Modeling Wave Interference Using Phasors – ++ Prep

Modeling Wave Interference Using Phasors

00:00 9 marks

Consider two waves with identical angular frequency but different phases.

(a) Represent a wave with amplitude A and phase \(\phi\) using a complex exponential.

(b) Given two waves represented by \(Ae^{i\phi_1}\) and \(Ae^{i\phi_2}\), express their sum.

(c) Determine the amplitude of the resultant wave in terms of A, \(\phi_1\), and \(\phi_2\).

Part a

5 minutes 3 marks

Question

Write the complex exponential representation of a wave with amplitude A and phase \(\phi\).

Hint Hint

Hint

Recall that Euler’s formula expresses complex exponentials in terms of sine and cosine.

Answer Answer
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Complete Answer

\(Ae^{i\phi}\)

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Part b

5 minutes 3 marks

Question

Sum the phasors \(Ae^{i\phi_1}\) and \(Ae^{i\phi_2}\).

Hint Hint

Hint

Factor out the common amplitude A.

Answer Answer
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Complete Answer

\(A(e^{i\phi_1} + e^{i\phi_2})\)

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Part c

5 minutes 3 marks

Question

Find the amplitude of the resultant wave in terms of A, \(\phi_1\), and \(\phi_2\).

Hint Hint

Hint

Use the modulus of the summed phasor and trigonometric identities.

Answer Answer
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Complete Answer

\(A\sqrt{2+2\cos(\phi_1-\phi_2)}\)

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