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Chapter 17: Problem 25

Two Identical Traveling Waves Two identical traveling waves, moving in the same direction, are out of phase by \(\pi / 2\) rad. What is the amplitude of the resultant wave in terms of the common amplitude \(Y\) of the combining waves?

Short Answer

Expert verified
The amplitude of the resultant wave is \(Y \sqrt{2}\).

Step by step solution

01

- Identify the given information

The problem states that there are two identical traveling waves moving in the same direction. These waves have a common amplitude, denoted as \(Y\), and are out of phase by \(\frac{\pi}{2}\) radians.
02

- Understand the phase relationship

Since the waves are out of phase by \(\frac{\pi}{2}\) radians, we know that one wave can be written as \(Y \sin(kx - \omega t)\) and the other as \(Y \sin(kx - \omega t + \frac{\pi}{2}) = Y \cos(kx - \omega t)\).
03

- Apply the principle of superposition

The resultant wave will be the sum of the two waves: \[Y_{resultant} = Y \sin(kx - \omega t) + Y \cos(kx - \omega t)\].
04

- Use trigonometric identities to combine the waves

Using the identity \(\sin(A) + \cos(A) = \sqrt{2} \cos(A - \frac{\pi}{4})\), the resultant wave amplitude can be found: \[Y_{resultant} = Y \sqrt{2} \cos(kx - \omega t - \frac{\pi}{4}) \].
05

- Extract the amplitude of the resultant wave

From the resultant wave expression, the amplitude is \[Y_{resultant amplitude} = Y \sqrt{2}\].

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Key Concepts

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

traveling waves
Traveling waves are disturbances that move through a medium from one location to another. These waves carry energy and information without transferring matter. For example, sound waves travel through air, and light waves travel through space.
In our problem, we have two identical traveling waves with the same amplitude, denoted by the symbol Y. These waves are described mathematically by sinusoidal functions, such as sine and cosine functions.
The mathematical representation helps us understand various properties of the waves, including their amplitude, wavelength, frequency, and phase.
phase relationship
The phase relationship between waves describes how much one wave is shifted relative to another. This shift is measured in radians or degrees.
In this problem, the two waves are out of phase by 蟺/2 radians (or 90 degrees). This means one wave reaches its maximum or minimum value a quarter of a cycle after the other wave.
In mathematical terms, we represent one wave as Y sin(kx - 蠅t) and the other as Y cos(kx - 蠅t). The cosine function is the sine function shifted by 蟺/2 radians.
trigonometric identities
Trigonometric identities are mathematical tools that simplify the addition and multiplication of trigonometric functions. These identities help us solve complex wave problems more easily.
In this exercise, we used the identity: \(\text{sin}(A) + \text{cos}(A) = \text{鈭�2 cos}(A - 蟺/4)\).
Using this identity, we combined the two out-of-phase waves into a single resultant wave: Y sin(kx - 蠅t) + Y cos(kx - 蠅t). The result is a wave with an amplitude 鈭�2 times the original amplitude Y, represented as Y 鈭�2 cos(kx - 蠅t - 蟺/4).
This identity allows us to find the amplitude of the resultant wave in terms of the common amplitude Y of the original waves.

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Most popular questions from this chapter

Tuning Fork Oscillation of a \(600 \mathrm{~Hz}\) tuning fork sets up standing waves in a string clamped at both ends. The wave speed for the string is \(400 \mathrm{~m} / \mathrm{s}\). The standing wave has four loops and an amplitude of \(2.0 \mathrm{~mm}\). (a) What is the length of the string? (b) Write an equation for the displacement of the string as a function of position and time.

Second Harmonic A rope, under a tension of \(200 \mathrm{~N}\) and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by $$ y(x, t)=(0.10 \mathrm{~m})\left(\sin \left(\left(\frac{\pi}{2} \mathrm{rad} / \mathrm{m}\right) x\right) \sin ((12 \pi \mathrm{rad} / \mathrm{s}) t)\right. $$ where \(x=0.00 \mathrm{~m}\) at one end of the rope, \(x\) is in meters, and \(t\) is in seconds. What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c) the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

Electromagnetic Waves The speed of electromagnetic waves (which include visible light, radio, and \(x\) -rays) in vacuum is \(3.0 \times\) \(10^{8} \mathrm{~m} / \mathrm{s}\). (a) Wavelengths of visible light waves range from about \(400 \mathrm{~nm}\) in the violet to about \(700 \mathrm{~nm}\) in the red. What is the range of frequencies of these waves? (b) The range of frequencies for shortwave radio (for example, FM radio and VHF television) is \(1.5\) to \(300 \mathrm{MHz}\). What is the corresponding wavelength range? (c) X-ray wavelengths range from about \(5.0 \mathrm{~nm}\) to about \(1.0 \times\) \(10^{-2} \mathrm{~nm}\). What is the frequency range for \(\mathrm{x}\) -rays?

The Fastest Wave What is the fastest transverse wave that can be sent along a steel wire? For safety reasons, the maximum tensile stress to which steel wires should be subjected is \(7.0 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}\). The density of steel is \(7800 \mathrm{~kg} / \mathrm{m}^{3}\). Show that your answer does not depend on the diameter of the wire.

Wire Clamped The tension in a wire clamped at both ends is doubled without appreciably changing the wire's length between the clamps. What is the ratio of the new to the old wave speed for transverse waves traveling along this wire?
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