Two sinusoidal waves with the same polarization
$E_1\~\sin{[\omega(t-z/c)+\varphi_1]}$ $E_2\~\sin{[\omega(t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?
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<h3> Statement </h3>
<p>
$12.1.4$
Two sinusoidal waves with the same polarization $E_1\~\sin{[\omega(t-z/c)+\varphi_1]}$, $E_2\~\sin{[\omega(t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?
</p>
<h3>Solution</h3>
<p>
Since waves are superimposed,
$$E_R = E_{w1} + E_{w2}$$
$$E_R = E_1\~\sin{[\omega(t-z/c)+\varphi_1]} + E_2\~\sin{[\omega(t-z/c)+\varphi_2]}$$
$$E_R = (E_1\cos{\varphi_1}+E_2\cos{\varphi_2})\sin{[\omega(t-z/c)]} + (E_1\sin{\varphi_1}+E_2\sin{\varphi_2})\cos{[\omega(t-z/c)]}$$
Let's suppose that
$$E_1\cos{\varphi_1}+E_2\cos{\varphi_2} = E \cos{\varphi} = E_x$$
and
$$E_1\sin{\varphi_1}+E_2\sin{\varphi_2} = E \sin{\varphi} = E_y$$
and considering the trigonometric identity $\sin{(x+y)} = \sin{x}\cos{y} + \cos{x}\sin{y}$,
$$E_R = E \sin{[\omega(t-z/c)+\varphi]}$$
As $E = \sqrt{{E_x}^2 + {E_y}^2}$ and taking in account that $\cos{(x-y)} = \cos{x}\cos{y} + \sin{x}\sin{y}$
</p>
<h4>Answer 1</h4>
<p>
$$E = \sqrt{E_1^2+E_2^2+2E_1E_2\cos{(\varphi_1-\varphi_2)}}$$
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<p>
Phase difference is
$$\tan{\varphi} = \frac{E_y}{E_x} = \frac{E_1\sin{\varphi_1}+E_2\sin{\varphi_2}}{E_1\cos{\varphi_1}+E_2\cos{\varphi_2}}$$
Finally, the phase is,
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<h4>Answer 2</h4>
<p>
$$\Phi = \omega(t-z/c) + \arctan{\frac{E_1\sin{\varphi_1}+E_2\sin{\varphi_2}}{E_1\cos{\varphi_1}+E_2\cos{\varphi_2}}}$$
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<p style="text-align: right; font-style: italic; font-size: 14;">
BSc. Luis Daniel Fernández Quintana<br>
Physics Department (FCNE)<br>
Universidad de Oriente, Cuba<br>
</p>
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