problem 14.57

a pair of in-phase stereo speakers is placed side by side, 0.784 m apart. you stand directly in front of one of the speakers, 2.54 m from the speaker.

part a
what is the lowest frequency that will produce constructive interference at your location?
express your answer using two significant figures.
(f = ) hz

Question

problem 14.57

a pair of in-phase stereo speakers is placed side by side, 0.784 m apart. you stand directly in front of one of the speakers, 2.54 m from the speaker.

part a
what is the lowest frequency that will produce constructive interference at your location?
express your answer using two significant figures.
(f = ) hz

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Wave Interference", "Path Difference", "Constructive Interference" ], "current_concepts": [ "Wave Interference", "Path Difference", "Constructive Interference" ] } ## Calculate the path lengths from each speaker to the observer $$ \begin{aligned} d &= 0.784\text{ m} \ x_1 &= 2.54\text{ m} \ x_2 &= \sqrt{x_1^2 + d^2} = \sqrt{2.54^2 + 0.784^2} \approx 2.6582\text{ m} \end{aligned} $$ ## Determine the path difference and lowest frequency for constructive interference $$ \begin{aligned} \Delta x &= x_2 - x_1 = 2.6582 - 2.54 = 0.1182\text{ m} \ \Delta x &= m \lambda = m \frac{v}{f} \quad (m = 1 \text{ for lowest non-zero frequency}) \ f &= \frac{v}{\Delta x} = \frac{343\text{ m/s}}{0.1182\text{ m}} \approx 2901.8\text{ Hz} \approx 2.9 \times 10^3\text{ Hz} \end{aligned} $$ What is the lowest frequency that will produce constructive interference at your location? Express your answer using two significant figures. $f =$ $2.9 \times 10^3$ $\text{Hz}$ { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "Speaker 1", "color": "#8C55F2", "size": 4, "withLabel": true } }, { "type": "point", "params": [[0, 0.784]], "properties": { "name": "Speaker 2", "color": "#8C55F2", "size": 4, "withLabel": true } }, { "type": "point", "params": [[2.54, 0]], "properties": { "name": "Observer", "color": "#F2557F", "size": 4, "withLabel": true } }, { "type": "segment", "params": [[0, 0], [2.54, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "dash": 1 } }, { "type": "segment", "params": [[0, 0.784], [2.54, 0]], "properties": { "strokeColor": "#F28A55", "strokeWidth": 2, "dash": 1 } }, { "type": "segment", "params": [[0, 0], [0, 0.784]], "properties": { "strokeColor": "#583C87", "strokeWidth": 2 } } ] } { "subject": "Natural Science", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Natural Science", "Physics", "Wave Interference" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the path lengths from each speaker to the observer

Determine the path difference and lowest frequency for constructive interference

Answer: