Question

If two loudspeakers at points \(A\) and \(B\) emit identical sine waves at the same frequency and constructive interference is observed at point \(C\), then the a) distance from \(A\) to \(C\) is the same as that from \(B\) to \(C\). b) points \(A, B,\) and \(C\) form an equilateral triangle. c) difference between the distance from \(A\) to \(C\) and the distance from \(B\) to \(C\) is an integer multiple of the wavelength of the emitted waves. d) difference between the distance from \(A\) to \(C\) and the distance from \(B\) to \(C\) is a half-integer multiple of the wavelength of the emitted waves.

Short answer

Answer: c) The difference between the distance from A to C and the distance from B to C is an integer multiple of the wavelength of the emitted waves.

Step-by-step solution

Understanding constructive interference

Constructive interference occurs when two or more waves combine in such a way that the resulting wave has a larger magnitude than the individual waves. This only happens when the waves are in phase, or reach a maximum or minimum at the same time.

Analyzing statement (a)

Statement a contradicts the fact that constructive interference occurs only when waves have the same phase. If A and C are at the same distance, the maximum and minimum points of the sound waves could not happen at the same time, leading to destructive interference instead of constructive interference.

Analyzing statement (b)

Constructive interference does not necessarily mean that points A, B, and C form an equilateral triangle. This statement is true only if the distance from A to C is the same as the distance from B to C, which does not satisfy the interference conditions discussed earlier.

Analyzing statement (c)

Constructive interference occurs when the difference in distance from A to C and B to C is an integer multiple of the wavelength (nλ, n = 0, 1, 2,...). This means that as long as the difference is a multiple of the wavelength, the waves will combine constructively at point C. So statement c) is true.

Analyzing statement (d)

In this statement, it is said that the difference between the distance from A to C and the distance from B to C is a half-integer multiple of the wavelength. This condition would actually lead to destructive interference instead of constructive interference, as the waves would be out of phase. Therefore, statement d) is false. The correct answer is c) the difference between the distance from A to C and the distance from B to C is an integer multiple of the wavelength of the emitted waves.