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Chapter 15: Problem 15

Why do circular water waves on the surface of a pond decrease in amplitude as they travel away from the source?

Short Answer

Expert verified
Answer: The amplitude of circular water waves decreases as they travel away from the source because the energy of the waves is distributed over expanding circular wavefronts. As the wavefronts expand in radius, they cover a larger surface area, causing the energy density to decrease, which in turn leads to a decrease in amplitude.

Step by step solution

01

Define amplitude

Amplitude refers to the maximum displacement of a wave from its equilibrium position. It is a measure of the wave's energy, as a wave with a higher amplitude has more energy than a wave with a lower amplitude.
02

Understand the circular wavefront

A circular wavefront is produced when a disturbance (e.g., a stone thrown into a pond) occurs at a single point on the surface of the water. The energy from this disturbance produces circular wavefronts, centered on the point of disturbance. As the wavefronts travel away from the source, they expand in a concentric and roughly circular pattern.
03

Explain energy distribution in circular wavefront

The energy from the disturbance is spread evenly over the surface of each successive wavefront. As a wavefront expands, its surface area increases; specifically, the surface area of a wavefront is proportional to the square of its radius (A = 4蟺r^2). Consequently, at a greater distance from the source, the same amount of energy is distributed over a larger area, resulting in a reduced energy density.
04

Relate energy density to amplitude

Since the amplitude of a wave is directly related to its energy, a decrease in energy density leads to a decrease in amplitude. As the wavefronts expand and travel away from the source, they encounter a larger area to cover. This causes the energy to be spread thinner, leading to a reduction in the amplitude of the waves.
05

Conclusion

Circular water waves on the surface of a pond decrease in amplitude as they travel away from the source because the energy of the waves is distributed over expanding circular wavefronts. As the wavefronts expand in radius, they cover a larger surface area, causing the energy density to decrease, which in turn leads to a decrease in amplitude.

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Most popular questions from this chapter

a) Starting from the general wave equation (equation 15.9 ), prove through direct derivation that the Gaussian wave packet described by the equation \(y(x, t)=(5.00 m) e^{-0.1(x-5 t)^{2}}\) is indeed a traveling wave (that it satisfies the differential wave equation). b) If \(x\) is specified in meters and \(t\) in seconds, determine the speed of this wave. On a single graph, plot this wave as a function of \(x\) at \(t=0, t=1.00 \mathrm{~s}, t=2.00 \mathrm{~s},\) and \(t=3.00 \mathrm{~s}\) c) More generally, prove that any function \(f(x, t)\) that depends on \(x\) and \(t\) through a combined variable \(x \pm v t\) is a solution of the wave equation, irrespective of the specific form of the function \(f\)

If two traveling waves have the same wavelength, frequency, and amplitude and are added appropriately, the result is a standing wave. Is it possible to combine two standing waves in some way to give a traveling wave?

A particular steel guitar string has mass per unit length of \(1.93 \mathrm{~g} / \mathrm{m}\). a) If the tension on this string is \(62.2 \mathrm{~N},\) what is the wave speed on the string? b) For the wave speed to be increased by \(1.0 \%\), how much should the tension be changed?

A string is made to oscillate, and a standing wave with three antinodes is created. If the tension in the string is increased by a factor of 4 a) the number of antinodes increases. b) the number of antinodes remains the same. c) the number of antinodes decreases. d) the number of antinodes will equal the number of nodes.

Write the equation for a sinusoidal wave propagating in the negative \(x\) -direction with a speed of \(120 . \mathrm{m} / \mathrm{s}\), if a particle in the medium in which the wave is moving is observed to swing back and forth through a \(6.00-\mathrm{cm}\) range in \(4.00 \mathrm{~s}\). Assume that \(t=0\) is taken to be the instant when the particle is at \(y=0\) and that the particle moves in the positive \(y\) -direction immediately after \(t=0\).
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