Worksheet: Modeling One-Dimensional Transverse Waves

In this worksheet, we will practice using amplitude, wave number, angular frequency, and phase shift to model the transverse motion of a one-dimensional wave.

Q1:

A transverse wave moves along a string at 3 . 0 0 × 1 0 2 m/s. The wave’s amplitude is 20.0 cm and its wavelength is 0.500 m. How much time is required for a point on the string to move through a vertical distance of 5.00 km?

Q2:

A wave is modelled with the function 𝑦 ( 𝑥 , 𝑡 ) = 0 . 2 5  0 . 3 0 𝑥 − 0 . 9 0 𝑡 + 𝜋 3  m c o s , where 𝑥 is measured in meters and 𝑡 is measured in seconds.

What is the wave’s amplitude?

What is the wave’s wave number?

What is the wave’s angular frequency?

What is the wave’s speed?

What is the phase shift of the wave?

  • A 𝜋 rad
  • B 2 𝜋 / 3 rad
  • C 𝜋 / 2 rad
  • D 𝜋 / 3 rad
  • E 3 𝜋 / 2 rad

What is the wave’s wavelength?

What is the wave’s period?

Q3:

A transverse wave on a string is modeled with the wave function 𝑦 ( 𝑥 , 𝑡 ) = 0 . 2  2 𝑥 − 3 𝑡 + 𝜋 1 6  s i n , where 𝑦 is measured in centimeters. What is the height of the string with respect to the equilibrium position at a position 𝑥 = 4 . 0 0 m and a time 𝑡 = 1 0 . 0 0 s ? Give your answer to 2 significant figures

Q4:

A motorboat is traveling across a lake at a speed of 15 m/s. Waves are propagating on the lake, and the boat bounces up and down when it passes a wave. The boat bounces up and down every 0.50 s while moving in the same direction that the waves propagate, and it bounces up and down every 0.30 s while moving in the opposite direction opposite to the propagation of the waves.

What is the speed of the waves?

What is the wavelength of the waves?

Q5:

Waves on a swimming pool propagate at 0.750 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.00 s. How far away is the other end of the pool?

Q6:

A wave is modeled by the wave function where 𝑥 is measured in meters and 𝑡 is measured in seconds.

Find the amplitude of the wave.

Find the wavelength of the wave.

Find the speed of propagation of the wave.

Find the frequency of the wave.

Find the period of the wave.

Q7:

The Voyager spacecraft transmits radio waves through empty space at the speed of light, 3 . 0 0 × 1 0 8 m/s. The wavelength of these waves is 0.120 m. What is the frequency of the waves?

  • A 4 . 3 2 × 1 0 9 Hz
  • B 5 . 0 0 × 1 0 9 Hz
  • C 2 . 1 6 × 1 0 9 Hz
  • D 2 . 5 0 × 1 0 9 Hz
  • E 4 . 8 0 × 1 0 8 Hz

Q8:

An electromagnetic wave has a frequency of 12 MHz. What is its wavelength in vacuum?

Q9:

Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is

What is the angular frequency?

What is the wave number?

What amplitude do the waves both have?

What is the phase difference between the waves?

  • A 𝜋 / 8 rad
  • B 𝜋 / 2 rad
  • C 𝜋 / 4 rad
  • D 𝜋 / 1 6 rad
  • E 𝜋 / 1 2 rad

Q10:

How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?

Q11:

A sunbather stands waist deep in the ocean and observes that six crests of periodic surface waves pass each 60 seconds. The crests are 16.00 meters apart.

What is the wavelength of the waves?

What is the frequency of the waves?

What is the period of the waves?

What is the speed of the waves?

Q12:

The electric field of an electromagnetic wave traveling in vacuum is described by the following wave function: where is the wavenumber in rad/m, is in m, is in s.

Find the amplitude of the wave.

Find the frequency of the wave.

  • A Hz
  • B Hz
  • C Hz
  • D Hz
  • E Hz

Find the wavelength of the wave.

Q13:

What is the wavelength of X-rays of frequency 2 . 0 × 1 0 1 7 Hz?

What is the wavelength of yellow light of frequency 5 . 1 × 1 0 1 4 Hz?

  • A 5 . 9 × 1 0 − 7 m
  • B 7 . 0 × 1 0 − 7 m
  • C 6 . 5 × 1 0 − 7 m
  • D 5 . 5 × 1 0 − 7 m
  • E 7 . 3 × 1 0 − 7 m

What is the wavelength of gamma-rays of frequency 1 . 0 × 1 0 2 3 Hz?

  • A 4 . 3 × 1 0 − 1 5 m
  • B 3 . 3 × 1 0 − 1 5 m
  • C 3 . 0 × 1 0 − 1 5 m
  • D 3 . 5 × 1 0 − 1 5 m
  • E 3 . 9 × 1 0 − 1 5 m

Q14:

How many times longer is the wavelength of a 1 0 3 0 kHz radio wave in air than the wavelength of a 20.0 Hz sound wave in air? (take the speed of sound as 340 m/s)

Q15:

AM radio has a frequency range of 540 kHz to 1 6 0 0 kHz, while FM radio has a frequency range of 88.0 MHz to 108 MHz.

