Question

(a) A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed \(v\), frequency \(f\), amplitude \(A\), and wavelength \(\lambda\). Calculate the maximum transverse velocity and maximum transverse acceleration of points located at (i) \(x = \lambda/2\), (ii) \(x = \lambda\)/4, and (iii) \(x = \lambda\)8, from the left-hand end of the string. (b) At each of the points in part (a), what is the amplitude of the motion? (c) At each of the points in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement?

Short answer

(a) Max velocity/acceleration depends on \(x\); (b) Amplitude is \(A\); (c) Time for displacement change is \(\frac{1}{2f}\).

Step-by-step solution

Understanding the Fundamental Mode of Vibration

In the fundamental mode of a string fixed at both ends, the string vibrates with a single antinode in the middle and nodes at both ends. The wavelength \( \lambda \) of the fundamental mode is twice the length of the string. The wave equation for a point \( x \) on the string is given by:\[ y(x, t) = A \sin\left(\frac{2\pi}{\lambda} x \right) \cos(2\pi f t) \]where \( y(x, t) \) is the displacement of the point at position \( x \) at time \( t \).

Calculating Maximum Transverse Velocity

The maximum transverse velocity \( v_{\text{max}} \) occurs when the time derivative of the displacement \( y(x, t) \) reaches its maximum. The velocity at point \( x \) is \[ v(x, t) = \frac{\partial y}{\partial t} = -2\pi f A \sin\left(\frac{2\pi}{\lambda} x \right) \sin(2\pi f t) \]Thus, the maximum transverse velocity at each point occurs when \( \sin(2\pi f t) = \pm 1 \), so:\[ v_{\text{max}}(x) = 2\pi f A \left| \sin\left(\frac{2\pi}{\lambda} x \right) \right| \]

Calculating Maximum Transverse Acceleration

The maximum transverse acceleration \( a_{\text{max}} \) occurs when the second time derivative of the displacement \( y(x, t) \) reaches its maximum. The acceleration is given by:\[ a(x, t) = \frac{\partial^2 y}{\partial t^2} = - (2\pi f)^2 A \sin\left(\frac{2\pi}{\lambda} x \right) \cos(2\pi f t) \]Thus, the maximum transverse acceleration at each point occurs when \( \cos(2\pi f t) = \pm 1 \), so:\[ a_{\text{max}}(x) = (2\pi f)^2 A \left| \sin\left(\frac{2\pi}{\lambda} x \right) \right| \]

Identifying the Amplitude for Each Point

The amplitude of motion at any point \( x \) on the string is the same and equal to \( A \) (the amplitude of the wave), because the entire wave moves uniformly across the string. So, the amplitude at all points \( x = \lambda/2, \lambda/4, \lambda/8 \) is the amplitude \( A \).

Determining Time for Largest Displacement Change

The time it takes for the string to go from its largest upward displacement to its largest downward displacement is half the period of the wave. The period \( T \) is the reciprocal of the frequency \( f \):\[ T = \frac{1}{f} \]The time taken for this change in displacement is:\[ \Delta t = \frac{T}{2} = \frac{1}{2f} \]

Key concepts

Transverse Velocity

In wave mechanics, transverse velocity refers to the speed at which a point on the string moves up and down as the wave travels. It's crucial to distinguish this from the wave speed, which is the speed at which the wave propagates along the string.

The formula for transverse velocity is derived from the time derivative of the displacement function, given as:

• Velocity, \( v(x, t) = \frac{\partial y}{\partial t} = -2\pi f A \sin\left(\frac{2\pi}{\lambda} x \right) \sin(2\pi f t) \)

This expression tells us how fast a point on the string moves perpendicularly to the wave's travel direction. The maximum transverse velocity occurs when \( \sin(2\pi f t) = \pm 1 \), leading to the maximum expression:

• \( v_{\text{max}}(x) = 2\pi f A \left| \sin\left(\frac{2\pi}{\lambda} x \right) \right| \)

This highlights that the transverse movement of the string depends on both the amplitude \( A \) of the vibration and the frequency \( f \) of the wave.

Transverse Acceleration

Transverse acceleration in the context of string vibrations measures how quickly the velocity of a point on the string changes. Like velocity, it acts perpendicular to the direction of wave propagation.

The maximum transverse acceleration is found by taking the second derivative of the displacement function:

• Acceleration, \( a(x, t) = \frac{\partial^2 y}{\partial t^2} = - (2\pi f)^2 A \sin\left(\frac{2\pi}{\lambda} x \right) \cos(2\pi f t) \).

The maximum value occurs when \( \cos(2\pi f t) = \pm 1 \):

• \( a_{\text{max}}(x) = (2\pi f)^2 A \left| \sin\left(\frac{2\pi}{\lambda} x \right) \right| \).

The factors affecting the transverse acceleration are similar to those affecting transverse velocity: the wave's frequency \( f \) and its amplitude \( A \). These tell us how energetic or vigorous the vibration is, influencing how quickly points on the string accelerate during their movement.

String Vibration

String vibration is a fundamental concept in wave mechanics where waves travel along a string fixed at both ends. This brings out standing waves, characterized by specific frequencies and patterns.

These waves are a result of the string's tension and its length, affecting the speed \( v \) at which waves move. In a vibrating string:

• Nodes: Points that remain stationary, as seen at the ends of the string.
• Antinodes: Points with maximum displacement, typically found in the middle sections between nodes.

The fundamental mode of vibration is the simplest possible wave on the string, with a single antinode in its center. It establishes the primary frequency of the string as it vibrates.

Fundamental Mode

The fundamental mode represents the simplest standing wave pattern a string can form when vibrated. It is the lowest possible frequency at which the string vibrates naturally.

In this mode:

• There is only one antinode, lying midway between the two nodes at the ends of the string.
• The wavelength \( \lambda \) of this mode is precisely twice the length of the string.
• The frequency of the fundamental mode is the lowest resonance frequency the string can support.

The importance of the fundamental mode lies in its role as the basis for more complex vibrations, making it crucial for understanding harmonics and overtones in musical instruments.

Wavelength

Wavelength is a key parameter in wave mechanics that represents the spatial period of a wave—the distance over which the wave's shape repeats.

For a string vibrating in a standing wave pattern, the wavelength is determined by the number of nodes and antinodes:

• In the fundamental mode, the wavelength is twice the length of the string.
• Each subsequent mode of vibration (harmonics) divides the string into more segments, changing the effective wavelength.

Wavelength is associated with frequency \( f \) through the wave speed \( v \):

• \( v = f\lambda \)

This relationship is crucial for calculating how waves travel on the string, important for both musical acoustics and general wave mechanics.