Week 13: Waves & Sound -- NO SOLUTIONS

For solutions, please visit: edu.phyzk.net/203/content/13

\(\Delta\) ← if you see the word "Delta" instead of a triangle, please refresh the page! Thanks :)

Overview⚓︎

I recommend reviewing the following concepts, they will probably come up in recitation:

Reading wave information from graphs
- from a displacement vs time graph: amplitude, period, frequency
- from a displacement vs position graph: amplitude, wavelength
- being careful about what each axis means
Wave speed on a string
- \(v = \dfrac{\lambda}{T} = f\lambda\)
- combining information from a time graph and a snapshot of the string
- using fractions of a cycle or fractions of a wavelength when the graph does not show a full period directly
Sound intensity and spherical spreading
- \(I = \dfrac{P}{4\pi r^2}\) for a source radiating uniformly in all directions
- intensity decreases like \(1/r^2\)
- adding intensities from multiple independent sound sources
Sound intensity level
- \(\beta = 10\log_{10}\!\left(\dfrac{I}{I_0}\right)\)
- \(I_0 = 1.0 \times 10^{-12}\ \mathrm{W/m^2}\)
- equal sources add in intensity, not in decibels
- for \(N\) identical sources: \(I_{\mathrm{tot}} = NI_{\mathrm{one}}\)
Doppler effect
- for a moving source and stationary observer:
- approaching: \(f' = f\dfrac{v}{v-v_s}\)
- receding: \(f' = f\dfrac{v}{v+v_s}\)
- for air near room temperature: \(v \approx 331 + 0.60T_C \ \mathrm{m/s}\)
Interference from two in-phase speakers
- wavelength from \(\lambda = v/f\)
- constructive interference: path difference \(= m\lambda\)
- destructive interference: path difference \(= \left(m+\tfrac12\right)\lambda\)
- on the line between the speakers, carefully writing each distance to the point

Practice Problems⚓︎

If you're comfortable with the "★★★" problems, you should do great during recitation.
To print these questions, simply press Ctrl + P while on this page, and it should come out formatted nicely -- just make sure to refresh first so all the math renders properly.

Difficulty key:

★☆☆ = beginner
★★☆ = standard
★★★ = challenging / multi-step

Waves and Sound Constants / Reminders

\(I_0 = 1.0 \times 10^{-12}\ \mathrm{W/m^2}\)
For a spherical wave from a point source: \(I = \dfrac{P}{4\pi r^2}\)
Sound intensity level: \(\beta = 10\log_{10}\!\left(\dfrac{I}{I_0}\right)\)
Wave speed: \(v = f\lambda = \dfrac{\lambda}{T}\)
Frequency and period: \(f = \dfrac{1}{T}\)
Speed of sound in air: \(v \approx 331 + 0.60T_C \ \mathrm{m/s}\)
Moving source, stationary observer:
- approaching: \(f' = f\dfrac{v}{v-v_s}\)
- receding: \(f' = f\dfrac{v}{v+v_s}\)
Two-source interference:
- constructive: \(\Delta r = m\lambda\)
- destructive: \(\Delta r = \left(m+\tfrac12\right)\lambda\)

[1] Wave Graphs and Speed on a String⚓︎

★☆☆
A displacement vs time graph is taken at one end of a string. The graph shows a maximum displacement of \(+3.0\ \mathrm{cm}\), a minimum displacement of \(-3.0\ \mathrm{cm}\), and the time between successive crests is \(0.40\ \mathrm{s}\).
- (a) What is the amplitude?
- (b) What is the period?
- (c) What is the frequency?
★☆☆
A snapshot of a sinusoidal wave on a string shows a maximum displacement of \(2.5\ \mathrm{cm}\) and a distance of \(1.8\ \mathrm{m}\) between two neighboring crests.
- (a) What is the amplitude?
- (b) What is the wavelength?
★★☆
At one point on a string, a displacement vs time graph shows that the wave has a period of \(0.25\ \mathrm{s}\). A snapshot of the string shows a wavelength of \(1.50\ \mathrm{m}\).
- (a) Find the frequency.
- (b) Find the wave speed.
★★☆
A displacement vs time graph at a fixed point on a string shows troughs at \(t=0.10\ \mathrm{s}\) and \(t=0.46\ \mathrm{s}\). A snapshot of the string at one instant shows that four full wavelengths occupy \(2.40\ \mathrm{m}\) of the string.
- (a) Find the period.
- (b) Find the frequency.
- (c) Find the wavelength.
- (d) Find the wave speed.
★★★
A point on a string is observed as the wave passes by. A crest occurs at \(t=0.18\ \mathrm{s}\), and the next time that same point passes through equilibrium moving downward is at \(t=0.27\ \mathrm{s}\). In a snapshot of the string taken at one instant, a crest is located at \(x=0.20\ \mathrm{m}\) and the nearest trough to its right is at \(x=0.80\ \mathrm{m}\).
- (a) Find the period.
- (b) Find the frequency.
- (c) Find the wavelength.
- (d) Find the wave speed.
★★★
A displacement vs time graph for a point on a string shows that the point goes from \(y=0\) moving upward to the next crest in \(0.060\ \mathrm{s}\). A snapshot of the string shows that the distance from one zero crossing with positive slope to the next zero crossing with positive slope is \(0.96\ \mathrm{m}\).
- (a) Find the period.
- (b) Find the frequency.
- (c) Find the wavelength.
- (d) Find the wave speed.
- (e) How long does it take a crest to travel \(7.2\ \mathrm{m}\) along the string?

