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Chapter 16: Problem 94

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 \(x=0.45 \mathrm{m} ?\).

Short Answer

Expert verified
The height of the resultant wave at time \(t=3.15s\) and position \(x=0.45m\) is given by adding the displacements of the first and second waves: \[y = y_1 + y_2 = 6 \sin\left(\frac{2\pi}{4.3}(0.45) - \frac{2\pi}{6}(3.15)\right) + 6 \sin\left(\frac{2\pi}{4.3}(0.45) - \frac{2\pi}{6}(3.15) + 0.5\right)\] Calculate the value of \(y\) to find the height of the resultant wave at the given time and position.

Step by step solution

01

Write the wave functions for the first and second waves

Since both waves have the same amplitude, wavelength, and period, we can express them as: \[y_1 = A \sin(kx - \omega t)\] \[y_2 = A \sin(kx - \omega t + \phi)\] Where - \(A = 6.00\) cm is the amplitude - \(\lambda = 4.3\) m is the wavelength - \(T = 6.00\) s is the period - \(\phi = 0.50\) rad is the phase shift #Step 2: Calculating \(k\) and \(\omega\)#
02

Calculate the wave number \(k\) and the angular frequency \(\omega\)

From the given parameters, we can determine the wave number \(k\) and the angular frequency \(\omega\) using the formulas: \[k = \frac{2\pi}{\lambda}\] \[\omega = \frac{2\pi}{T}\] Plug the values of \(\lambda\) and \(T\): \[k = \frac{2\pi}{4.3}\] \[\omega = \frac{2\pi}{6.00}\] #Step 3: Determine the displacements#
03

Calculate the displacements of the first and second waves

Use the given values of \(t = 3.15\) s and \(x = 0.45\) m to find the displacements of the first and second waves: \[y_1 = 6 \sin\left(\frac{2\pi}{4.3}(0.45) - \frac{2\pi}{6}(3.15)\right)\] \[y_2 = 6 \sin\left(\frac{2\pi}{4.3}(0.45) - \frac{2\pi}{6}(3.15) + 0.5\right)\] #Step 4: Calculating the resultant wave height#
04

Calculate the height of the resultant wave

Use the principle of superposition to find the height of the resultant wave by adding the displacements of the first and second waves: \[y = y_1 + y_2\] Solve for \(y\): \[y = 6 \sin\left(\frac{2\pi}{4.3}(0.45) - \frac{2\pi}{6}(3.15)\right) + 6 \sin\left(\frac{2\pi}{4.3}(0.45) - \frac{2\pi}{6}(3.15) + 0.5\right)\] Calculate the value of \(y\) to find the height of the resultant wave at the given time and position. #Completion# The height of the resultant wave at time \(t=3.15s\) and position \(x=0.45m\) can be found using the principle of superposition and the provided wave parameters. Once the displacements of both waves are calculated, they can simply be added to find the height of the resultant wave.

