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Chapter 16: Problem 38

Two sinusoidal waves of the same frequency are to be sent in the same direction along a taut string. One wave has an amplitude of \(5.0 \mathrm{~mm}\), the other \(8.0 \mathrm{~mm}\). (a) What phase difference \(\phi_{1}\) between the two waves results in the smallest amplitude of the resultant wave? (b) What is that smallest amplitude? (c) What phase difference \(\phi_{2}\) results in the largest amplitude of the resultant wave? (d) What is that largest amplitude? (e) What is the resultant amplitude if the phase angle is \(\left(\phi_{1}-\phi_{2}\right) / 2 ?\)

Short Answer

Expert verified
(a) Phase difference for smallest amplitude is \(\pi\). (b) Smallest amplitude is 3 mm. (c) Phase difference for largest amplitude is 0. (d) Largest amplitude is 13 mm. (e) Resultant amplitude for \((\phi_{1} - \phi_{2})/2\) is about 9.4 mm.

Step by step solution

01

Understanding the Problem

When two sinusoidal waves interfere, the resultant amplitude depends on their phase difference. The amplitude of the resultant wave can vary between the sum and the difference of individual amplitudes based on this phase difference.
02

Finding the Smallest Amplitude

The smallest amplitude occurs when the waves are completely out of phase (\(\phi_{1} = \pi, 3\pi, 5\pi, ...\). The formula for the amplitude of the resultant wave is given by: \(A_{resultant} = |A_1 - A_2|\). Substituting the given amplitudes, \(A_1 = 5.0\, \text{mm}\) and \(A_2 = 8.0\, \text{mm}\), we get:\[A_{min} = |5 - 8| = 3 \text{ mm}\]
03

Finding the Largest Amplitude

The largest amplitude occurs when the waves are in phase (\(\phi_{2} = 0, 2\pi, 4\pi, ...\)). The formula for the amplitude of the resultant wave when in phase is \(A_{resultant} = A_1 + A_2\). Substituting the given values,\[A_{max} = 5 + 8 = 13 \text{ mm}\]
04

Analyzing the Given Phase Angle

The phase angle \((\phi_{1} - \phi_{2})/2\) is halfway between the phase differences required for the smallest and largest amplitudes. It represents the case where the waves are neither fully in-phase nor out-of-phase.
05

Calculating the Resultant Amplitude for Given Phase Angle

With phase angle \( \phi = (\phi_{1} - \phi_{2})/2 = \pi/2 \) for destructive interference, the amplitude can be determined using:\[A_{resultant} = \sqrt{A_1^2 + A_2^2 - 2A_1A_2\cos(\phi)}\]Substituting \( \phi = \pi/2 \), \( \cos(\pi/2) = 0 \), we simplify to:\[A = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.4 \text{ mm}\]

