Question

Suppose a speaker is at (0, 2.3) m and an identical speaker, connected to the same amplifier, is at (6.6, 0) m. Both speakers are putting out a sound with the same frequency and phase. Find the x-value of a point of constructive interference between (0,0) and the speaker on the x-axis, regardless of frequency.

Answer 1

To find the x-value of a point of constructive interference between (0,0) and the speaker on the x-axis, we need to consider the path difference between the two speakers.

Step-by-step explanation:

To find the x-value of a point of constructive interference between (0,0) and the speaker on the x-axis, we need to consider the path difference between the two speakers.

The formula for path difference is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, x1 = 0, y1 = 2.3, x2 = 6.6, and y2 = 0. Plugging these values into the formula, we get:

d = sqrt((6.6 - 0)^2 + (0 - 2.3)^2) = sqrt(43.56 + 5.29) = sqrt(48.85) = 6.99 m

Since the path difference is equal to one wavelength for constructive interference, we can equate it to the wavelength formula:

d = λ * n, where n is an integer.

Therefore, we can solve for the wavelength:

λ = d / n = 6.99 m / n

To find the x-value of the point of constructive interference between (0,0) and the speaker on the x-axis, we need to find the value of n that makes the x-value equal to zero.

Setting x = 0, we get:

0 = 6.99 m / n

Solving for n, we get:

n = 6.99 m / 0 = undefined

Therefore, there is no point of constructive interference with an x-value equal to zero between (0,0) and the speaker on the x-axis.

Answer 2

To find a point of constructive interference between the two speakers, we need to ensure that the sound waves from both speakers arrive at this point in phase. Constructive interference occurs when the path length difference between the waves from the two speakers is an integer multiple of the wavelength
`(\(n\lambda\)`, where
`\(n\)` is an integer and
`\(\lambda\)` is the wavelength).

Let's consider a point
`\(P(x, 0)\)` on the x-axis between the origin (0,0) and the speaker on the x-axis. We need to find the value of
`\(x\)` where the path length difference between the sound waves from the two speakers to point
`\(P\)` meets the condition for constructive interference.

1. Calculate the Distance from Each Speaker to Point P :

- Distance from the first speaker at (0, 2.3) to P:
`\(d_1 = √((x-0)^2 + (0-2.3)^2)\)`

- Distance from the second speaker at (6.6, 0) to P:
`\(d_2 = |6.6 - x|\)`

2. Path Length Difference :

The path length difference
`\(\Delta d\) is \(d_2 - d_1\).`

3. Constructive Interference Condition :

For constructive interference,
`\(\Delta d = n\lambda\)`. Since we are not given the wavelength and the frequency, we assume the simplest case where
`\(n=0\)` for the first point of constructive interference. Thus,
`\(\Delta d = 0\)`.

4. **Setting Up the Equation**:

`\[ |6.6 - x| - √(x^2 + (2.3)^2) = 0 \]`

5. Solving for x :

This requires solving the equation to find the value of
`\(x\).`

Let's perform the calculation.

The symbolic solver encountered a difficulty with the absolute value function in the equation. To resolve this, we can consider the problem in two separate cases based on the range of
`\(x\):`

1. Case 1 :
`\(x \leq 6.6\), where \(|6.6 - x| = 6.6 - x\).`

2. Case 2 :
`\(x > 6.6\), where \(|6.6 - x| = x - 6.6\).`

Since we are looking for a point between (0,0) and the speaker on the x-axis (6.6, 0), we will focus on Case 1. The equation becomes:

`\[ 6.6 - x - √(x^2 + 2.3^2) = 0 \]`

Now, let's solve this equation to find the value of
`\(x\).`

For the case where
`\(x \leq 6.6\),`the value of
`\(x\)`at a point of constructive interference between the two speakers is approximately
`\(2.90\)` meters.

This is the x-coordinate of the first point on the x-axis between (0,0) and the speaker at (6.6, 0) where constructive interference occurs, assuming the simplest case of
`\(n=0\).`