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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 16.9 Practice - Waves

Subsection A*R*C*S Practice

A*R*C*S 16.9.1. Wave Calculation.

A sinusoidal wave travels along a stretched string. A particle on the string has a maximum velocity of \(1.40 \mathrm{~m/s}\) and a maximum acceleration of \(270 \mathrm{~m/s^2}\text{.}\) Find the frequency and amplitude of the wave.
Solution.
1. Analyze and Represent
  1. List quantities.
    \begin{equation*} v_{\text{max}} = 1.4 \mathrm{~m/s} \end{equation*}
    \begin{equation*} a_{\text{max}} = 270 \mathrm{~m/s^2} \end{equation*}
    \begin{equation*} f = ? \end{equation*}
    \begin{equation*} y_{\text{max}} = ? \end{equation*}
  2. Identify assumptions. No losses in the string and sinusoidal wave motion
  3. Represent the situation physically. A history graph of the wave can be seen below. The maximum speed happens when the displacement is zero, and the maximum acceleration happens when the displacement is at a maximum
Figure 16.9.1. A sinusoidal wave with the instants of maximum velocity and acceleration shown.
2. Calculate
  1. Represent principles symbolically.
    Angular frequency: \(\omega = 2\pi f\)
    Wave equation: \(y(x,t) = y_{\text{max}}\cos{(2𝜋x/\lambda + \omega t + 𝜙)}\)
    \begin{equation*} v_{\text{max}} = \omega y_{\text{max}} \end{equation*}
    \begin{equation*} a_{\text{max}} = \omega^2 y_{\text{max}} \end{equation*}
  2. Solve unknown(s) symbolically. You can solve for \(\omega\) using the relationships of the amplitudes:
    \begin{equation*} \omega = \frac{a_{\text{max}}}{v_{\text{max}}} \end{equation*}
    Then the frequency can be written as:
    \begin{equation*} f = \frac{a_{\text{max}}}{2\pi v_{\text{max}}} \end{equation*}
    Finally, the amplitude can be written as:
    \begin{equation*} y_{\text{max}} = \frac{v_{\text{max}}}{\omega} \end{equation*}
    \begin{equation*} y_{\text{max}} = \frac{v_{\text{max}}^2}{a_{\text{max}}} \end{equation*}
  3. \begin{equation*} f = 31 \mathrm{~Hz} \end{equation*}
    \begin{equation*} y_{\text{max}} = 7 \times 10^{-3} \mathrm{~m} \end{equation*}
3. Sensemake
  1. Are the units correct?
    \begin{equation*} \mathrm{1/s} = \frac{\mathrm{m/s^2}}{\mathrm{m/s}} \end{equation*}
    \begin{equation*} \mathrm{m} = \frac{\mathrm{m^2/s^2}}{\mathrm{m/s^2}} \end{equation*}
  2. Is your numerical answer reasonable? This string has a frequency of 30 Hertz. For a string with this frequency, it is not surprising the maximum displacement is only a few millimeters. This could likely be built in a lab, so these numbers do seem reasonable.
  3. Does your symbolic answer make physical sense?
    If the maximum speed increases, the frequency decreases and the displacement increases. With more speed, a particle on the string is travelling faster up and down in a larger period.

A*R*C*S 16.9.2. Jet Engine.

During takeoff, the sound intensity level of a jet engine is \(150 \mathrm{~dB}\) at a distance of \(22 \mathrm{~m}\text{.}\) What is the sound intensity level at a distance of \(1.0 \mathrm{~km}\text{?}\)

Subsection Numerical Practice

Calculation 16.9.3. Wave Speed.

The speed of a wave in a medium depends on which of the following quantities?
  1. The amplitude of the source
  2. The frequency of the source
  3. The period of the source
  4. The characteristics of the medium
  5. The power of the wave
Answer.
D.

Calculation 16.9.4. Wave Frequency.

The frequency of a wave depends on which of the following quantities?
  1. The amplitude of the source
  2. The frequency of the source
  3. The period of the source
  4. The characteristics of the medium
  5. The power of the wave
Answer.
B., E.

