Question

An incompressible fluid of density \(\rho\) and negligible viscosity flows with velocity \(v\) along a thin straight tube, perfectly light and flexible, of cross-section \(A\) and held under tension \(T\). Assume that small transverse displacements \(u\) of the tube are governed by $$ \frac{\partial^{2} u}{\partial t^{2}}+2 v \frac{\partial^{2} u}{\partial x \partial t}+\left(v^{2}-\frac{T}{\rho A}\right) \frac{\partial^{2} u}{\partial x^{2}}=0 $$ (a) Show that the general solution consists of a superposition of two waveforms travelling with different speeds. (b) The tube initially has a small transverse displacement \(u=a \cos k x\) and is suddenly released from rest. Find its subsequent motion.

Short answer

Waveforms travel with speeds \(c_1 = v + \sqrt{\frac{T}{\rho A}}\) and \(c_2 = v - \sqrt{\frac{T}{\rho A}}\). The motion is given by \( u(x,t) = \frac{a}{2} [ \cos(k(x - c_1 t)) + \cos(k(x - c_2 t))].\)

Step-by-step solution

- Identify the Wave Equation

Recognize that the given equation is a second-order partial differential equation:\[\frac{\partial^{2} u}{\partial t^{2}}+2 v \frac{\partial^{2} u}{\partial x \partial t}+\left(v^{2}-\frac{T}{\rho A}\right) \frac{\partial^{2} u}{\partial x^{2}}=0.\]

- Seek a Solution of the Form

Consider a solution of the form \(u = f(x - c_1 t) + g(x - c_2 t)\), where \(c_1\) and \(c_2\) are wave speeds. We aim to show these represent waveforms traveling with different speeds.

- Derive Characteristic Equation

Substitute \(u = e^{i(kx - \omega t)}\) into the given partial differential equation and simplify to derive the characteristic equation:

- Solve for Wave Speeds

Solve the characteristic equation for \(\omega\):\(\omega = vk \pm k \sqrt{\frac{T}{\rho A}.\)Thus, the wave speeds are \(c_1 = v + \sqrt{\frac{T}{\rho A}}\) and \(c_2 = v - \sqrt{\frac{T}{\rho A}}\).

- Solution in Terms of Superposition

Conclude that the general solution can be written as a superposition of two traveling waves:\( u = f(x - c_1 t) + g(x - c_2 t).\)

- Apply Initial Condition

Use the initial condition \(u = a \cos k x\) to find the specific forms of \(f\) and \(g\):Since the tube is initially at rest, \(\frac{\partial u}{\partial t} = 0\) when \(t = 0\).

- Use Initial Velocity Condition

Since \(\frac{\partial u}{\partial t} = 0\) at \(t = 0\), differentiate \(u = f(x - c_1 t) + g(x - c_2 t)\) with respect to \(t\), set \(t = 0\), and combine the resulting equations to solve for \(f\) and \(g\).

- Derive Explicit Form

Combine the results from the previous step to explicitly write the function in terms of \(x\) and \(t\) after substituting \(f\) and \(g\) back in:

- Final Solution

The subsequent motion is:\( u(x,t) = \frac{a}{2} [ \cos(k(x - c_1 t)) + \cos(k(x - c_2 t))].\)

Key concepts

partial differential equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. They are fundamental in describing various physical phenomena such as heat, sound, and fluid dynamics. In the given exercise, we deal with a second-order PDE:\[\frac{\partial^{2} u}{\partial t^{2}}+2 v \frac{\partial^{2} u}{\partial x \partial t}+\left(v^{2}-\frac{T}{\rho A}\right) \frac{\partial^{2} u}{\partial x^{2}}=0.\]This equation describes the transverse displacements (\

waveforms

Waveforms represent the shape and form of signals traveling through a medium, which can be described mathematically or visually. In our exercise, we focus on the transverse displacement waveform moving along a flexible tube.
The general solution for our PDE is a combination of two traveling waveforms:\[u = f(x - c_1 t) + g(x - c_2 t),\]where the functions f and g can be any shape that represents the wave, and \(c_1\) and \(c_2\) are the wave speeds.
The waveform's shape is determined by boundary and initial conditions. For example, if the initial displacement is given as \(u = a \cos(kx)\), the resulting motion would produce a specific waveform for the system. The waveform evolution over time comes from synthesizing these different speeds and initial shapes.

boundary conditions

Boundary conditions are constraints necessary for solving PDEs to ensure the solutions are physically meaningful. They describe how the wave behaves at the edges of the domain we're considering.

• Dirichlet Boundary Condition: Specifies the value of a function on a boundary.
• Neumann Boundary Condition: Specifies the value of the derivative of a function on a boundary.
• Mixed Boundary Condition: Combination of both Dirichlet and Neumann.

In our problem, the initial displacement provided, \(u = a \cos(k x)\), acts as the initial boundary condition. This setup allows us to solve the PDE, by applying these conditions to synthesize the final waveforms.

superposition of waves

Superposition of waves is a key concept wherein two or more waves overlap, resulting in a combined waveform. This concept is vital in understanding how the waves described by our PDE interact.In our equation:\[u = f(x - c_1 t) + g(x - c_2 t),\] we see a direct example of this. Here, the overall displacement is a sum of two individual waveforms traveling at different speeds \(c_1\) and \(c_2\). Each of these speeds is derived from the PDE characteristics, representing different velocities at which the disturbances propagate along the tube.The principle of superposition enables complex wave patterns to be constructed simply by adding simpler waves together, letting us see how initial condition reshapes into the composite waveform observed.