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Key Concepts

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

Sinusoidal Waves
Imagine a smooth and continuous wave moving up and down like a gentle ripple on water. That's a sinusoidal wave. It gets its name from the 'sine' function in trigonometry, which accurately describes its shape. These waves are commonly found in nature and physics.
Examples include sound waves, light waves, and even ocean waves. The key characteristics of sinusoidal waves include their wavelength, frequency, and amplitude.
  • Wavelength is the distance between two consecutive peaks or troughs of the wave.
  • Frequency tells us how many waves pass a point in one second.
  • Amplitude is the wave's height from its central axis to its peak.
Understanding these properties helps us delve deeper into how these waves interact, leading to wave interference.
Phase Difference
When two sinusoidal waves meet, their positions relative to each other matter greatly. This concept is known as phase difference. Phase difference tells us how synchronized two waves are. Even tiny shifts in synchronization can cause noticeable effects.
  • If two waves are in phase, their peaks and troughs align perfectly, enhancing their overall effect.
  • When out of phase, the peak of one wave aligns with the trough of another, potentially canceling each other out.
Measuring this difference in synchronization is often expressed in terms of degrees or radians, with 0 radians (or 0 degrees) meaning perfect alignment, and 蟺 radians (or 180 degrees) indicating complete misalignment. This alignment or misalignment significantly impacts the resultant wave when two waves interfere.
Amplitude
Amplitude is a crucial aspect of waves, representing the maximum displacement from the wave's equilibrium position. It reflects the wave's energy: larger amplitudes imply more energy. Whether it's the loudness of a sound or the brightness of light, amplitude plays a vital role.
For two interfering waves, changes in their amplitude can tell us about how much they add to or cancel each other.
  • The largest amplitude occurs when two waves are perfectly in phase, meaning their amplitudes add together.
  • The smallest amplitude appears when the waves are perfectly out of phase, and their amplitudes subtract.
The resultant amplitude depends on both the amplitudes of the individual waves and their phase difference, showing the importance of both factors in wave interference.
Interference Patterns
Wave interference occurs when two or more waves overlap. The result is an interference pattern鈥攁 new wave form created from the summation of individual waves.
There are two primary types of interference:
  • Constructive Interference: This happens when waves meet in phase, leading to a wave with amplitude equal to the sum of individual amplitudes.
  • Destructive Interference: Occurs when waves are out of phase, leading to cancellation, reducing the overall amplitude.
Through interference patterns, we can visualize the way waves interact in various mediums. For instance, in physics labs, students often observe interference patterns using light, which creates colorful bands. These patterns not only deepen our understanding of wave behavior but are also foundational in various technologies, such as noise-cancelling headphones or optical instruments.

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Most popular questions from this chapter

A sinusoidal wave of frequency \(500 \mathrm{~Hz}\) has a speed of \(350 \mathrm{~m} / \mathrm{s}\). (a) How far apart are two points that differ in phase by \(\pi / 3\) rad? (b) What is the phase difference between two displacements at a certain point at times \(1.00 \mathrm{~ms}\) apart?

A standing wave pattern on a string is described by $$ y(x, t)=0.040(\sin 5 \pi x)(\cos 40 \pi t) $$ where \(x\) and \(y\) are in meters and \(t\) is in seconds. For \(x \geq 0\), what is the location of the node with the (a) smallest, (b) second smallest, and (c) third smallest value of \(x\) ? (d) What is the period of the oscillatory motion of any (nonnode) point? What are the (e) speed and (f) amplitude of the two traveling waves that interfere to produce this wave? For \(t \geq 0\), what are the \((\mathrm{g})\) first, \((\mathrm{h})\) second, and (i) third time that all points on the string have zero transverse velocity?

A transverse sinusoidal wave is generated at one end of a long, horizontal string by a bar that moves up and down through a distance of \(1.00 \mathrm{~cm}\). The motion is continuous and is repeated regularly 120 times per second. The string has linear density 120 \(\mathrm{g} / \mathrm{m}\) and is kept under a tension of \(90.0 \mathrm{~N}\). Find the maximum value of (a) the transverse speed \(u\) and (b) the transverse component of the tension \(\tau\). (c) Show that the two maximum values calculated above occur at the same phase values for the wave. What is the transverse displacement \(y\) of the string at these phases? (d) What is the maximum rate of energy transfer along the string? (e) What is the transverse displacement \(y\) when this maximum transfer occurs? (f) What is the minimum rate of energy transfer along the string? (g) What is the transverse displacement \(y\) when this minimum transfer occurs?

If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing wind. The air pressure variations in the vortexes tend to cause the line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. In long lines, the resonant frequencies are so close that almost any wind speed can set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. If a transmission line has a length of \(347 \mathrm{~m}\), a linear density of \(3.35 \mathrm{~kg} / \mathrm{m}\), and a tension of \(65.2 \mathrm{MN}\), what are (a) the frequency of the fundamental mode and (b) the frequency difference between successive modes?

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?
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