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Chapter 16: Problem 148

The wave function that models a standing wave is given as \(y_{R}(x, t)=6.00 \mathrm{cm} \sin \left(3.00 \mathrm{m}^{-1} x+1.20 \mathrm{rad}\right)\) \(\cos \left(6.00 \mathrm{s}^{-1} t+1.20 \mathrm{rad}\right) .\) What are two wave functions that interfere to form this wave function? Plot the two wave functions and the sum of the sum of the two wave functions at \(t=1.00 \mathrm{s}\) to verify your answer.

Short Answer

Expert verified
Two wave functions that interfere to form this standing wave function are \(y_{1}(x, t) = 6.00 cm \cos (3.00 m^{-1} x - 6.00 s^{-1} t + 2.40 rad)\) and \(y_{2}(x, t) = 6.00 cm \cos (3.00 m^{-1} x + 6.00 s^{-1} t + 2.40 rad)\). The plots of these wave functions and their sum at \(t=1.00s\) confirm this solution.

Step by step solution

01

Identify Wave Function Parameters

The given wave function is \(y_{R}(x, t)=6.00 cm \sin (3.00m^{-1}x+1.20 rad)\cos (6.00s^{-1}t+1.20 rad)\). The wave function is a product of a spatial part \(\sin(3.00m^{-1}x+1.20 rad)\) and a temporal part \(\cos(6.00s^{-1}t+1.20 rad)\). The amplitude is 6.00 cm, the wavenumber is 3.00 m^{-1}, and the frequency is 6.00 s^{-1}.
02

Construct the Two Wave Functions

Two waves with same amplitudes, frequencies and phases but moving in opposite directions interfere to form a standing wave. Thus the two wave functions can be given as \(y_{1}(x, t) = 6.00 cm \cos (3.00 m^{-1} x - 6.00 s^{-1} t + 2.40 rad)\) and \(y_{2}(x, t) = 6.00 cm \cos (3.00 m^{-1} x + 6.00 s^{-1} t + 2.40 rad)\).
03

Plot the Two Wave Functions and Their Sum

Plot \(y_{1}(x,1.00s)\), \(y_{2}(x,1.00s)\), and the sum \(y_{1}(x,1.00s) + y_{2}(x,1.00s)\) to verify the answer. This step is performed on graphing software like Desmos or GeoGebra. The correct plots should show \(y_{1}(x,1.00s)\) and \(y_{2}(x,1.00s)\) as waves traveling in opposite directions, and the sum \(y_{1}(x,1.00s) + y_{2}(x,1.00s)\) as a standing wave.

