Question

Two coherent sound waves have intensities of $0.040 \mathrm{W} / \mathrm{m}^{2}\( and \)0.090 \mathrm{W} / \mathrm{m}^{2}$ where you are listening. (a) If the waves interfere constructively, what is the intensity that you hear? (b) What if they interfere destructively? (c) If they were incoherent, what would be the intensity? [Hint: If your answers are correct, then (c) is the average of (a) and (b).]

Short answer

Answer: (a) In the constructive interference case, the intensity = 0.18218 W/m². (b) In the destructive interference case, the intensity = 0.00782 W/m². (c) In the incoherent waves case, the intensity = 0.130 W/m².

Step-by-step solution

Calculate amplitudes of given sound waves

Firstly, we need to find the amplitude of each sound wave using intensity formula: $$I = \frac{1}{2} p_0 v A^2$$. Here, \(I\) is the intensity, \(p_0\) is the air density (given by 1.2 kg/m³), \(v\) is the speed of sound (340 m/s), and \(A\) is the amplitude. Rearranging the formula to find amplitude, we get: $$A = \sqrt{\frac{2 \cdot I}{p_0\cdot v}}$$. Let's find the two amplitudes using the given intensities.

Find amplitudes A1 and A2

For the first wave with intensity \(I_1\), calculate the amplitude \(A_1\): $$A_1 = \sqrt{\frac{2 \cdot 0.040}{1.2 \cdot 340}} = 0.009223$$ For the second wave with intensity \(I_2\), calculate the amplitude \(A_2\): $$A_2 = \sqrt{\frac{2 \cdot 0.090}{1.2 \cdot 340}} = 0.013816$$

Calculate the resulting intensity in case of constructive interference

In the case of constructive interference, the amplitudes add up directly: $$A_\text{total} = A_1 + A_2 = 0.009223 + 0.013816 = 0.023039$$ Now, use the intensity formula to find the resulting intensity, \(I_\text{total}\): $$I_\text{total} = \frac{1}{2} p_0v (A_\text{total})^2$$ $$I_\text{total} = \frac{1}{2} \cdot 1.2 \cdot 340 \cdot (0.023039)^2 = 0.18218\ \text{W}/\text{m}^2$$ So, the resulting intensity in constructive interference is 0.18218 W/m².

Calculate the resulting intensity in case of destructive interference

In the case of destructive interference, the difference in amplitudes determines the resulting amplitude: $$A_\text{total} = \lvert A_1 - A_2 \rvert = 0.013816 - 0.009223 = 0.004593$$ Now, use the intensity formula to find the resulting intensity, \(I_\text{total}\): $$I_\text{total} = \frac{1}{2} p_0v (A_\text{total})^2$$ $$I_\text{total} = \frac{1}{2} \cdot 1.2 \cdot 340 \cdot (0.004593)^2 = 0.00782\ \text{W}/\text{m}^2$$ So, the resulting intensity in destructive interference is 0.00782 W/m².

Calculate the resulting intensity in case of incoherent waves

For incoherent waves, the intensities add up directly: $$I_\text{total} = I_1 + I_2 = 0.040 + 0.090 = 0.130\ \text{W}/\text{m}^2$$ Thus, the resulting intensity in case of incoherent waves is 0.130 W/m². Now let's summarize the results: (a) In the constructive interference case, the intensity = 0.18218 W/m² (b) In the destructive interference case, the intensity = 0.00782 W/m² (c) In the incoherent waves case, the intensity = 0.130 W/m²