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?
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To find the wavelength of the sound, we can use the interference pattern created by the two loudspeakers.
The maximum intensity occurs when the path difference between the two wave sources is an integer multiple of the wavelength. In other words, the path difference should be equal to an integer multiple of the wavelength (nλ, where n is an integer).
Given that the maximum intensity occurs when the speakers are 14.0 cm apart and the intensity becomes zero at a separation of 54.0 cm, we can calculate the path difference between these two points.
Path difference = 54.0 cm - 14.0 cm = 40.0 cm.
Now, we need to find the smallest value of n (integer) for which the path difference equals nλ.
Since we have two loudspeakers emitting sound waves along the x-axis, there will be two paths for the sound waves to travel. For simplicity, let's assume that the two paths are equal in length when the speakers are 14.0 cm apart. This means that when the speakers are 14.0 cm apart, the path difference will be equal to one wavelength.
Therefore, the path difference of 40.0 cm corresponds to 40.0 cm / 14.0 cm = 2.86 wavelengths (approximately).
Since path difference = nλ, we can say that 2.86 wavelengths = nλ.
To find the value of n, we can divide both sides of the equation by the number of wavelengths (2.86) to get:
n = 2.86 / λ.
Since n should be an integer, the value of λ should be such that 2.86 is approximately an integer. In this case, the closest integer value to 2.86 is 3.
So, n = 3.
Now, we can rearrange the equation to solve for the wavelength (λ):
λ = 2.86 / n = 2.86 / 3 = 0.953 cm.
Therefore, the wavelength of the sound is approximately 0.953 cm.