Question

I would like to know how 1.41m was made in the equation r1=r2 I can send the full problem.

Answer 1

Answer:

To cause destructive interference at point O, the upper speaker must be raised by a minimum vertical distance of 1.66m

Explanations:

The distance between the upper and lower speakers before destructive interference is h

Since the height between the two speakers, h = 2.82 m

h/2 = 2.82/2 = 1.41 m

Using the Pythagora's theorem

`\begin{gathered} r^2_1=8^2+1.41^2 \\ r^2_1\text{ = 64 + }1.9881 \\ r^2_1\text{ = }65.9881 \\ r_1\text{ = }\sqrt[]{65.9881} \\ r_1\text{ = }8.12\text{ m} \end{gathered}`

Since r₁ = r₂

r₂ = 8.12 m

To cause a destructive interference, let us move the speaker on the top by a distance x

We can redraw the diagram as shown below:

The new distance from the midpoint = x + 1.41

The speed of sound in air, v = 343 m/s

The frequency of sound, f = 380 Hz

To calculate the wavelength of the sound produced by the speakers, use the equation:

v = λf

343 = λ x 380

λ = 343/380

λ = 0.903 m

Note that the value of r₁ now will change because the upper speaker has been shifted by a distance x. The value of r₂ remains 8.12m because the speaker at the bottom is not shifted

To find the new value of r₂ use the equation:

r₁ - r₂ = λ / 2

r₁ - 8.12 = 0.903/2

r₁ - 8.12 = 0.45

r₁ = 0.45 + 8.12

r₁ = 8.57 m

To know the value of x, use Pythagora's theorem for the second diagram above

`\begin{gathered} r^2_1=8^2+(x+1.41)^2 \\ 8.57^2=8^2\text{ }+(x+1.41)^2 \\ 73.44\text{ - 64 = }(x+1.41)^2 \\ (x+1.41)^2\text{ = }9.44 \\ x\text{ + 1.41 = }\sqrt[]{9.44} \\ x\text{ + 1.41 = }3.07 \\ x\text{ = 3.07 - 1.41} \\ x\text{ = }1.66\text{ m} \end{gathered}`

To cause destructive interference at point O, the upper speaker must be raised by a minimum vertical distance of 1.66m