Calculate the wavelength range for AM radio.

  • A 178 m to 535 m
  • B 181 m to 652 m
  • C 196 m to 636 m
  • D 187 m to 556 m
  • E 191 m to 542 m

Calculate the wavelength range for FM radio.

  • A 2.78 m to 3.41 m
  • B 2.67 m to 3.30 m
  • C 2.31 m to 3.42 m
  • D 2.21 m to 3.02 m
  • E 2.83 m to 3.66 m

Q16:

If you wish to detect details of size 0.20 nm with electromagnetic radiation, the radiation must have a wavelength of this size. What is its frequency?

  • A 2 . 2 × 1 0   Hz
  • B 1 . 8 × 1 0   Hz
  • C 2 . 6 × 1 0   Hz
  • D 1 . 5 × 1 0   Hz
  • E 3 . 0 × 1 0   Hz

Q17:

What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at a wave speed of 0.800 m/s?

Q18:

A computer user finds that his wireless router transmits data at a rate of 75 Mbps (megabits per second).

How much time is required for the router to transmit a single bit of data?

  • A s
  • B s
  • C s
  • D s
  • E s

If a reflected wifi signal has to travel 8.00 m further than a directly received wifi signal, what will be the time difference between receiving the direct signal and the reflected signal?

  • A s
  • B s
  • C s
  • D s
  • E s

Q19:

A pulse can be described as a single wave disturbance that moves through a medium. Consider a pulse that is defined at time 𝑡 = 0 . 0 0 s by the equation 𝑦 ( 𝑥 ) = 4 . 0 0 𝑥 + 3 . 0 0 m m 3 2 2 , centered around 𝑥 = 0 . 0 0 m. The pulse moves with a velocity 𝑣 = 4 . 0 0 m/s in the positive 𝑥 -direction.

What is the amplitude of the pulse?

What value of 𝑥 is the pulse centered around when 𝑡 = 4 . 0 0 s?

Q20:

Consider the wave function 𝑦 ( 𝑥 , 𝑡 ) = ( 5 . 0 0 )  0 . 6 0 0 𝑥 + 2 . 5 0 𝑡 + 𝜋 5  c m s i n , where 𝑥 is measured in meters and 𝑡 is measured in seconds.

What is the period of the wave?

What is the wavelength of the wave?

At what speed does the wave propagate?

What is the initial phase shift of the wave?

Q21:

Consider a wave described by the wave function 𝑦 ( 𝑥 , 𝑡 ) = 0 . 4 5 0 ( 3 . 5 2 𝑥 − 6 5 . 9 𝑡 ) s i n , where 𝑥 is measured in meters and 𝑡 is measured in seconds.

How many crests pass by an observer at a fixed location in 2.50 minutes?

How far has the wave traveled in 2.50 minutes?

Q22:

Consider two sinusoidal waves traveling along a string, modeled as 𝑦 ( 𝑥 , 𝑡 ) = 0 . 2 ( 5 𝑥 + 4 𝑡 ) 1 s i n and 𝑦 ( 𝑥 , 𝑡 ) = 0 . 4 ( 1 0 𝑥 − 8 𝑡 ) 2 s i n , where 𝑥 is measured in meters and 𝑡 is measured in seconds. What is the displacement of the resultant wave formed by the interference of the two waves at the position 𝑥 = 0 . 4 m at time 𝑡 = 0 . 3 s ?

Q23:

Consider two sinusoidal waves traveling along a string modeled as 𝑦 ( 𝑥 , 𝑡 ) = 0 . 7 ( 5 𝑥 − 4 𝑡 ) 1 s i n and 𝑦 ( 𝑥 , 𝑡 ) = 0 . 7 ( 5 𝑥 + 4 𝑡 ) 2 s i n , where 𝑥 is measured in meters and 𝑡 in seconds. The resultant of these waves can be found using s i n s i n c o s c o s s i n ( 𝑢 ± 𝑣 ) = 𝑢 𝑣 ± 𝑢 𝑣 .

What is the wave number of the resultant wave?

  • A 3 m−1
  • B 4 m−1
  • C 10 m−1
  • D 5 m−1
  • E 0.8 m−1

What is the angular frequency of the resultant wave?

Q24:

Two sinusoidal waves are moving through a medium in the positive 𝑥 -direction, both having an amplitude of 8.0 cm, a wavelength of 4.0 m, and a period of 5.0 s, but one has a phase shift 𝜙 = 0 . 3 0 r a d . What is the height of the resultant wave at a time 𝑡 = 4 . 5 s and a position 𝑥 = 0 . 6 0 m ?

Q25:

Consider two waves, and , that are identical except for a phase shift propagating in the same medium. The resultant wave of and has 1.5 times the amplitude of the individual waves.

What is the phase shift measured in radians?

What is the phase shift measured in degrees?

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