Question

Two identical loudspeakers 2.00 m apart are emitting sound waves into a room where the speed of sound is 340 m/s. Abby is standing 5.50 m in front of one of the speakers perpendicular to the line joining the speakers, and hears a maximum in the intensity of the sound. What is the lowest possible frequency of sound for which this is possible? Express your answer with the appropriate units.

Answer 1

Answer:

242.85 Hz

Explanation:

For maximum intensity of sound, the path difference,ΔL = (n + 1/2)λ/2 where n = 0,1,2...

Since Abby is standing perpendicular to one speaker, the path length for the sound from the other speaker to him is L₁ = √(2.00² + 5.50²) = √(4.00 + 30.25) = √34.25 = 5.85 m.

The path difference to him is thus ΔL = 5.85 m - 5.50 m = 0.35 m.

Since ΔL = (n + 1/2)λ/2 and for lowest frequency n = 0,

ΔL = (n + 1/2)λ/2 = (0 + 1/2)λ/2 = λ/4

ΔL = λ = v/f and f = v/4ΔL where f = frequency of wave and v = velocity of sound wave = 340 m/s.

f = 340/(4 × 0.35) = 242.85 Hz

Answer 2

The condition of constructive interference and the speed of a wave allows finding the frequency for the first constructive interference that Abby hears is:

         f = 971.4 Hz

The interference of the coherent sound wave occurs when two wave fronts are added at a point and the intensity depends on the path difference of these waves, we have two extreme cases:

• Destructive. In this case the sum of the intensity gives a resultant of zero.

• Constructive. The sum of the intensities gives a maximum and is described by the expression.

          Δr = $2n \ \frac{\lambda}{2}$

Where Δr is the path difference, λ is the wavelength and n is an integer.

In the attachment we can see the distance from Abby is to the speakers for the closest y = 5.50 m, the distance to the furthest speaker.

Let's use the Pythagoras' theorem,

             r =$\sqrt{x^2 + y^2}$  

Let's calculate.

            r = $\sqrt{2^2 + 5.5^2}$  

            r = 5.85 m

The difference in path is:

           Δr = r-y

           Δr = 5.85 - 5.5

           Δr = 0.35 m

let's find the wavelength.

          Δr = 2n $\frac{\lambda}{2}$  

          λ = $\frac{\Delta r}{n}$

The first constructive interference occurs for n = 1

       

           λ= Δr

           λ = 0.35 m

Wave speed is proportional to wavelength and frequency.

          v = λ f

          f = $\frac{v}{\lambda }$

Let's calculate.

          f = $\frac{340}{0.35}$  

          f = 971.4 Hz

In conclusion using the condition of constructive interference and the speed of a wave we can find the frequency for the first constructive interference that Abby hears is:

          f = 971.4 Hz

Learn more here:  brainly.com/question/3648655