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Key Concepts

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

Wave Function
A wave function is a mathematical description that captures the behavior of a wave. It's essentially a formula that tells us the displacement of points along the wave over time and space. In our example, the wave function is given by \(y_{R}(x, t)=6.00 \mathrm{cm} \sin \left(3.00 \mathrm{m}^{-1} x+1.20 \mathrm{rad}\right) \cos \left(6.00 \mathrm{s}^{-1} t+1.20 \mathrm{rad}\right)\).
This function is expressed as a product of two components: a sine function and a cosine function.
  • The sine function \(\sin(3.00 \mathrm{m}^{-1}x+1.20 \mathrm{rad})\) describes how the wave varies in space.
  • The cosine function \(\cos(6.00 \mathrm{s}^{-1}t+1.20 \mathrm{rad})\) describes how the wave changes over time.
Together, these functions create a standing wave鈥攁 wave that appears to stand still despite being the result of two waves traveling in opposite directions.
Superposition Principle
The superposition principle is a fundamental concept in physics, especially in wave theory.
This principle asserts that when two or more waves overlap, the resulting wave displacement is the sum of the individual wave displacements. This is particularly important for understanding standing waves.
In our problem, the wave function is expressed as a product of sine and cosine functions, representing two overlapping waves:
  • These individual waves are given by \(y_{1}(x, t) = 6.00 \mathrm{cm} \cos (3.00 \mathrm{m}^{-1} x - 6.00 \mathrm{s}^{-1} t + 2.40 \mathrm{rad})\) and \(y_{2}(x, t) = 6.00 \mathrm{cm} \cos (3.00 \mathrm{m}^{-1} x + 6.00 \mathrm{s}^{-1} t + 2.40 \mathrm{rad})\) .
  • By superimposing these waves, a standing wave is formed.
This emphasizes that the entire wave behavior is a combination of simpler wave actions.
Wave Interference
Wave interference is another critical concept when dealing with overlapping waves. It describes how waves interact with each other.
  • Constructive interference happens when waves align their peaks and troughs, reinforcing each other and creating a larger amplitude.
  • Destructive interference occurs when waves' peaks align with the other wave's troughs, canceling each other out and reducing amplitude.
In this exercise, the standing wave arises from the interference of two traveling waves moving in opposite directions. When these waves overlap:
  • They continuously undergo constructive and destructive interference.
  • This creates nodes (points of zero amplitude) and antinodes (points of maximum amplitude).
This repeating pattern gives the appearance of the wave 'standing' still.
Wavenumber
The wavenumber is a measure of the number of wave cycles, or wavelengths, that exist over a given distance.
It is often denoted by \(k\) and is calculated as the number of cycles per unit distance. In our wave function,
  • we see that the spatial part is \(\sin(3.00 \mathrm{m}^{-1}x+1.20 \mathrm{rad})\).
  • The coefficient of \(x\), 3.00 \( \mathrm{m}^{-1}\), is the wavenumber.
A higher wavenumber indicates more compressions in the wave over a distance, reflecting a shorter wavelength. Understanding the wavenumber is crucial for visualizing how tightly packed the wave crests and troughs are.
Frequency
Frequency refers to how often the wave oscillates in time. It's the number of complete wave cycles that pass a given point per second, most commonly measured in Hertz (Hz).
In our wave function,
  • the temporal part \(\cos(6.00 \mathrm{s}^{-1}t+1.20 \mathrm{rad})\) shows the wave's frequency.
  • The coefficient of \(t\), 6.00 \(\mathrm{s}^{-1}\), represents the frequency.
A higher frequency means the wave oscillates more quickly over time, indicative of faster repeating cycles. Understanding frequency helps us grasp how swiftly the wave pattern repeats and is essential for analyzing wave behavior over time.

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Most popular questions from this chapter

Two sinusoidal waves are moving through a medium in the positive \(x\) -direction, both having amplitudes of 6.00 \(\mathrm{cm},\) a wavelength of \(4.3 \mathrm{m},\) and a period of \(6.00 \mathrm{s},\) but one has a phase shift of an angle \(\phi=0.50\) rad. What is the height of the resultant wave at a time \(t=3.15 \mathrm{s}\) and a position \(x=0.45 \mathrm{m} ?\).

A wave traveling on a Slinky() that is stretched to \(4 \mathrm{m}\) takes \(2.4 \mathrm{s}\) to travel the length of the slinky and bach again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes At what frequency must the Slinky be oscillating?

A standing wave is produced on a string under a tension of \(70.0 \mathrm{N}\) by two sinusoidal transverse waves that are identical, but moving in opposite directions. The string is fixed at \(x=0.00 \mathrm{m}\) and \(x=10.00 \mathrm{m} .\) Nodes appear at \(x=0.00 \mathrm{m}, 2.00 \mathrm{m}, 4.00 \mathrm{m}, 6.00 \mathrm{m}, 8.00 \mathrm{m},\) and 10.00 m. The amplitude of the standing wave is \(3.00 \mathrm{cm} .\) It takes \(0.10 \mathrm{s}\) for the antinodes to make one complete oscillation. (a) What are the wave functions of the two sine waves that produce the standing wave? (b) What are the maximum velocity and acceleration of the string, perpendicular to the direction of motion of the transverse waves, at the antinodes?

A string is fixed at both end. The mass of the string is \(0.0090 \mathrm{kg}\) and the length is \(3.00 \mathrm{m}\). The string is under a tension of 200.00 N. The string is driven by a variable frequency source to produce standing waves on the string. Find the wavelengths and frequency of the first four modes of standing waves.

Consider what is shown below. A 20.00-kg mass rests on a frictionless ramp inclined at \(45^{\circ} .\) A string with a linear mass density of \(\mu=0.025 \mathrm{kg} / \mathrm{m}\) is attached to the 20.00-kg mass. The string passes over a frictionless pulley of negligible mass and is attached to a hanging mass \((m)\) The system is in static equilibrium. A wave is induced on the string and travels up the ramp. (a) What is the mass of the hanging mass (m)? (b) At what wave speed does the wave travel up the string?
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