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Key Concepts

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

Sinusoidal Waves
Sinusoidal waves are fundamental waves characterized by their smooth, periodic oscillations. They are named after the sine function, which mathematically describes their shape. These waves are vital in various fields, including physics, engineering, and signal processing. The basic form of a sinusoidal wave is expressed through the wave equation:
  • Amplitude (\( A \)): The maximum displacement from the rest position. In this exercise, the amplitude is 6.00 cm.
  • Wavelength (\( \lambda \)): The distance between consecutive peaks of the wave. Here, the wavelength is 4.3 m.
  • Period (\( T \)): The time it takes for one complete cycle of the wave to pass a point, which is 6.00 s in this case.
These parameters define how the wave moves through space and time, creating the repeating, smooth pattern of sinusoidal waves.
Phase Shift
The phase shift in wave motion is a measure of how far two waves have moved out of phase with each other. It determines the position of the wave at a specific time relative to a reference wave. In the context of sinusoidal waves, a phase shift is represented by the term \( \phi \) in the wave equation.
  • A phase shift of \( \phi = 0.50 \) rad in this exercise indicates that the second wave leads the first wave by a specific amount.
  • This shift changes the point where the wave reaches its maximum (peak) and minimum values (trough).
  • Phase shifts are crucial when combining waves using superposition, as they affect the interference pattern and amplitude at given points.
Understanding phase shift allows for better manipulation of waves, as seen in technologies like noise-cancelling headphones, where phase shifts are used to cancel unwanted sounds.
Wave Equation
The wave equation is a mathematical formula that describes how waves propagate through space and time. It captures the relationship between various properties of a wave, including amplitude, wavelength, and angular frequency. The general form for a sinusoidal wave moving in one dimension is:
\[ y(x, t) = A \sin(kx - \omega t + \phi) \]
  • By adjusting \( A \), \( k \), \( \omega \), and \( \phi \), we can describe different waves and predict their behavior under various conditions.
  • For superposition, two or more wave equations are added to find a resultant wave, as seen in this exercise.
  • This equation helps predict how waves interfere, leading to constructive or destructive interference patterns.
By comprehending the wave equation, one can better understand and manipulate wave phenomena, applicable in fields such as acoustics, optics, and beyond.
Wave Number
The wave number, represented as \( k \), is an important concept in wave mechanics. It indicates how many wavelengths fit into a unit distance. The formula to calculate the wave number is:
\[ k = \frac{2\pi}{\lambda} \]
  • In this exercise, the wave number tells us how densely packed the waves are per meter, with \( \lambda = 4.3 \) m.
  • A higher wave number means more oscillations per unit distance, indicating a shorter wavelength.
  • The wave number is intertwined with the phase velocity and frequency of the wave, affecting how the wave propagates through a medium.
Grasping the concept of wave number is essential for understanding wave propagation, diffraction, and interference, making it key to studying wave phenomena in various scientific fields.

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Most popular questions from this chapter

A pulse can be described as a single wave disturbance that moves through a medium. Consider a pulse that is \(\begin{array}{llllll}\text { defined } & \text { at } & \text { time } & t=0.00 \mathrm{s} & \text { by } & \text { the equation }\end{array}\) \(y(x)=\frac{6.00 \mathrm{m}^{3}}{x^{2}+2.00 \mathrm{m}^{2}}\) centered around \(x=0.00 \mathrm{m} .\) The pulse moves with a velocity of \(v=3.00 \mathrm{m} / \mathrm{s}\) in the positive \(x\) -direction. (a) What is the amplitude of the pulse? (b) What is the equation of the pulse as a function of position and time? (c) Where is the pulse centered at time \(t=5.00 \mathrm{s} ?\)

A wave is modeled by the wave function a me a nunction \(y(x, t)=(0.30 \mathrm{m}) \sin \left[\frac{2 \pi}{4.50 \mathrm{m}}\left(x-18.00 \frac{\mathrm{m}}{\mathrm{s}} t\right)\right] . \quad\) What are the amplitude, wavelength, wave speed, period, and frequency of the wave?

Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is $$y_{R}(x, t)=0.70 \mathrm{m} \sin \left(3.00 \mathrm{m}^{-1} x-6.28 \mathrm{s}^{-1} t+\pi / 16 \mathrm{rad}\right)$$,What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

In this section, we have considered waves that move at a constant wave speed. Does the medium accelerate?

A string has a mass of \(150 \mathrm{g}\) and a length of 3.4 m. One end of the string is fixed to a lab stand and the other is attached to a spring with a spring constant of \(k_{s}=100 \mathrm{N} / \mathrm{m} .\) The free end of the spring is attached to another lab pole. The tension in the string is maintained by the spring. The lab poles are separated by a distance that stretches the spring \(2.00 \mathrm{cm} .\) The string is plucked and a pulse travels along the string. What is the propagation speed of the pulse?
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