Wave Function Transformation and Intersection Analysis

00:00 9 marks

A sound wave in a narrow hallway is modeled by \(y=3\sin(2x)+4\cos(2x)\).

Part a

5 minutes 3 marks

Question

Express \(y=3\sin(2x)+4\cos(2x)\) in the form \(R\sin(2x+\alpha)\), where \(R>0\) and \(0\leq \alpha < 2\pi\). Give the exact value of \(R\) and find \(\alpha\) to 3 significant figures.

Hint Hint

Hint

Calculate \(R = \sqrt{3^2+4^2}\). Use \(\cos\alpha = \frac{3}{R}\) and \(\sin\alpha = \frac{4}{R}\) to find \(\alpha\) in the correct quadrant.

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\(R=5\) and \(\alpha=0.927\) radians (3 s.f.), so \(y=5\sin(2x+0.927)\).

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Part b

5 minutes 3 marks

Question

Determine the smallest positive value of \(x\) for which the wave attains a value of 3, using the R-form from part (a). Give your answer to 3 significant figures.

Hint Hint

Hint

Solve \(5\sin(2x+\alpha)=3\) for \(x\). Isolate \(\sin(2x+\alpha)\). Find the principal value of \(2x+\alpha\) and use general solutions to find the smallest positive \(x\). Use \(\alpha \approx 0.927\) radians.

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\(x = \frac{1}{2}(\arcsin(\frac{3}{5}) – \arctan(\frac{4}{3}) + \pi) \approx 0.785\) radians (3 s.f.).

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Part c

5 minutes 3 marks

Question

Discuss the effect of the phase shift \(\alpha = \arctan(\frac{4}{3}) \approx 0.927\) radians on the wave’s peaks in the hallway sound wave context. Explain the physical implication of this phase shift.

Hint Hint

Hint

Positive phase shift in \(R\sin(Bx+\alpha)\) is a leftward shift by \(\frac{\alpha}{B}\). Consider peak timing and spatial location, relate to sound wave propagation.

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Phase shift \(\alpha \approx 0.927\) radians in \(y=5\sin(2x+\alpha)\) shifts peaks to smaller \(x\) values compared to \(y=5\sin(2x)\). This is a left shift by \(\frac{\alpha}{2} \approx \frac{0.927}{2} \approx 0.464\) units. In hallway sound waves, peaks of compression/rarefaction are spatially shifted earlier. Physically, phase shift arises from sound generation initial conditions, interference, or hallway reflections, altering wave starting position and peak locations.

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