Question:medium
For a travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi(10t – 0.0080 x + 0.35)$, where $x$ and $y$ are in cm and $t$ in s. The phase difference between oscillatory motion of two points separated by a distance of 0.5 m is: ____.
For a travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi(10t – 0.0080 x + 0.35)$, where $x$ and $y$ are in cm and $t$ in s. The phase difference between oscillatory motion of two points separated by a distance of 0.5 m is: ____.
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Always ensure units are consistent. In wave problems, the most common mistake is mixing meters (distance) with centimeters (from the wave equation).
Updated On: May 28, 2026
- 0.8 $\pi$ rad
- 8 $\pi$ rad
- 0.008 $\pi$ rad
- 0.08 $\pi$ rad
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Topic:
This problem belongs to the study of "Waves," specifically focusing on travelling harmonic waves. A wave represents the propagation of a disturbance through a medium. In a harmonic wave, every point in the medium undergoes simple harmonic motion. The "phase" of this oscillation describes the specific state of motion (position and direction) at any given time. When we look at two different points in space, they reach the same state of motion at different times, creating a "phase difference" that depends on the distance between them.
Step 2: Key Formulas and Approach:
The standard equation for a travelling wave moving in the positive x-direction is: \[ y(x, t) = A \cos(\omega t - kx + \phi_0) \] Where:
$A$ is the amplitude.
$\omega$ is the angular frequency ($2\pi f$).
$k$ is the wave number ($2\pi / \lambda$).
The phase difference ($\Delta \phi$) between two points separated by a distance ($\Delta x$) is given by: \[ \Delta \phi = k \cdot \Delta x \]
Step 3: Detailed Explanation:
Analyze the given equation: The wave is described by $y = 2.0 \cos [2\pi(10t - 0.0080x + 0.35)]$. To find $k$, we distribute the $2\pi$ inside the brackets: \[ y = 2.0 \cos (20\pi t - 0.016\pi x + 0.7\pi) \]
Identify the wave number ($k$): By comparing this to the standard form, the coefficient of $x$ is $k$. Thus, $k = 0.016\pi \text{ rad/cm}$. Note that the units for $x$ are in centimeters.
Prepare the path difference ($\Delta x$): The distance between the two points is given as $0.5 \text{ m}$. To maintain consistency with the wave equation (which uses cm), we must convert this distance: \[ \Delta x = 0.5 \text{ m} = 50 \text{ cm} \]
Calculate the phase difference: Substitute the values into the phase difference formula: \[ \Delta \phi = k \cdot \Delta x \] \[ \Delta \phi = (0.016\pi \text{ rad/cm}) \times 50 \text{ cm} \] \[ \Delta \phi = 0.8\pi \text{ rad} \]
Step 4: Final Answer:
The phase difference between the two points is 0.8 $\pi$ rad.
This problem belongs to the study of "Waves," specifically focusing on travelling harmonic waves. A wave represents the propagation of a disturbance through a medium. In a harmonic wave, every point in the medium undergoes simple harmonic motion. The "phase" of this oscillation describes the specific state of motion (position and direction) at any given time. When we look at two different points in space, they reach the same state of motion at different times, creating a "phase difference" that depends on the distance between them.
Step 2: Key Formulas and Approach:
The standard equation for a travelling wave moving in the positive x-direction is: \[ y(x, t) = A \cos(\omega t - kx + \phi_0) \] Where:
$A$ is the amplitude.
$\omega$ is the angular frequency ($2\pi f$).
$k$ is the wave number ($2\pi / \lambda$).
The phase difference ($\Delta \phi$) between two points separated by a distance ($\Delta x$) is given by: \[ \Delta \phi = k \cdot \Delta x \]
Step 3: Detailed Explanation:
Analyze the given equation: The wave is described by $y = 2.0 \cos [2\pi(10t - 0.0080x + 0.35)]$. To find $k$, we distribute the $2\pi$ inside the brackets: \[ y = 2.0 \cos (20\pi t - 0.016\pi x + 0.7\pi) \]
Identify the wave number ($k$): By comparing this to the standard form, the coefficient of $x$ is $k$. Thus, $k = 0.016\pi \text{ rad/cm}$. Note that the units for $x$ are in centimeters.
Prepare the path difference ($\Delta x$): The distance between the two points is given as $0.5 \text{ m}$. To maintain consistency with the wave equation (which uses cm), we must convert this distance: \[ \Delta x = 0.5 \text{ m} = 50 \text{ cm} \]
Calculate the phase difference: Substitute the values into the phase difference formula: \[ \Delta \phi = k \cdot \Delta x \] \[ \Delta \phi = (0.016\pi \text{ rad/cm}) \times 50 \text{ cm} \] \[ \Delta \phi = 0.8\pi \text{ rad} \]
Step 4: Final Answer:
The phase difference between the two points is 0.8 $\pi$ rad.
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