Question

Audible beats can be produced by two sine waves with frequencies \(f_{1}\) and \(f_{2},\) if a) the two frequencies are close to each other. b) \(f_{1}\) is greater than \(f_{2}\). c) \(f_{1}\) is smaller than \(f_{2}\). d) the two frequencies are identical.

Short answer

a) Frequencies are close to each other b) \(f_{1}\) greater than \(f_{2}\) c) \(f_{1}\) smaller than \(f_{2}\) d) Frequencies are identical Answer: a) Frequencies are close to each other

Step-by-step solution

(Condition a: Frequencies close to each other)

When the two frequencies are close to each other, they can interfere constructively and destructively, which results in the changing pattern of loudness - this is the phenomenon of audible beats. Therefore, under the condition (a), two sine waves can produce audible beats.

(Condition b: \(f_{1}\) greater than \(f_{2}\))

If \(f_{1}\) is greater than \(f_{2}\) but the frequencies are not close to each other, the interference between the two sine waves will not produce the changing pattern of loudness, so no audible beats will be heard. However, if the frequencies are close enough, as described in condition (a), then audible beats can be produced.

(Condition c: \(f_{1}\) smaller than \(f_{2}\))

Similar to condition (b), if \(f_{1}\) is smaller than \(f_{2}\) but the frequencies are not close to each other, the interference between the two sine waves will not produce the changing pattern of loudness, so no audible beats will be heard. However, if the frequencies are close enough, as described in condition (a), then audible beats can be produced.

(Condition d: Frequencies are identical)

When the two frequencies are identical (\(f_{1} = f_{2}\)), the sine waves will always interfere constructively, and there will be no changing pattern of loudness. Therefore, under condition (d), two sine waves cannot produce audible beats.

(Conclusion)

Among the given conditions, only when the two frequencies are close to each other (condition a) can two sine waves produce audible beats.