Modeling Wave Interference Using Trigonometric Functions

00:00 4 marks

In a physics experiment, two sinusoidal waves of the same angular frequency \(\omega\) travel in the same direction and interfere. The waves are described by the functions \(y_1 = 3\sin(\omega t)\) and \(y_2 = 2\sin\left(\omega t + \frac{\pi}{3}\right)\).

(a) Show that the resulting wave can be expressed in the form \(y = R\sin(\omega t + \delta)\).

(b) Determine the values of \(R\) and \(\delta\) in exact form.

Hint Hint

Hint

Use the sine addition formula \(\sin(A+B) = \sin A \cos B + \cos A \sin B\) to expand \(y_2\) and then combine terms to express in the form \(R\sin(\omega t + \delta)\).

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Complete Answer

The resulting wave is \(y = \sqrt{19}\sin\left(\omega t + \arctan\left(\frac{\sqrt{3}}{4}\right)\right)\), where \(R = \sqrt{19}\) and \(\delta = \arctan\left(\frac{\sqrt{3}}{4}\right)\).

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