Question

A sound wave with intensity \(25 \mathrm{mW} / \mathrm{m}^{2}\) interferes constructively with a sound wave that has an intensity of $15 \mathrm{mW} / \mathrm{m}^{2} .$ What is the intensity of the superposition of the two? (tutorial: superposition)

Short answer

Answer: The intensity of the superposition of the two sound waves is approximately 40 mW/m².

Step-by-step solution

Understanding the relationship between intensity and amplitude

The intensity of a sound wave is proportional to the square of its amplitude (A). The following formula relates intensity (I) to amplitude (A): I = kA², where k is a constant of proportionality.

Finding the amplitude of each sound wave

Using the formula I = kA², we can find the amplitude of the individual sound waves. We can write this as: A₁² = I₁/k A₂² = I₂/k Where A₁ and A₂ are the amplitudes, and I₁ and I₂ are the intensities of the two waves.

Adding the amplitudes

As given, the two sound waves interfere constructively, which means the amplitudes should be added to find the amplitude of the superposition. We can denote the amplitude of the resulting wave as A: A = A₁ + A₂

Finding the intensity of the superposition

Now, to find the intensity of the superposition, we can use I = kA² again. We will need to square the amplitude of the resulting wave to find the intensity: I = k(A₁ + A₂)²

Eliminating the constant 'k'

From Step 2, we can write k in terms of A₁² and I₁: k = I₁/A₁² Substituting this into the equation from Step 4, we get: I = I₁/A₁² * (A₁ + A₂)²

Completing the calculation

We now have an equation that relates the intensity of the superposition, Intensity I, to the amplitudes and intensities of the original waves. Plug in the given values of I₁ (25 mW/m²) and I₂ (15 mW/m²): I = (25/((25/mW/m²))²) * (A₁ + A₂)² Solving for I, we find the intensity of the superposition to be: I ≈ 40 mW/m² The intensity of the superposition of the two sound waves is approximately 40 mW/m².