Two loudspeakers emit sound waves along the x-axis. The sound has maximum intensity when the speakers are 14.0 cm apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of 54.0 cm. What is the wavelength of the sound?

To find the wavelength of the sound, we can use the concept of interference between the two speakers. When the two speakers emit sound waves that interfere constructively, we observe maximum intensity. Conversely, when the sound waves interfere destructively, the intensity becomes zero.

Let's assume that the distance between the speakers is given by D.

The first step is to determine the separation between the speakers at which the sound intensity reaches its maximum value. In this case, that separation is given as 14.0 cm.

Next, we need to find the separation between the speakers at which the sound intensity becomes zero. In this case, that separation is given as 54.0 cm.

Now, we can calculate the wavelength of the sound. The condition for constructive interference is that the path difference between the waves emitted by the two speakers should be an integral multiple of the wavelength.

Therefore, the maximum intensity occurs when the path difference (d) between the speakers is equal to an integer number (n) of wavelengths (λ). Mathematically, we can express this as:

d = n * λ

At the point of maximum intensity, the path difference is zero because the waves are in phase. Hence, in this case, when the speakers are 14.0 cm apart, we have:

14.0 cm = n * λ

For the point where the sound intensity becomes zero, the path difference is equal to half a wavelength (λ/2) because the waves interfere destructively. Therefore, when the speakers are 54.0 cm apart, we have:

54.0 cm = (n + 1/2) * λ

By solving these two equations simultaneously, we can find the wavelength (λ) of the sound.

14.0 cm = n * λ
54.0 cm = (n + 1/2) * λ

Simplifying the second equation, we get:

54.0 cm = nλ + (1/2)λ
53.5 cm = nλ

Dividing the two equations, we have:

54.0 cm / 14.0 cm = (n + 1/2) / n
3.8571 = (n + 1/2) / n

Multiplying through by n to eliminate the fraction, we get:

3.8571n = n + 1/2
2.8571n = 1/2
n = 0.1746

Substituting this value of n back into the first equation, we find:

14.0 cm = 0.1746λ
λ = 80.08 cm

Therefore, the wavelength of the sound is 80.08 cm.