[2] Sound Intensity and Sound Level⚓︎

★☆☆
A small speaker emits sound uniformly in all directions with a power output of \(18.0\ \mathrm{W}\).
- What is the sound intensity at a distance of \(6.00\ \mathrm{m}\) from the speaker?
★☆☆
A sound wave has an intensity level of \(72.0\ \mathrm{dB}\).
- What is its intensity?
★★☆
At your location, one room fan produces a sound intensity level of \(79.0\ \mathrm{dB}\). How many identical fans would be needed to produce a sound intensity level of \(88.0\ \mathrm{dB}\) at that same location?
★★☆
One student speaking alone produces a sound intensity level of \(56.0\ \mathrm{dB}\) at the back of a classroom. If \(12\) students are all talking independently with the same intensity at that same location, what sound intensity level results?
★★★
Two identical outdoor speakers are separated by \(24.0\ \mathrm{m}\). Each emits sound uniformly in all directions with a power output of \(0.400\ \mathrm{W}\).
- (a) Find the total sound intensity at the midpoint between the speakers.
- (b) A person then walks \(5.00\ \mathrm{m}\) directly toward one of the speakers. What total sound intensity do they hear there?
★★★
Thirty-six identical machines together produce a sound intensity level of \(96.0\ \mathrm{dB}\) at a certain point in a workshop.
- (a) Find the total sound intensity produced by all \(36\) machines.
- (b) Find the intensity produced by one machine.
- (c) What sound intensity level would result if only \(4\) of these machines were running?

[3] Doppler Effect and Two-Speaker Interference⚓︎

★☆☆
What is the speed of sound in air on a day when the temperature is \(22.0^\circ\mathrm{C}\)?
★☆☆
A sound wave in air has frequency \(680\ \mathrm{Hz}\). If the speed of sound is \(340\ \mathrm{m/s}\), what is the wavelength?
★★☆
A police car moves at \(31.0\ \mathrm{m/s}\) while sounding a siren of frequency \(720\ \mathrm{Hz}\). The air temperature is \(25.0^\circ\mathrm{C}\).
- What frequency is heard by a stationary observer when the police car is approaching?
★★☆
Two in-phase loudspeakers, \(A\) and \(B\), face each other and are separated by \(1.50\ \mathrm{m}\). They emit sound of frequency \(680\ \mathrm{Hz}\), and the speed of sound is \(340\ \mathrm{m/s}\).
- (a) What is the wavelength?
- (b) Locate all points on the line between the speakers, measured from speaker \(A\), where destructive interference occurs.
★★★
An ambulance travels at \(27.0\ \mathrm{m/s}\) and emits a siren of frequency \(950\ \mathrm{Hz}\). The air temperature is \(15.0^\circ\mathrm{C}\).
- (a) Find the speed of sound in air.
- (b) What frequency is heard by a stationary observer as the ambulance approaches?
- (c) What frequency is heard by the observer after the ambulance passes and is moving away?
★★★
Two in-phase loudspeakers, \(A\) and \(B\), face each other and are separated by \(1.50\ \mathrm{m}\). They emit sound of frequency \(660\ \mathrm{Hz}\), and the speed of sound is \(330\ \mathrm{m/s}\).
- (a) What is the wavelength?
- (b) Locate all points on the line between the speakers, measured from speaker \(A\), where constructive interference occurs.