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Chapter 2: Problem 2

A distributed force per unit area \(F(x, y, t)\) acts on a flat stretched membrane in a direction normal to its surface. Show how to modify the wave equation (2.1.1) to include the presence of this force density.

Short Answer

Expert verified
Question: Write down the modified wave equation that includes the presence of a distributed force per unit area acting in the direction normal to the surface of a flat stretched membrane. Answer: The modified wave equation is given by: $$ \frac{\partial^2 u}{\partial t^2} = \frac{c^2}{\rho A} \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + F(x, y, t), $$ where \(u(x, y, t)\) denotes the displacement of the membrane, \(c\) denotes the speed of propagation, \(\rho\) is the mass density of the membrane, \(A\) is the area of the membrane, and \(F(x, y, t)\) is the distributed force per unit area acting in the direction normal to the surface of the membrane.

Step by step solution

01

Write Down the General Wave Equation (2.1.1)

The general wave equation for a flat stretched membrane can be represented as: $$ \frac{\partial^2 u}{\partial t^2} = c^2 \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right), $$ where \(u(x, y, t)\) denotes the displacement of the membrane, \(c\) denotes the speed of the propagation, and \(t\) is the time variable.
02

Add the Distributed Force Term in the Equation

Now, we need to modify the wave equation to include the presence of the force density, F(x, y, t). According to Newton's second law, the force acting on the membrane is equal to the mass of the membrane multiplied by its acceleration. The distributed force (force per unit area) can be expressed as: $$ F(x, y, t) = \rho A \frac{\partial^2 u}{\partial t^2}, $$ where \(\rho\) is the mass density of the membrane, and \(A\) is the area of the membrane. Now, dividing both sides of the equation by \(\rho A\), we get: $$ \frac{F(x, y, t)}{\rho A} = \frac{\partial^2 u}{\partial t^2}. $$
03

Modify the Wave Equation to Include the Distributed Force Term

Now, substituting the modified acceleration term into the general wave equation (2.1.1), we get: $$ \frac{F(x, y, t)}{\rho A} = c^2 \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right). $$
04

Rearrange and Simplify the Equation

Rearranging the terms to make the equation more readable, we have: $$ \frac{\partial^2 u}{\partial t^2} = \frac{c^2}{\rho A} \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + F(x, y, t). $$ This equation represents the modified wave equation, including the presence of the distributed force per unit area, F(x, y, t), acting in the direction normal to the surface of a flat stretched membrane.

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Most popular questions from this chapter

A plane wave of the form \(\zeta(x, y, t)=F(y) e^{i\left(x_{x} x-\omega t\right)}\) is traveling in the \(x\) direction on a membrane. Investigate what restrictions the wave equation (2.1.1) puts on the form that the function \(F(y)\) may take.

Explain, in physical terms, why the method of separation of variables applied to the wave equations that we have considered always leads to a sinusoidal time function, though the spatial functions may take a variety of forms.

Prove that the two-dimensional gradient \(\nabla_{2} \zeta(x, y)\) of the surface \(\zeta=\zeta(x, y)\) at any point has a magnitude giving the maximum slope of the surface at that point and a direction giving the direction in which the maximum slope occurs.

From the arguments leading to the wave equation (2.1.1), show that in static equilibrium the displacement \(\zeta(x, y)\) of an elastic membrane obeys the two-dimensional Laplace equation $$ \nabla_{2}^{2} \zeta=\frac{\partial^{2} \zeta}{\partial x^{2}}+\frac{\partial^{2} \zeta}{\partial y^{2}}=0 $$ [Thus, given a uniformly stretched membrane, one may place blocks or clamps so as to fix some desired displacement along a closed boundary contour of arbitrary geometry. The apparatus is then an analog computer for solving Laplace's equation (in two dimensions) for geometries too complicated to handle analytically. This scheme is particularly useful in electrostatic problems, where the displacement \(\zeta\) represents the electrostatic potential \(V .\) Moreover, the trajectories of electrons in vacuum tubes can be plotted by rolling small ball bearings on such a model made from a rubber membrane.]

Find the form of the wave equation (2.1.1) with respect to new axes \(x^{\prime}\) and \(y^{\prime}\), which are rotated an angle \(\theta\) with respect to the \(x y\) axes. Hint: \(x^{\prime}=x \cos \theta+y \sin \theta, y^{\prime}=-x \sin \theta+\) \(y \cos \theta\).
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