A water wave is called a deep-water wave if the water's...

A water wave is called a deep-water wave if the water's depth is more than one-quarter of the wavelength. Unlike the waves we've considered in this chapter, the speed of a deep-water wave depends on its wavelength: $$ v=\sqrt{\frac{g \lambda}{2 \pi}} $$ Longer wavelengths travel faster. Let's apply this to standing waves. Consider a diving pool that is $5.0 \mathrm{m}$ decp and $10.0 \mathrm{m}$ wide. Standing water waves can set up across the width of the pool. Because water sloshes up and down at the sides of the pool, the boundary conditions require antinodes at $x=0$ and $x=L$. Thus a standing water wave resembles a standing sound wave in an open-open tube. a. What are the wavelengths of the first three standing-wave modes for water in the pool? Do they satisfy the condition for being deep-water waves? Draw a graph of each. b. What are the wave speeds for each of these waves? c. Derive a general expression for the frequencies $f_{m}$ of the possible standing waves. Your expression should be in terms of $m, g,$ and $L$ d. What are the oscillation periods of the first three standing wave modes?

A water wave is called a deep-water wave if the water's depth is more than one-quarter of the wavelength. Unlike the waves we've considered in this chapter, the speed of a deep-water wave depends on its wavelength:
$$
v=\sqrt{\frac{g \lambda}{2 \pi}}
$$
Longer wavelengths travel faster. Let's apply this to standing waves. Consider a diving pool that is $5.0 \mathrm{m}$ decp and $10.0 \mathrm{m}$ wide. Standing water waves can set up across the width of the pool. Because water sloshes up and down at the sides of the pool, the boundary conditions require antinodes at $x=0$ and $x=L$. Thus a standing water wave resembles a standing sound wave in an open-open tube.
a. What are the wavelengths of the first three standing-wave modes for water in the pool? Do they satisfy the condition for being deep-water waves? Draw a graph of each.
b. What are the wave speeds for each of these waves?
c. Derive a general expression for the frequencies $f_{m}$ of the possible standing waves. Your expression should be in terms of $m, g,$ and $L$
d. What are the oscillation periods of the first three standing wave modes?

Key Concepts

Deep-Water Waves

These are water waves that occur when the depth of the water is greater than one-quarter of the wavelength. In deep-water waves, the motion of water particles is influenced only by the surface, and the bottom of the water body does not affect the wave propagation. This concept is crucial when determining whether the dispersion relation for deep-water waves applies.

Dispersion Relation for Deep-Water Waves

In deep-water waves the speed of wave propagation is not constant but depends on the wavelength. The dispersion relation is given by the formula v = ?(g?/(2?)), where g is the acceleration due to gravity and ? is the wavelength. This relationship indicates that longer wavelengths travel faster, which is a key factor in analyzing wave speeds under deep-water conditions.

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere. This interference creates a pattern of nodes (points of zero amplitude) and antinodes (points of maximum amplitude). Understanding standing waves is important for analyzing resonant modes in finite systems, such as water in a pool or sound in a tube.

Boundary Conditions for Standing Waves

When a wave is confined within a medium with fixed boundaries, the physical constraints require specific patterns at the edges. For example, in an open-open system like the given pool, the boundary conditions force antinodes at both ends of the medium. These conditions determine the allowed wavelengths and the spatial distribution of nodes and antinodes for the standing wave modes.

Frequency and Harmonic Modes

For a system supporting standing waves, the frequencies of the modes are determined by the allowed wavelengths and the corresponding wave speed. The general expression for the frequency of a mode often relates the mode number, the wave speed, and the size of the system. Moreover, the period of oscillation is the reciprocal of the frequency. These relationships provide a framework for understanding how resonances and harmonics are established in physical systems such as water waves in a diving pool.

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