi 3 = R3_y /R3_{\scriptstyle x \hfill \atop
  \scriptstyle  \hfill}  \\
 4.2\phi 3 = \tan ^{ - 1} (R3_y /R3_{\scriptstyle x \hfill \atop
  \scriptstyle  \hfill} ) \\
 \Pr oof \\
 1.LHS = R1*\sin (\theta  + \phi 1) + R2*\sin (\theta  + \phi 2)\, \\
 2.\,\,\,\,\,\,\,\,\,\,\, = R1*[\sin \theta \cos \phi 1 + \cos \theta \sin \phi 1) + R2*(\sin \theta \cos \phi 2 + \cos \theta \sin \phi 2) \\
 3.\,\,\,\,\,\,\,\,\,\,\, = (R1\cos \phi 1 + R2\cos \phi 2)*\sin \theta  + (R1\sin \phi 1 + R2\sin \phi 2)\cos \theta  \\
 4.\,\,\,\,\,\,\,\,\,\,\, = R3_x *\sin \theta  + R3_y *\cos \theta  \\
 5.\,\,\,\,\,\,\,\,\,\,\, = \sqrt {R3_x ^2  + R3_y ^2 } [\frac{{R3_x }}{{\sqrt {R3_x ^2  + R3_y ^2 } }}*\sin \theta  + \frac{{R3_y }}{{\sqrt {R3_x ^2  + R3_y ^2 } }}*\cos \theta ] \\
 6.\,\,\,\,\,\,\,\,\,\, = R3*(\cos \phi 3\sin \theta  + \sin \phi 3\cos \theta ) \\
 7.\,\,\,\,\,\,\,\,\, = R3*\sin (\theta  + \phi 3) \equiv RHS\,\,Then \\
 8.R3 = \sqrt {R3_x ^2  + R3_y ^2 } \,;Q.E.D.1 \\
 9.R3_x  = \sum\limits_{i = 1}^{i = 2} {Ri_x }  = \sum\limits_{i = 1}^{i = 2} {R_i } \cos \phi _i  = (R1\cos \phi 1 + R2\cos \phi 2) \\
 10.R3_y  = \sum\limits_{i = 1}^{i = 2} {Ri_y }  = \sum\limits_{i = 1}^{i = 2} {R_i } \sin \phi _i  = (R1\sin \phi 1 + R2\sin \phi 2) \\
 11.\tan \phi 3 = R3_y /R3_x  \\
 12.\phi 3 = \tan ^{ - 1} (R3_y /R3_x )\,\,\,Q.E.D.2 \\
 Example \\
 1.y1 = \sin \theta ;y2 = \sin (\theta  + 90^o );if\,y1 + y2 = y3 = R3*\sin (\theta  + \phi 3) \\
 2.Then\,2.1R3(Total\,Amplitude) = ?\,and\,\phi 3(Total\,Amplitude) = ? \\
 Ans \\
 3.1\,R3x = R1*\cos 0^o  + R2*\cos 90^o  = 1 + 0 = 1 \\
 3.2\,R3y = R1*\sin 0^o  + R2*\sin 90^o  = 0 + 1 = 1 \\
 3.3\,R3 = \sqrt {R3_x ^2  + R3_y ^2 }  = \sqrt {1^2  + 1^2 }  = \sqrt 2 \,\,Ans1 \\
 4.\tan \phi 3 = R3_y /R3_x  = 1/1 = 1 \\
 5.\phi 3 = \tan ^{ - 1} (1) = 45^o \,\,Ans2 \\
 6.Y3 = \sqrt 2 \sin (\theta  + 45^o )\,\,Ans3 \\
 \end{array}
\]