To find the distances at which you hear minimum sound intensity, we need to consider destructive interference between the two speakers.

Step 1: Calculate the Speed of Sound

The speed of sound in air at 20°C is given by:

$v = 343 \text{ m/s}$

Step 2: Determine the Wavelength

The wavelength ($\lambda$) of the sound wave is:

$\lambda = \frac{v}{f} = \frac{343}{686} = 0.50 \text{ m}$

Step 3: Condition for Destructive Interference

Destructive interference occurs when the path difference ($\Delta d$) between the two waves is:

$\Delta d = \left(m + \frac{1}{2} \right) \lambda$

for $m = 0, 1, 2, \dots$.

Step 4: Define Distances and Setup Equation

• Let speaker 1 be at $(0,0)$.
• Let speaker 2 be at $(3,0)$.
• The listener starts at (0, 2.5) and moves along the positive $y$-axis.
• Distance from speaker 1: $d_1 = y$.
• Distance from speaker 2 (using the distance formula):

$d_2 = \sqrt{3^2 + y^2} = \sqrt{9 + y^2}$

The path difference is:

$\Delta d = d_2 - d_1 = \sqrt{9 + y^2} - y$

Setting it equal to destructive interference conditions:

$\sqrt{9 + y^2} - y = \left(m + \frac{1}{2} \right) (0.50)$

Step 5: Solve for $y$

Solving this equation for $m = 0, 1, 2, 3$, we get values for $y$. Let's compute these values. sympy as sp

Define variable

y = sp.symbols('y')

Given values

lambda_ = 0.50 # Wavelength in meters

Equation: sqrt(9 + y^2) - y = (m + 1/2) * lambda

solutions = [] for m in range(4): # Finding the first few values eq = sp.Eq(sp.sqrt(9 + y**2) - y, (m + 1/2) * lambda_) sol = sp.solve(eq, y) # Extracting positive real solutions real_sol = [s.evalf() for s in sol if s.is_real and s > 0] solutions.extend(real_sol)

Sorting solutions in increasing order

solutions = sorted(solutions) solutionsThe distances from the speaker at which you hear minimum sound intensity are:

$1.70, 2.98, 5.63, 17.88 \text{ meters}$

(rounded to two decimal places).

Would you like a step-by-step explanation for solving the equation?

Here are five related questions:

1. How would the results change if the speakers were 4 meters apart instead of 3?
2. What happens to the interference pattern if the sound waves are out of phase?
3. How does temperature affect the speed of sound and interference pattern?
4. What if the listener moves along a different path—how would the equations change?
5. How can this concept be applied to noise-canceling headphones?

Tip: Destructive interference occurs at odd multiples of $\lambda/2$, while constructive interference occurs at integer multiples of $\lambda$.