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Worksheet: Modeling Longitudinal Waves

Q1:

Consider a graph of a compression wave. Shown are snapshots of the wave function for (blue) and (red).

What is the wavelength of the compression wave?

What is the maximum displacement of the compression wave?

What is the velocity of the compression wave?

What is the period of the compression wave?

Q2:

Consider a sound wave modeled with the equation 𝑠 ( π‘₯ , 𝑑 ) = 4 . 0 0 ( 3 . 6 6 π‘₯ βˆ’ 1 2 5 6 𝑑 ) n m c o s , where π‘₯ is measured in meters and 𝑑 is measured in seconds.

What is the maximum displacement of the sound wave?

What is the wavelength?

What is the frequency of the sound wave?

What is the speed of the sound wave?

Q3:

A sound wave is modeled with the wave function Ξ” 𝑃 = 1 . 2 0 ο€Ή π‘˜ π‘₯ βˆ’ 6 . 2 8 Γ— 1 0 𝑑  P a s i n 4 and the sound wave travels in air at a speed of 𝑣 = 3 4 3 . 0 0 / m s .

What is the wave number of the sound wave?

What is the value for Ξ” 𝑃 ( 3 . 0 0 , 2 0 . 0 0 ) m s ?

Q4:

A speaker is placed at the opening of a long horizontal tube. The speaker oscillates, creating a sound wave that moves down the tube. The wave moves through the tube at a speed of 340 m/s. The sound wave is modeled with the wave function 𝑠 ( π‘₯ , 𝑑 ) = 𝑠 ( π‘˜ π‘₯ βˆ’ πœ” 𝑑 + πœ™ ) m a x c o s . At time 𝑑 = 0 . 0 0 s, an air molecule at the point π‘₯ = 2 . 3 0 m is at the maximum displacement 𝑠 = 6 . 3 4 m a x nm. At 𝑑 = 0 . 0 0 s, another air molecule that is at the point π‘₯ = 2 . 7 0 m has a displacement of 2.30 nm.

What is the wave number of the wave?

What is the angular frequency of the wave?

What is the initial phase angle of the wave?

Q5:

Consider the beats shown in this graph of the gauge pressure on the vertical axis versus time on the horizontal axis for the position π‘₯ = 0 . 0 m. The wave moves with a speed of 340 m/s.

How many beats are there per second?

How many times does the wave oscillate per second?

Q6:

Consider a sound wave, moving through air, modeled with the equation 𝑠 ( π‘₯ , 𝑑 ) = 1 2 . 0 0 nm  c o s ( 7 2 . 1 5 π‘₯ – 2 0 . 9 3 Γ— 1 0 𝑑 ) 3 , where π‘₯ is measured in meters and 𝑑 in seconds. What is the shortest time required for an air molecule to move between 5.000 nm and βˆ’ 5 . 0 0 0 nm?

Q7:

A speaker is placed at the opening of a long horizontal tube. The speaker oscillates, creating a sound wave that moves through the tube at a speed of 340 m/s. The sound wave is modeled with the wave function 𝑠 ( π‘₯ , 𝑑 ) = 𝑠 ( π‘˜ π‘₯ βˆ’ πœ” 𝑑 + πœ™ ) m a x c o s , where π‘₯ is measured in meters and 𝑑 is measured in seconds. At time 𝑑 = 0 . 0 0 s , an air molecule at the position π‘₯ = 3 . 5 0 m is at the maximum displacement of 7.00 nm. At the same time, another molecule at the position π‘₯ = 3 . 7 0 m has a displacement of 3.00 nm. What is the frequency at which the speaker is oscillating?

Q8:

The displacement of certain gas molecules in a sound wave is modeled with the wave function 𝑠 ( π‘₯ , 𝑑 ) = 3 . 0 0 ( 6 4 . 7 5 π‘₯ βˆ’ 1 . 8 4 Γ— 1 0 𝑑 ) n m c o s 4 , where π‘₯ is measured in meters and 𝑑 in seconds.

What is the speed of the sound wave?

What is the maximum speed of the air molecules as they oscillate in simple harmonic motion?

  • A 5 . 5 2 Γ— 1 0 βˆ’ 5 m/s
  • B 4 . 2 3 Γ— 1 0 βˆ’ 3 m/s
  • C 5 . 5 2 Γ— 1 0 βˆ’ 3 m/s
  • D 4 . 2 3 Γ— 1 0 βˆ’ 5 m/s
  • E 3 . 1 8 Γ— 1 0 βˆ’ 5 m/s

What is the magnitude of the maximum acceleration of the air molecules as they oscillate in simple harmonic motion?

Q9:

A wave traveling on a Slinky that is 5.0 m in length takes 2.8 s to travel the length of the Slinky and back again.

What is the speed of the wave?

Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?

Q10:

A sound wave is modeled as Ξ” 𝑃 = 1 . 3 0 0 ( 3 5 . 9 4 π‘₯ βˆ’ 1 9 1 3 7 𝑑 ) P a s i n , where π‘₯ is measured in meters and 𝑑 in seconds.

Find the maximum change in pressure.

Find the wavelength of the wave.

Find the frequency of the wave.

Find the speed of the wave.

Q11:

A seismograph records a 23.0 s difference in the arrival times of 𝑆 waves and 𝑃 waves from an earthquake epicenter. If the waves traveled the same path at constant wave speeds 𝑉 = 5 . 0 0  km/s and 𝑉 = 6 . 7 0  km/s, how far away is the epicenter of the earthquake?