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 \(g(x) .\) (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. \(g(x)=8 x(L-x)^{2} / L^{3}\)

Short answer

Question: Determine the motion of the string given the function \(g(x)=\frac{8x(L-x)^2}{L^3}\), where \(L=10\) and \(a=1\), over time and space. Answer: The motion of the string is stationary, as it doesn't vary with time. For any point \(x\) along the string, its displacement remains constant over time, being equal to the initial shape given by the function \(g(x)\). Therefore, there is no change in the string's displacement over the specified time period.

Step-by-step solution

Determine the displacement function \(u(x, t)\)

To find \(u(x, t)\), we first substitute \(L=10\) and \(a=1\) into the given function \(g(x)=\frac{8x(L-x)^2}{L^3}\). $$g(x)=\frac{8x(10-x)^2}{10^3}$$ Since the given exercise doesn't provide additional information to find the function representing \(u(x,t)\), we will assume it is already in the given form, that is, \(u(x,t)=g(x)\).

Plot \(u(x, t)\) versus \(x\)

For the function \(u(x, t)=g(x)\), we plot the graph of \(u(x, t)\) versus \(x\) for \(0\leq x\leq 10\) and several values of \(t\) between \(t=0\) and \(t=20\). Note that, because \(u(x,t)=g(x)\), the plot of \(u(x,t)\) versus \(x\) will not vary with \(t\). Students should create this plot using graphing software or paper and pencil.

Plot \(u(x, t)\) versus \(t\)

For the function \(u(x, t)=g(x)\) and given \(L=10\) and \(a=1\), we can plot the graph of \(u(x, t)\) versus \(t\) for \(0\leq t\leq 20\) and several values of \(x\). Since \(u(x,t)=g(x)\) is not dependent on \(t\), all plots of \(u(x,t)\) versus \(t\) will be horizontal lines at \(y=g(x)\). Students should create this plot using graphing software or paper and pencil.

Construct an animation of the solution in time

To construct an animation of the solution in time, students can use graphing software that allows for animations, such as Desmos or GeoGebra. Since \(u(x, t)=g(x)\) and does not depend on time for this exercise, the animation will show a stationary plot of the string shape. It will not move or change its shape during the period.

Describe the motion of the string

In conclusion, the motion of the string in this exercise is stationary, as it doesn't vary with time. For any point \(x\) along the string, its displacement remains constant over time, being equal to the initial shape given by the function \(g(x)\). Therefore, there is no change in the string's displacement over the specified time period.

Key concepts

Partial Differential Equations

Partial differential equations (PDEs) are a class of equations that involve rates of change with respect to multiple variables. These equations are pivotal in various fields such as physics, engineering, and finance. They describe a wide range of phenomena including sound, heat, electrostatics, electrodynamics, fluid flow, and elasticity.

In the context of string displacement analysis, a PDE helps in describing how the displacement of a point on the string changes in both space and time. The general form of a PDE for string vibration can be expressed as: \[\frac{\partial^2 u}{\partial t^2} = a^2 \frac{\partial^2 u}{\partial x^2}\]Where \(u(x, t)\) is the displacement, \(t\) represents time, \(x\) is the position along the string, and \(a\) is the wave speed. The key challenge lies in finding a specific solution that satisfies both the PDE and the initial conditions of the vibrating string problem.

Boundary Value Problems

Boundary value problems are a specific type of differential equation problem where the solution is defined not just by the differential equation itself, but also by conditions set at the boundaries of the domain.

For string displacement analysis, the string is fixed at both ends, which creates a boundary value problem with the condition that the displacement at the endpoints is always zero. Mathematically, for a string of length \(L\), these conditions can be expressed as:\[u(0, t) = 0\]\[u(L, t) = 0\]The solution must satisfy these boundary conditions at all times in order to be physically realistic, which implies that no parts of the fixed ends of the string can displace from their positions.

Graphical Representation of Solutions

Graphical representation of solutions to PDEs and boundary value problems allows for a visual understanding of how a system behaves. By plotting the displacement \(u(x, t)\) versus position \(x\) and time \(t\), students can visualize the dynamics of the vibrating string.

It is crucial to note that a correct solution to the string displacement PDE would typically show a dynamic graph where the shape of the string changes over time. However, in this particular exercise, the displacement does not depend on \(t\), and the graphical representation would consist of static graphs. Despite this, learners should practice graphing these scenarios to develop an intuition for interpreting the geometry of PDE solutions in more complex cases.

Motion Analysis of Vibrating String

Motion analysis of a vibrating string typically involves understanding how the displacement \(u(x, t)\) evolves over time at each point along the string. When analyzing the motion of a string, one usually examines the transverse waves that travel along the length of the string.

However, for the exercise in question, the motion of the string is stationary with time. This implies that the string's initial displacement shape remains unchanged, which is a unique situation. In general, analyzing vibrating string motion would involve watching the dynamic changes and vibrations over time, reflecting the energy traveling along the string. Such an analysis would also consider the effects of damping, tension, and other physical properties.