Calculation 16.9.5. Wave Amplitude.

The amplitude of a wave depends on which of the following quantities?
  1. The power coming from the source
  2. The frequency of the source
  3. The period of the source
  4. The characteristics of the medium
  5. The wavelength of the wave
Answer.
A.

Calculation 16.9.6. Two Wave.

Waves A and B are traveling in the same medium, and they have the same wavelength and frequency. But Wave A has an amplitude twice that of Wave B. How does the speed of Wave A compare to the speed of wave B?
  1. The speed of Wave A is 1/4 the speed of Wave B.
  2. The speed of Wave A is 1/2 the speed of Wave B.
  3. The speed of Wave A is the same as the speed of Wave B.
  4. The speed of Wave A is twice the speed of Wave B.
  5. The speed of Wave A is four times the speed of Wave B.
  6. None of the above are correct.
Answer.
C.

Calculation 16.9.7. Wavelength of the Ocean.

Sitting on the dock of the bay, you notice that the crests of an ocean wave pass a pier every \(12.0 \mathrm{~s}\text{.}\) You know also that there are two buoys \(28 \mathrm{~m}\) apart, and that the crest travels between the buoys in about \(5 \mathrm{~s}\text{.}\) What is the wavelength of the ocean wave?
  1. 16.8 m
  2. 24.7 m
  3. 31.6 m
  4. 42.2 m
  5. 59.4 m
  6. 67.2 m
Answer.
F.

Calculation 16.9.8. Wave Intensity.

Consider a logarithmic function like that found in the equation for decibels (also used for the Richter scale, by the way). If the sound intensity level increases from 30 dB to 60 dB, what can you say about the intensity of the wave?
  1. The intensity must have decreased by a factor greater than 2
  2. The intensity must have decreased by a factor of 2
  3. The intensity must have decreased by a factor less than 2
  4. The intensity must have remained constant
  5. The intensity must have increased by a factor greater than 2
  6. The intensity must have increased by a factor of 2
  7. The intensity must have increased by a factor less than 2
Answer.
E.

Calculation 16.9.9. Sound Intensity.

If the sound intensity level at a certain point increases from \(30 \mathrm{~dB}\) to \(60 \mathrm{~dB}\text{,}\) by what factor did the intensity change?
  1. 0.5
  2. 2
  3. 30
  4. 100
  5. 1000
Answer.
E.

Calculation 16.9.10. Sound at a Concert.

You are attending a concert by your favorite band, but you can barely hear them, because only 2 of the 20 speakers are working. The sound intensity level at your location is \(60 \mathrm{~dB}\text{.}\) If all 20 speakers suddenly start working, what is the new sound intensity level at your location? Assume that you don’t change your location and that all of the speakers are the same distance from you.
  1. 70 dB
  2. 78 dB
  3. 100 dB
  4. 120 dB
  5. 600 dB
Answer.
A.

Calculation 16.9.11. Ambulance Siren.

You hear the siren from an ambulance and the frequency you hear is decreasing. Which of the following can you conclude are plausible?
  1. The car is receding from you at a constant speed.
  2. The car is receding from you at an increasing speed.
  3. The car is receding from you at a decreasing speed.
  4. The car is approaching you at a constant speed.
  5. The car is approaching you at an increasing speed.
  6. The car is approaching you at a decreasing speed.
Answer.
B., F.

Calculation 16.9.12. Dolphin Frequency.

Dolphins emit clicks of sound for communicating and echo-location. A marine biologist, standing at rest in shallow seawater, is monitoring a dolphin swimming directly away at 8 m/s. The biologist measures the number of clicks occurring per second to be at a frequency of 2500 Hz. The speed of sound in calm seawater is 1522 m/s. What is the frequency of the clicks that the dolphin sends out?
  1. 1522 Hz
  2. 2464 Hz
  3. 2487 Hz
  4. 2500 Hz
  5. 2513 Hz
  6. 2536 Hz
Answer.
E.

References References

[1]
Numerical practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
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