Question

1. the frequency of the third harmonic of an open pipe is 900 hz. what is the length of the pipe?

a. 0.19 m
b. 0.28 m
c. 0.57 m
d. 1.1 m

1. what determines whether two overlapping waves produce constructive or destructive interference?

a. their speed.
b. the directions that the waves are traveling.
c. their amplitudes
d. the phase relationship between the waves.

Explanation:

Step1: Recall the formula for the frequency of harmonics in an open - pipe

The formula for the frequency of the \(n\)th harmonic in an open - pipe is \(f_n=\frac{n v}{2L}\), where \(v = 343\ m/s\) (speed of sound in air at room temperature), \(n\) is the harmonic number, \(f_n\) is the frequency of the \(n\)th harmonic, and \(L\) is the length of the pipe.
For the third harmonic \(n = 3\) and \(f_3=900\ Hz\).
We can re - arrange the formula for \(L\): \(L=\frac{n v}{2f_n}\)

Step2: Substitute the values into the formula

Substitute \(n = 3\), \(v = 343\ m/s\), and \(f_n=900\ Hz\) into the formula \(L=\frac{n v}{2f_n}\)

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For question 16:

• Option A: The speed of waves (\(v=\lambda f\)) does not determine the type of interference (constructive or destructive). It is related to the wavelength (\(\lambda\)) and frequency (\(f\)) of the wave.
• Option B: The direction of wave travel does not determine constructive or destructive interference. Interference is mainly about the superposition of waves at a point in space.
• Option C: The amplitudes of waves determine the intensity of the resultant wave after interference (\(I\propto A^{2}\)), but not whether the interference is constructive or destructive.
• Option D: For two overlapping waves \(y_1 = A_1\sin(\omega t +\phi_1)\) and \(y_2=A_2\sin(\omega t+\phi_2)\), the phase difference \(\Delta\phi=\phi_2 - \phi_1\). If \(\Delta\phi = 2m\pi\) (\(m = 0,1,2,\cdots\)), constructive interference occurs (\(y=y_1 + y_2\) and \(A = A_1+A_2\) if \(A_1 = A_2\)). If \(\Delta\phi=(2m + 1)\pi\) (\(m=0,1,2,\cdots\)), destructive interference occurs (\(y=y_1 + y_2\) and \(A=\vert A_1 - A_2\vert\) if \(A_1 = A_2\))

Answer:

1. C. \(0.57\ m\)
2. D. The phase relationship between the waves.