Two identical loudspeakers separated by distance △ x each emit sound waves of wavelength λ and amplitude a along the and amplitude a along the x-axis.What is the minimum value of the ratio △ x / λ for which the amplitude of their superposition is also a?

Question

Two identical loudspeakers separated by distance   △  x each emit sound waves of wavelength   λ and amplitude a along the and amplitude a along the x-axis.What is the minimum value of the ratio   △  x      /    λ for which the amplitude of their superposition is also a?

Jamari Morgan · Accepted Answer

We know that the amplitude of the superposition A will be given by  A  =  2  a  cos  ⁡  (            △      ϕ        2    )On the other hand, the phase difference is given by  △  ϕ  =  2  π            △      x        λ  Combining these two results, we have  A  =  2  a  cos  ⁡  (  π            △      x        λ    )Therefore, for A=a we will have  2  a  cos  ⁡  (  π            △      x        λ    )  =  a  ⇒  cos  ⁡  (  π            △      x        λ    )  =      1    2    ⇒  (  π            △      x        λ        )          m      i      n        =      π    3  Finally,            △      x        λ    =      1    3  Result:1/3

Two identical loudspeakers separated by distance triangle x each emit sound waves of wavelength lambda and amplitude a along the and amplitude a along the x-axis. What is the minimum value of the ratio triangle x/lambda for which the amplitude of their superposition is also a?

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