Question 1

A harmonic oscillator with mass $$m$$ and natural frequency $$\$$omega_0$$ is subjected to a periodic force F(t$$) = $$F_0$$ \sin(\omega t)$$. At resonance, which of the following statements is TRUE about the phase relationship between the driving force and the oscillator's motion?

Accepted Answer

The driving force is always in phase with the oscillator's motion.

Question 2

For a pipe with both ends open, the fundamental frequency $$f_1 is $$given by $$f_1$$ = \frac{v}{2L}$$, where v$ is the speed of sound and L$ is the length of the pipe. What are the possible resonant frequencies for this pipe?

Accepted Answer

$$f_n = n f_1$ with n = 1$$, 2, 3, \ldots$$

Question 3

A pipe with one end closed and one end open has a length $$L. $$What is the formula for its fundamental frequency $$f_1$$?

Accepted Answer

$$f_1$$ = \frac{v}{4L}$$

Question 4

Which of the following factors does NOT contribute to the finite width and height of resonant peaks in a pipe?

Accepted Answer

The speed of sound in air

Question 5

Two sound waves of the same frequency are generated at points $$x_1$$ and $$x_2. If$$ $$\$$phi_1$$ = \$$phi_2$$ = 0$$, what is the condition for constructive interference at a point $$x$$?

Accepted Answer

$$\Delta x = n \lambda$$

Question 6

In a two-source interference experiment, the phase difference at a point is $$\Delta \phi = k(r_2$ - r_1$). $$What is the condition for destructive interference?

Accepted Answer

$$\Delta \phi = (2n+1) \pi$$

Question 7

A listener moves from point O to point P, a perpendicular distance of 0.35 m, and hears the first cancellation of sound from two speakers placed 3.0 m apart. If the speakers are driven in phase, what is the frequency of the oscillator if the speed of sound is 343 m/s?

Accepted Answer

$$f = 490 Hz$$

Question 8

Two oscillatory processes $$y_1(t) = A$$ \sin(\$$omega_1 t$$)$$ and y_2$(t) = A \sin(\$$omega_2$$ t)$$ are superposed. What is the beat frequency $$f_b$$?

Accepted Answer

$$f_b = |f_1$ - f_2$|$$

Question 9

A piano tuner hears 2.00 beats/s when tuning a string to a reference note at 523 Hz. What are the possible frequencies of the string?

Accepted Answer

$$525 Hz $$and $$521 Hz$$

Question 10

If the piano tuner tightens the string and hears 3.00 beats/s, what is the new frequency of the string?

Accepted Answer

$$526 Hz$$

Question 11

The beat period $$T_b is $$given by $$T_b = \frac{2\pi}{|\$$omega_1$$ - \$$omega_2$$|}. If$$ $$\$$omega_1$$ = 100$$ rad/s and $$\$$omega_2$$ = 98$$ rad/s, what is $$T_b$$?

Accepted Answer

$$T_b = \pi s$$

Question 12

A human auditory canal is about 25 mm long. Using $$f_1$$ = \frac{v}{4L}$$ and v = 343$ m/s, what is the fundamental frequency?

Accepted Answer

$$f_1 = 3430$ Hz

Question 13

Which of the following best explains why sounds of 3-4 kHz appear loudest to the human ear?

Accepted Answer

The resonant frequency of the auditory canal is in the 3-4 kHz range.

Question 14

If two sources emit sound waves with amplitudes $$\Delta P_1$ and $$\Delta $$P_2$$, what is the expression for the excess pressure at a distance $$r$$ from the sources?

Accepted Answer

$$\Delta P(r, t) = \Delta P_1$ \sin(kr_1$ - \omega t) + \Delta P_2$ \sin(kr_2$ - \omega t)$$

Question 15

In the context of beats, what happens to the amplitude of the resulting wave when $$\cos\left(\frac{\$$omega_1$$ - \$$omega_2$$}{2} t\right) = 0$$?

Accepted Answer

The amplitude is at a minimum (zero).

Question 16

A pipe with one end closed and length $$L$$ supports resonant frequencies at $$f_n = n f_1$ with n = 1$$, 3, 5, \ldots$$. What is the physical reason for this pattern?

Accepted Answer

Only odd harmonics are allowed because the closed end is a node and the open end is an antinode.

Question 17

If the length of a pipe is doubled, what happens to its fundamental frequency?

Accepted Answer

It is halved.

Question 18

Which scenario best illustrates the concept of beats in a real-world context?

Accepted Answer

A piano tuner hears a periodic variation in loudness when two notes are played together.

Question 19

If two waves have a path difference $$\Delta r = r_2$ - r_1$ = \lambda/2$$, what type of interference occurs?

Accepted Answer

Destructive interference

Resonance, Interference, and Beats in Sound Waves

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