Question:
A string is stretched tightly between two fixed points, labeled as "A" and "B." Point "C" is located at the midpoint of the string, equidistant from points "A" and "B." The string is oscillating with a frequency of 120 Hz.
a) If the wavelength of the standing wave on the string is 2.4 meters, calculate the speed of the wave.
b) The string is vibrating in its fifth harmonic. Calculate the length of the string.
c) A second identical string is placed parallel to the original string, with a separation distance of 0.6 meters between them. If both strings are vibrating with the same frequency of 120 Hz, determine the locations of constructive and destructive interferences. Explain your answer.
Answer:
a) The speed of a wave can be calculated using the equation:
\(v = f \times \lambda \)
Where: \( v \) = speed of the wave (m/s) \( f \) = frequency of the wave (Hz) \( \lambda \) = wavelength of the wave (m)
Given: Frequency \( f = 120 , \text{Hz} \) Wavelength \( \lambda = 2.4 , \text{m} \)
Substituting the given values into the equation:
\( v = 120 , \text{Hz} \times 2.4 , \text{m} = \boxed{288 , \text{m/s}} \)
b) The length of a string vibrating in its nth harmonic can be calculated using the equation:
\( L = \frac{n}{2} \times \lambda \)
Where: \( L \) = length of the string (m) \( n \) = harmonic number (unitless) \( \lambda \) = wavelength of the wave on the string (m)
Given: Harmonic number \( n = 5 \) Wavelength \( \lambda = 2.4 , \text{m} \)
Substituting the given values into the equation:
\( L = \frac{5}{2} \times 2.4 , \text{m} = \boxed{6 , \text{m}} \)
c) In a standing wave system, constructive interference occurs at the points where two waves have a phase difference of zero (i.e., the crests and troughs of the waves align). Destructive interference occurs at the points where two waves have a phase difference of 180 degrees (i.e., the crests of one wave align with the troughs of the other wave).
Therefore, to determine the locations of constructive and destructive interferences, we need to consider the phase difference between the waves on the two parallel strings.
The phase difference \( \Delta \phi \) between two waves can be calculated using the equation:
\( \Delta \phi = \frac{{2 \pi \Delta x}}{\lambda} \)
Where: \( \Delta \phi \) = phase difference (radians) \( \Delta x \) = distance between the sources of the waves (m) \( \lambda \) = wavelength (m)
Given: Separation distance between the strings \( \Delta x = 0.6 , \text{m} \) Wavelength of the waves \( \lambda = 2.4 , \text{m} \)
Substituting the given values into the equation:
\( \Delta \phi = \frac{{2 \pi \times 0.6 , \text{m}}}{2.4 , \text{m}} = \pi , \text{radians} \)
Constructive interference occurs when the phase difference is an integer multiple of \( 2 \pi \), while destructive interference occurs when the phase difference is an odd multiple of \( \pi \).
For constructive interference: \( \Delta \phi = 2 \pi n \), where \( n \) is an integer
\( \pi = 2 \pi n \) \( n = \frac{1}{2} \)
Therefore, constructive interference occurs at a separation distance of 0.6 meters from the original string.
For destructive interference: \( \Delta \phi = (2n + 1) \pi \), where \( n \) is an integer
\( \pi = (2n + 1) \pi \) \( n = 0 \)
Therefore, destructive interference occurs at the original string itself.
In summary: