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Chapter 16: Problem 49

A swimming duck paddles the water with its feet once every 1.6 s, producing surface waves with this period. The duck is moving at constant speed in a pond where the speed of surface waves is 0.32 m/s, and the crests of the waves ahead of the duck are spaced 0.12 m apart. (a) What is the duck's speed? (b) How far apart are the crests behind the duck?

Short Answer

Expert verified
The duck's speed is 0.245 m/s, and the crest distance behind is 0.904 m.

Step by step solution

01

Identify Known Quantities

We know the period of the duck's paddling is 1.6 s, and the speed of the surface waves is 0.32 m/s. The crests ahead of the duck are spaced 0.12 m apart.
02

Use Wave Speed Formula

To find the wavelength when the duck is not moving, use the formula for wave speed: \[v = \frac{\lambda}{T},\] where \(v = 0.32\, \text{m/s}\) (wave speed) and \(T = 1.6\, \text{s}\) (period). Solve for \(\lambda\):\[\lambda = v \cdot T = 0.32 \times 1.6 = 0.512\, \text{m}.\] This wavelength is the distance between wave crests when the duck is not moving.
03

Calculate Relative Speed of Duck to Wave Crests Ahead

The recorded wavelength ahead of the duck is 0.12 m, which is shorter because the duck moves towards these crests. Use the equation:\[ \frac{0.32 - v_d}{0.12} = \frac{0.32}{0.512},\]where \( v_d \) is the duck's speed. Solve for \(v_d\) (duck鈥檚 speed).
04

Solve for Duck's Speed

Using the proportion from the equation above:\[(0.32 - v_d) \times 0.512 = 0.32 \times 0.12,\]which leads to:\[0.32 - v_d = \frac{0.32 \times 0.12}{0.512},\]thus:\[v_d = 0.32 - \frac{0.0384}{0.512} = 0.32 - 0.075 = 0.245\, \text{m/s}.\] So, the duck's speed is 0.245 m/s.
05

Calculate Crest Distance Behind Duck

To find the crest distance behind the duck, note that the wavelength increases behind due to the duck moving away. Use:\[ \frac{0.32 + v_d}{\lambda_{behind}} = \frac{0.32}{0.512},\]so,\[\lambda_{behind} = \frac{0.32 + 0.245}{0.32/0.512}.\]
06

Solve for Distance Between Wave Crests Behind

Solve the earlier equation:\[\lambda_{behind} = \frac{0.565}{0.625},\]\[\lambda_{behind} = 0.904\, \text{m}.\] So, the distance between the crests behind the duck is 0.904 m.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Waves
Surface waves are disturbances that travel along the interface between two different media. In this context, they occur at the surface of water where the duck is paddling. These waves involve the movement of water molecules in circular orbits, decreasing in diameter with depth.
Surface waves are essential in understanding various phenomena in physics and nature. They can be seen not only in water but also in other surface contexts, such as where air meets liquid or solid surfaces.
Key characteristics of surface waves include:
  • Propagation parallel to the surface.
  • Movement of particles in retrograde circulatory paths.
  • Dependence on surface tension and gravitational effects.
Wave Speed
Wave speed refers to the distance a wave travels through a medium per unit time. In the given exercise, the wave speed is provided as 0.32 m/s. It is crucial for determining how fast the wave energy is being transmitted across the surface of the water.
Wave speed ( v ) is determined using the wavelength ( \lambda ) and the period ( T ) of the wave, following the formula:
\[v = \frac{\lambda}{T}\]
This formula helps in comprehending how wave characteristics relate to each other.
  • Faster wave speeds indicate quicker energy propagation.
  • Nature of the medium affects wave speed.
  • Wave speed is constant in uniform mediums.
Wavelength
The wavelength of a wave is the distance between consecutive crests or troughs. It's an important measure because it determines how spread out the wave energy is over an area. In the scenario of the duck, we calculated the wavelength where the duck is stationary to be 0.512 m.
Understanding wavelength is essential to grasp the relationship between different wave parameters. Longer wavelengths generally mean less frequent waves, while shorter ones suggest more rapid wave occurrences.
  • Wavelength influences wave behavior relative to obstacles.
  • Inversely related to frequency, as \(\lambda = \frac{v}{f}\) .
  • Helps categorize types of waves.
Period
The period of a wave is the time it takes for one complete cycle of a wave to pass a given point, measured in seconds. In this exercise, the duck paddles every 1.6 seconds, which is the wave period. This period is fundamental in determining how often waves are generated.
The period directly influences the wave speed along with wavelength. Lower periods mean more frequent wave cycles, and vice versa.
  • Related to frequency by: \( f = \frac{1}{T} \) .
  • Control over period determines wave production rate.
  • Affects the energy per wave cycle.
Relative Speed
Relative speed involves the motion of the duck compared to the moving wave crests. It indicates how the speed of one object is perceived relative to another. In the problem, the relative speed is vital in solving the apparent change in wave crest spacing either in front of or behind the duck due to its movement.
By considering relative speed, one can understand the shift in observed positions of wave crests:
  • Motion towards crests reduces the wavelength perceived ahead, calculated to be 0.12 m in the scenario.
  • Motion away from crests increases the perceived wavelength behind, determined to be 0.904 m.
  • Relative speed adjustments help in calculating real speeds and distances.
These effects are evident in both everyday and scientific phenomena, like sound pitch change due to the Doppler effect.

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