Four waves are described by equations as follow \[ \begin{array}{l}...
0Four waves are described by equations as follow
\[
\begin{array}{l}
Y_{1}=A \cos (\omega t-k x) \\
Y_{2}=\frac{A}{2} \cos \left(\omega t-k x+\frac{\pi}{2}\right) \\
Y_{3}=\frac{A}{4} \cos (\omega t-k x+\pi) \\
Y_{4}=\frac{A}{8} \cos \left(\omega t-k x+\frac{3 \pi}{2}\right)
\end{array}
\]
\( \mathrm{P} \)
W.
and their resultant wave is calculated as \( Y=Y_{1}+Y_{2}+Y_{3}+Y_{4} \) such as
\( Y=A^{1} \cos (\omega t-k x+\phi) \) then (symbols have their usual meanings)
(1) \( A^{1}=\frac{\sqrt{5} A}{8} \quad \phi=\tan ^{-1}\left(\frac{1}{4}\right) \)
(2) \( A^{1}=\frac{2 \sqrt{5} A}{8} \quad \phi=\tan ^{-1}\left(\frac{1}{3}\right) \)
(3) \( A^{1}=\frac{3 \sqrt{5} A}{8} \quad \phi=\tan ^{-1}\left(\frac{1}{2}\right) \)
(4) \( A^{1}=\frac{4 \sqrt{5} A}{8} \quad \phi=\tan ^{-1}(1) \)
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