Problem 16 - Classical Dynamics of Particles and Systems

Question

Consider a wave \(g(x-v t)\) propagating in the \(+x\) -direction with velocity \(v\). A rigid wall is placed at \(x=x_{0} .\) Describe the motion of the wave for \(x

Step-by-Step Solution

Answer

The motion of the wave for x

Step 1: 1. Determine the original wave

The original wave is given by the function g(x - vt), where x is the position and t is the time. This wave propagates in the +x direction with a velocity v.

Step 2: 2. Find the reflected wave

Since there is a rigid wall at x=x₀, the wave will be reflected upon meeting the wall. The reflected wave's function will be g(x₀ - x - vt), where x₀ is the position of the wall. Notice that the sign has changed to '-' in the argument of g to ensure that the wave is now propagating in the -x direction upon reflection.

Step 3: 3. Apply the principle of superposition

According to the principle of superposition, when two or more waves overlap, the resulting wave is the sum of the individual waves. Therefore, to find the total wave for x < x₀, we need to add the original wave and the reflected wave together.

Step 4: 4. Find the resulting wave function

The resulting wave function is the sum of the original wave and the reflected wave. So, we can write the total wave function as: \( W(x,t) = g(x - vt) + g(x₀ - x - vt) \) This function represents the motion of the wave for x < x₀, taking into account the reflection by the rigid wall at x = x₀.

Key Concepts

Reflection of Waves

When a wave encounters an obstacle, like a rigid wall, it cannot continue in its original direction. Instead, it is reflected back into the medium. In our exercise, this occurs at the position \(x = x_0\). Reflection involves a change in direction of the wave, which means the reflected wave travels in the opposite direction to the incident wave.

The incident wave is given by the function \(g(x-vt)\), which travels in the \(+x\) direction. Upon striking the wall at \(x = x_0\), the wave is reflected, reversing its direction to travel in the \(-x\) direction. This reversal is reflected in the new wave function \(g(x_0 - x - vt)\), where the sign before \(x\) becomes negative.

Reflection doesn't change the wave's frequency and speed, but it may affect the wave's phase. Understanding this concept is crucial when analyzing wave behaviors at boundaries, ensuring comprehension of phenomena like echoes or standing waves.

Superposition Principle

The superposition principle is a fundamental concept in wave propagation. It posits that when multiple waves traverse the same space, the resulting wave is simply the algebraic sum of the individual waves. In our context, we have both the original and the reflected waves present in the region \(x < x_0 \).

• Original Wave: \(g(x - vt)\)
• Reflected Wave: \(g(x_0 - x - vt)\)

The principle assists us in predicting the resulting wave's behavior by combining the incident and reflected waves. By adding these waves together mathematically using the principle of superposition, we arrive at the total wave function for the area where both waves are present:

\[W(x,t) = g(x - vt) + g(x_0 - x - vt)\]This approach simplifies complex wave interactions, providing a straightforward method to determine the resultant wave's characteristics.

Wave Function

A wave function is a mathematical representation describing a wave's motion. It encodes information about the wave's displacement at any point \(x\) and time \(t\). For mechanical waves, this function is typically expressed as \(g(x - vt)\), where \(v\) is the wave’s velocity.

From the exercise, the wave function for the total wave considering both the incident and reflected waves is expressed as:

\[W(x,t) = g(x - vt) + g(x_0 - x - vt)\]

This formula reflects both components of the wave present in the system. The first term \(g(x - vt)\) represents the original wave traveling towards the wall, while the second term \(g(x_0 - x - vt)\) denotes the wave reflected back from the wall.

Such functions provide insight into the possible displacements or disturbances of the wave at any point in time and space. Understanding wave functions is essential in predicting and elucidating phenomena like interference, diffraction, and reflection in various wave types, including sound, light, and water waves.

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Problem 15

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Problem 17

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Key Concepts

Reflection of Waves

When a wave encounters an obstacle, like a rigid wall, it cannot continue in its original direction. Instead, it is reflected back into the medium. In our exercise, this occurs at the position \(x = x_0\). Reflection involves a change in direction of the wave, which means the reflected wave travels in the opposite direction to the incident wave.

The incident wave is given by the function \(g(x-vt)\), which travels in the \(+x\) direction. Upon striking the wall at \(x = x_0\), the wave is reflected, reversing its direction to travel in the \(-x\) direction. This reversal is reflected in the new wave function \(g(x_0 - x - vt)\), where the sign before \(x\) becomes negative.

Reflection doesn't change the wave's frequency and speed, but it may affect the wave's phase. Understanding this concept is crucial when analyzing wave behaviors at boundaries, ensuring comprehension of phenomena like echoes or standing waves.

Superposition Principle

The superposition principle is a fundamental concept in wave propagation. It posits that when multiple waves traverse the same space, the resulting wave is simply the algebraic sum of the individual waves. In our context, we have both the original and the reflected waves present in the region \(x < x_0 \).

• Original Wave: \(g(x - vt)\)
• Reflected Wave: \(g(x_0 - x - vt)\)

The principle assists us in predicting the resulting wave's behavior by combining the incident and reflected waves. By adding these waves together mathematically using the principle of superposition, we arrive at the total wave function for the area where both waves are present:

\[W(x,t) = g(x - vt) + g(x_0 - x - vt)\]This approach simplifies complex wave interactions, providing a straightforward method to determine the resultant wave's characteristics.

Wave Function

A wave function is a mathematical representation describing a wave's motion. It encodes information about the wave's displacement at any point \(x\) and time \(t\). For mechanical waves, this function is typically expressed as \(g(x - vt)\), where \(v\) is the wave’s velocity.

From the exercise, the wave function for the total wave considering both the incident and reflected waves is expressed as:

\[W(x,t) = g(x - vt) + g(x_0 - x - vt)\]

This formula reflects both components of the wave present in the system. The first term \(g(x - vt)\) represents the original wave traveling towards the wall, while the second term \(g(x_0 - x - vt)\) denotes the wave reflected back from the wall.

Such functions provide insight into the possible displacements or disturbances of the wave at any point in time and space. Understanding wave functions is essential in predicting and elucidating phenomena like interference, diffraction, and reflection in various wave types, including sound, light, and water waves.