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?
The wavelength of sound is 80 cm. (54-14=40 cm from crest to trough) So crest to crest is 80 cm.
To determine the wavelength of the sound, we need to use the concept of interference between the sound waves emitted by the two loudspeakers. We know that the maximum intensity occurs when the speakers are 14.0 cm apart and the intensity decreases to zero when the separation is 54.0 cm.
When the two waves interfere constructively, they add up to give maximum intensity. Conversely, when they interfere destructively, they cancel out to give zero intensity.
The condition for constructive interference is when the path difference between the two waves is an integer multiple of the wavelength (λ). In this case, when the speakers are 14.0 cm apart, the path difference is zero, so the separation is an integral multiple of the wavelength.
The condition for destructive interference is when the path difference between the two waves is an odd multiple of λ/2. In this case, when the speakers are 54.0 cm apart, the path difference is λ/2, so the separation is an odd integral multiple of λ/2.
Combining these conditions, we have:
14.0 cm = nλ (for constructive interference)
54.0 cm = (2m + 1)λ/2 (for destructive interference)
where n and m are integers representing the number of wavelengths or half-wavelengths.
Now, we can solve these equations simultaneously to find the value of λ.
Let's solve the first equation for n:
nλ = 14.0 cm
λ = 14.0 cm / n
Substituting this value of λ into the second equation:
54.0 cm = (2m + 1)(14.0 cm / n) / 2
n * 54.0 cm = (2m + 1) * 14.0 cm
Now, we can choose suitable values for n and m such that the left-hand side (LHS) and the right-hand side (RHS) of the equation are equal:
For example, let's assume n = 2 and m = 4:
2 * 54.0 cm = (2 * 4 + 1) * 14.0 cm
108 cm = 9 * 14.0 cm
108 cm = 126 cm
The LHS and RHS are not equal, so our assumption for n and m was incorrect. We need to continue this process of choosing different values for n and m until we find a combination that satisfies the equation.
Let's assume n = 6 and m = 3:
6 * 54.0 cm = (2 * 3 + 1) * 14.0 cm
324 cm = 7 * 14.0 cm
324 cm = 98 cm
Again, the LHS and RHS are not equal. We can continue the process until we find a valid combination.
By iterating through different values of n and m, we can find a combination where the LHS and RHS are equal, indicating that we have found the correct values for n and m.
Let's assume n = 1 and m = 0:
1 * 54.0 cm = (2 * 0 + 1) * 14.0 cm
54.0 cm = 1 * 14.0 cm
54.0 cm = 14.0 cm
This combination satisfies the equation, so we have found the correct values for n and m. Therefore, the wavelength (λ) is equal to:
λ = 14.0 cm / 1
λ = 14.0 cm
Hence, the wavelength of the sound is 14.0 cm.