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Chapter 15: Problem 74

Two waves traveling along a stretched string have the same frequency, but one transports 2.5 times the power of the other. What is the ratio of the amplitudes of the two waves?

Short Answer

Expert verified
The ratio of the amplitudes is \( \sqrt{2.5} \).

Step by step solution

01

Understanding the Relationship Between Power and Amplitude

The power transported by a wave is proportional to the square of its amplitude. Mathematically, this can be expressed as \( P \propto A^2 \), where \( P \) is the power and \( A \) is the amplitude.
02

Setting Up the Equation for the Powers

Let's denote the amplitudes of the two waves as \( A_1 \) and \( A_2 \), with respective powers \( P_1 \) and \( P_2 \). We know from the problem that one wave transports 2.5 times the power of the other, so \( P_2 = 2.5P_1 \).
03

Relating Amplitude to Power

Since \( P \propto A^2 \), we can write \( P_1 = kA_1^2 \) and \( P_2 = kA_2^2 \), where \( k \) is a constant of proportionality. Since \( P_2 = 2.5P_1 \), we substitute:\[ kA_2^2 = 2.5kA_1^2 \] Canceling \( k \) from both sides gives us:\[ A_2^2 = 2.5A_1^2 \]
04

Solving for the Amplitudes Ratio

To find the ratio \( \frac{A_2}{A_1} \), take the square root of both sides:\[ \sqrt{A_2^2} = \sqrt{2.5A_1^2} \] This simplifies to:\[ A_2 = \sqrt{2.5}A_1 \] Thus, the ratio of the amplitudes is \( \frac{A_2}{A_1} = \sqrt{2.5} \).

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Key Concepts

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

Amplitude Ratio
The amplitude ratio is a critical concept when comparing two or more waves. It represents a comparison of their respective amplitudes. In the context of wave physics, amplitude refers to the maximum displacement of points on a wave. When considering two waves, it is common to compare their amplitudes by calculating their ratio.

This ratio can offer insights into the relative power and intensity of the waves. For example, if one wave is transporting more power than another, it often has a higher amplitude. In the given problem, we were asked to find the amplitude ratio of two waves where one carries 2.5 times the power of the other. By understanding the relationship between wave power and amplitude, we find that the amplitude ratio of the more powerful wave to the less powerful one is the square root of 2.5.
Wave Amplitude
Wave amplitude is a fundamental property of any wave, describing its peak height from its rest position. It is a measurement of a wave's energy, with higher amplitudes indicating more energetic waves. In many cases, like sound and light waves, the amplitude determines the wave's intensity, such as its loudness or brightness.

The amplitude is directly related to concepts of constructive and destructive interference in waves, affecting how waves overlap and interact when combined. In the exercise, knowing the amplitude allows us to infer other properties of the wave, such as its power, since there is a direct mathematical relationship between these quantities.
  • Amplitude is typically measured in meters for mechanical waves.
  • Higher amplitude means greater energy transported by the wave.
  • In the problem, we note that differences in amplitude directly on how much power a wave conveys.
Relationship Between Power and Amplitude
Understanding the relationship between power and amplitude is essential for mastering wave dynamics. In physics, the power of a wave is directly proportional to the square of its amplitude. This relationship can be expressed with the mathematical equation: \[ P \propto A^2 \]

where \( P \) is the power of the wave and \( A \) is its amplitude. This proportion indicates that if one wave has twice the amplitude of another, it will transport four times as much power, because the amplitude ratio is squared in terms of power.

In this exercise, when we dealt with one wave having 2.5 times the power of another, it implied a specific amplitude change: \[ \frac{P_2}{P_1} = \left(\frac{A_2}{A_1}\right)^2 \] Solving this allows us to find the ratio of their amplitudes, showing the squared nature relationship to power. Hence, understanding this proportionality is crucial for solving wave-related problems.

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Most popular questions from this chapter

For a spherical wave traveling uniformly away from a point source, show that the displacement can be represented by $$D=\left(\frac{A}{r}\right) \sin (k r-\omega t)$$ where \(r\) is the radial distance from the source and \(A\) is a constant.

(II) The intensity of an earthquake wave passing through the Earth is measured to be \(3.0 \times 10^{6} \mathrm{~J} / \mathrm{m}^{2} \cdot \mathrm{s}\) at a distance of \(48 \mathrm{~km}\) from the source. (a) What was its intensity when it passed a point only \(1.0 \mathrm{~km}\) from the source? \((b)\) At what rate did energy pass through an area of \(2.0 \mathrm{~m}^{2}\) at \(1.0 \mathrm{~km} ?\)

(II) A \(65-\mathrm{cm}\) guitar string is fixed at both ends. In the frequency range between 1.0 and \(2.0 \mathrm{kHz},\) the string is found to resonate only at frequencies \(1.2,1.5,\) and \(1.8 \mathrm{kHz} .\) What is the speed of traveling waves on this string?

(II) A transverse traveling wave on a cord is represented by \(D=0.22 \sin (5.6 x+34 t)\) where \(D\) and \(x\) are in meters and \(t\) is in seconds. For this wave determine (a) the wavelength, (b) frequency, (c) velocity (magnitude and direction), (d) amplitude, and (e) maximum and minimum speeds of particles of the cord.

Two strings on a musical instrument are tuned to play at \(392 \mathrm{~Hz}(\mathrm{G})\) and \(494 \mathrm{~Hz}(\mathrm{~B}),(a)\) What are the frequencies of the first two overtones for each string? \((b)\) If the two strings have the same length and are under the same tension, what must be the ratio of their masses \(\left(m_{\mathrm{G}} / m_{\mathrm{A}}\right) ?(c)\) If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths \(\left(\ell_{\mathrm{G}} / \ell_{\mathrm{A}}\right) ?(d)\) If their masses and lengths are the same, what must be the ratio of the tensions in the two strings?
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