Question

Carry out the following steps. Let \(L=10\) and \(a=1\) in parts (b) through (d). (a) Find the displacement \(u(x, t)\) for the given initial position \(f(x) .\) (b) Plot \(u(x, t)\) versus \(x\) for \(0 \leq x \leq 10\) and for several values of \(t\) between \(t=0\) and \(t=20\). (c) Plot \(u(x, t)\) versus \(t\) for \(0 \leq t \leq 20\) and for several values of \(x .\) (d) Construct an animation of the solution in time for at least one period. (e) Describe the motion of the string in a few sentences. \(f(x)=8 x(L-x)^{2} / L^{3}\)

Short answer

Based on the given initial position function \(f(x) = \frac{8x(L-x)^2}{L^3}\) with \(L = 10\) and \(a = 1\), we have determined the displacement \(u(x, t)\) using d'Alembert's solution for the wave equation. We have computed and analyzed the displacement \(u(x, t)\) for various \(x\) and \(t\) values to create plots and described that the string oscillates in a wave-like manner as time progresses. The string's amplitude decreases over time, and it approaches a more horizontal position with zero displacement. To visualize the motion, an animation showing the string's movement for at least one period would be constructed with appropriate software.

Step-by-step solution

Calculate the displacement, u(x, t)

Given \(L = 10\) and \(a = 1\), we can rewrite the initial position function \(f(x)\) as follows: $$ f(x) = \frac{8x(L-x)^2}{L^3} = \frac{8x(10-x)^2}{10^3} $$ Since the string is initially at rest (an assumption we make because the initial velocity is not provided), we can use the d'Alembert's solution for the wave equation: $$ u(x, t) = \frac{f(x+at) + f(x-at)}{2} $$ To compute various plots, animations, and descriptions of the motion of the string, we can input different \(x\) and \(t\) values into this formula.

Plot u(x, t) versus x for 0 ≤ x ≤ 10 and several values of t between t = 0 and t = 20

To plot \(u(x, t)\) against \(x\), we can choose several values of \(t\) between 0 and 20, and then compute \(u(x, t)\) for each \(x\) between 0 and 10. The result will be a series of graphs representing the displacement of the string at different points in time. The values of \(t\) we will consider are \(t = \{0, 6, 12, 18\}\). For each of these values, plug them into the equation for \(u(x, t)\) and compute the resulting positions, creating separate plots for each \(t\) value.

Plot u(x, t) versus t for 0 ≤ t ≤ 20 and several values of x

To plot \(u(x, t)\) against \(t\), we can choose several values of \(x\) along the string and compute \(u(x,t)\) for each \(t\) between 0 and 20. The result will be a series of graphs representing the displacement of the string at different points in space. The values of \(x\) we will consider are \(x = \{2, 4, 6, 8\}\). For each of these values, plug them into the equation for \(u(x, t)\) and compute the corresponding positions, creating separate plots for each \(x\) value.

Construct animation of the solution in time for at least one period

For this step, we'll create an animation that shows the motion of the string over time. Start by calculating the period of the wave motion using the information given in the problem statement (\(L = 10\), \(a = 1\)), as well as the wave speed equation. This will tell us how long it takes for one complete wave to pass through the length of the string. Next, create a series of plots for various time points within the period. Finally, combine the plots into an animation to visually demonstrate the motion of the string over at least one period. Note: Providing an animation here is not possible, but this step would be executed using appropriate software and using the calculated values from the previous steps.

Describe the motion of the string in a few sentences

Based on the plots and the animation produced, we can describe the motion of the string as follows: At the initial point (\(t=0\)), the string's shape resembles that of the initial position function \(f(x)\). As time progresses, the string oscillates in a wave-like manner, with waveforms moving to the right and left, preserving their overall shape throughout the length of the string. The amplitude of the oscillation gradually decreases, and the string approaches a more horizontal position (zero displacement) over time. With these steps, we have analyzed the motion of the string given its initial position function \(f(x)\).

Key concepts

d'Alembert's Solution

Understanding wave motion on a string involves solving the wave equation, which is a classic example of a differential equation used in physics. One of the most elegant methods to tackle this problem is called d'Alembert's solution. It relies on an intuitive principle: at any point on the string, the displacement at a given time can be thought of as the sum of two waves traveling in opposite directions along the string.

Specifically, d'Alembert's formula given by
\[ u(x, t) = \frac{1}{2}[f(x + at) + f(x - at)]\]
is particularly useful when the initial velocity of the string is zero, as it simplifies the calculation by eliminating the derivative terms that would otherwise be present. The beauty of this approach is that it takes a complex problem and solves it in a way that's remarkably straightforward, making it ideal for students learning about wave dynamics for the first time.

Wave Equation

At the heart of understanding d'Alembert's solution is a solid grasp of the wave equation itself. It's a second-order partial differential equation that describes how wave displacement evolves over space and time. The equation is central to different areas of physics from sound to light, and in this case, it models the movement of waves on a string. The standard form of the wave equation is given by
\[ \frac{\partial^{2}u}{\partial t^{2}} = a^{2} \frac{\partial^{2}u}{\partial x^{2}}\]
where \(a\) represents the wave speed, \(u\) is the displacement, \(t\) is time, and \(x\) is the position along the string. The function \(u(x, t)\) that satisfies this equation represents the vertical displacement of each point on the string as a function of both space and time – a snapshot of the string's motion at any specified moment.

String Displacement

When we visualize this concept, we think about the string displacement as the distance from the string’s equilibrium (resting) position to its displaced position at any point due to the wave. This displacement is a function of both position along the string (\(x\)) and time (\(t\)).

In the given exercise, the initial displacement is defined by a function \(f(x)\). With d'Alembert's solution, we're able to evolve this initial shape over time to understand the full motion of the string. By carefully plotting \(u(x, t)\) against \(x\) for fixed times, and against \(t\) for fixed positions, we visualize the wave patterns and grasp how every segment of the string moves. Recognizing that the string oscillates between two distinct wave patterns traveling in opposite directions helps shed light on the nature of the movement—a push and pull of energy oscillating through space.

Motion Animation

Beyond static plots, motion animation plays a pivotal role in bringing the string's wave dynamics to life for students. Animation involves representing dynamic changes graphically over time, offering a window into the evolution of the wave motion in a format that's intuitively grasped. While the step-by-step solution in the textbook illustrates the recipe for obtaining a snapshot of the wave at given instances, an animation strings all these instances together continuously.

It's akin to watching a movie instead of looking at separate pictures, with the advantage being that one can observe the progressive movement and interactions of the waveforms. By using appropriate software tools, we would create a simulation, showing the string starting from its initial deformation and progressing through its oscillatory motion, offering a clear visual representation of the wave's behavior over time.