Question

A string of length \(a\) is fixed at the left end, and the right end moves with displacement \(A \sin \omega t\). Find \(\psi(x, t)\) and a consistent set of initial conditions for the displacement and the velocity.

Short answer

The wave function \(\psi(x,t)\) is given by \(A \sin (\omega (t - \frac{x}{v}))\) and the initial conditions for displacement and velocity are \( \psi(x,0) = A \sin(kx) \) and \( \frac{\partial\psi}{\partial t}|_{t=0} = vA k \cos(kx)\), respectively.

Step-by-step solution

Identify Variables

In this problem, \(A\) represents amplitude of the motion, \(\omega\) is frequency of oscillation and \(t\) is the time variable. The position along the string is represented by \(x\). The equation \(A \sin \omega t\) represents simple harmonic motion of the free end of the string.

Formulate Wave Equation

The general solution for wave equation is written as \( \psi(x,t) = f(x ± vt) \). Since the wave is moving with a sinusoidal motion, we can incorporate the given motion equation into our general wave equation as \( \psi(x,t) = A \sin (\omega (t - \frac{x}{v})) \), where \(v\) represents velocity of the wave and can be written as the ratio \(\frac{\omega}{k}\), \(\(k\)\) is the wave number.

Initial conditions

The initial conditions for displacement (\(\psi(x,0)\)) and velocity (\(\frac{\partial\psi}{\partial t}|_{t=0}\)) will be consistent with the wave equation. Displacement at time zero: \( \psi(x,0) = A \sin(kx) \). Initial velocity is the derivative of displacement with respect to time, thus taking the derivative produces: \( \frac{\partial\psi}{\partial t}|_{t=0} = vA k \cos(kx) \).

Key concepts

Harmonic Motion

In the realm of wave dynamics, harmonic motion refers to a type of periodic motion where an object oscillates along a path and repeats itself after regular intervals of time. For instance, in a wave traveling along a string, the patterns of crests and troughs are an example of such repeating cycles.
To understand simple harmonic motion, think about how the displacement of a point on a string moving in a sine wave is described by the equation \[ \text{Displacement} = A \sin(\omega t + \phi),\]where:

• \(A\) is the amplitude, indicating the maximum distance from the equilibrium position that the wave reaches,
• \(\omega\) is the angular frequency, dictating how many oscillations occur per unit of time,
• \(\phi\) is the phase angle, determining the initial angle of the wave at time \(t = 0\).

For a string with one end displaced in this manner, simple harmonic motion is described by its sinusoidal structure, reflecting the ongoing back-and-forth movement along the string.

Initial Conditions

Initial conditions in the context of wave equations specify the state of the system at the starting point of observation. For wave problems, it's crucial to know both the initial displacement and the initial velocity, which will help construct a comprehensive picture of the wave's behavior over time.
In our scenario, the string initially has a displacement given by \[ \psi(x,0) = A \sin(kx), \]reflecting the wave's initial shape. Here, \(k\) represents the wave number, associated with the spatial frequency of the wave.
On the other hand, the initial velocity can be derived from the time derivative of the displacement function. It is computed as:\[\left. \frac{\partial \psi}{\partial t} \right|_{t=0} = vA k \cos(kx),\]where \(v\) is the wave velocity. These initial conditions are fundamental as they serve as the starting point for determining how the wave propagates and evolves with time.

Wave Velocity

Wave velocity is a key concept in the study of wave mechanics, representing the speed at which the wave propagates through a medium. It is determined by both the physical properties of the medium and the type of wave in question.
In mathematical terms, wave velocity \(v\) can be expressed as the ratio of angular frequency \(\omega\) to the wave number \(k\), that is:\[v = \frac{\omega}{k},\]which relates how fast the waveform itself travels along the string.
Consider how wave velocity affects dynamics: a higher wave velocity means faster dissemination of wave patterns along the string. It is a crucial factor in problems involving wave motion, ensuring that we understand how alterations in frequency or medium properties will affect the overall movement of the wave. This term is not just a measure of speed but deeply influences how energy is transferred through the wave system.