/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 The wave equation describing the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 19: Problem 3

The wave equation describing the transverse vibrations of a stretched membrane under tension \(T\) and having a uniform surface density \(\rho\) is $$ T\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)=\rho \frac{\partial^{2} u}{\partial t^{2}} $$ Find a separable solution appropriate to a membrane stretched on a frame of length \(a\) and width \(b\), showing that the natural angular frequencies of such a membrane are $$ \omega^{2}=\frac{\pi^{2} T}{\rho}\left(\frac{n^{2}}{a^{2}}+\frac{m^{2}}{b^{2}}\right) $$ where \(n\) and \(m\) are any positive integers.

Short Answer

Expert verified
The natural angular frequencies are \( \omega^2 = \frac{\pi^2 T}{\rho} \left( \frac{n^2}{a^2} + \frac{m^2}{b^2} \right) \).

Step by step solution

01

- Assume a Separable Solution

Assume a separable solution of the form \[ u(x, y, t) = X(x)Y(y)T(t) \]
02

- Substitute into the Wave Equation

Substitute the assumed solution into the wave equation: \[ T \frac{\text{d}^2 (X(x)Y(y)T(t))}{\text{d} x^2} + T \frac{\text{d}^2 (X(x)Y(y)T(t))}{\text{d} y^2} = \rho \frac{\text{d}^2 (X(x)Y(y)T(t))}{\text{d} t^2} \] Simplify to get: \[ XT \frac{\text{d}^2 Y}{\text{d} y^2} + YT \frac{\text{d}^2 X}{\text{d} x^2} = \rho X Y \frac{\text{d}^2 T}{\text{d} t^2} \]
03

- Separate Variables

Divide through by \(XYT\): \[ \frac{1}{X} \frac{\text{d}^2 X}{\text{d} x^2} + \frac{1}{Y} \frac{\text{d}^2 Y}{\text{d} y^2} = \frac{\rho}{T} \frac{1}{T} \frac{\text{d}^2 T}{\text{d} t^2} \] Set each part equal to a separation constant \( -\beta \): \[ \frac{1}{X} \frac{\text{d}^2 X}{\text{d} x^2} = -\beta^2, \;\frac{1}{Y} \frac{\text{d}^2 Y}{\text{d} y^2} = -\beta^2, \; \frac{\rho}{T} \frac{1}{T} \frac{\text{d}^2 T}{\text{d} t^2} = -\beta^2 \]
04

- Solve the Spatial Equations

Solve for \(X(x)\) and \(Y(y)\): \[ \frac{\text{d}^2 X}{\text{d} x^2} + \beta^2 X = 0 \] The general solution: \[ X(x) = A \sin(\beta x) + B \cos(\beta x) \] Apply boundary conditions for a membrane: \[ X(0) = 0 \Rightarrow B = 0, \; X(a) = 0 \Rightarrow A \sin( \beta a) = 0 \Rightarrow \beta a = n\pi \text{ leading to } \beta = \frac{n\pi}{a} \; Y(y) = C \sin(\gamma y) + D \cos(\gamma y) \Rightarrow \gamma = \frac{m\pi}{b} \]
05

- Solve the Time Equation

Solve for \(T(t)\): \[ \frac{\rho}{T} \frac{\text{d}^2 T}{\text{d} t^2} + \beta^2 T = 0 \] leading to: \[ \frac{\text{d}^2 T}{\text{d} t^2} + \frac{T}{\rho} \left( \frac{n^2 \pi^2}{a^2} + \frac{m^2 \pi^2}{b^2} \right) = 0 \] The natural angular frequencies are: \[ \omega^2 = \frac{T\pi^2}{\rho} \left( \frac{n^2}{a^2} + \frac{m^2}{b^2} \right) \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Separable Solution
A separable solution is a method used to simplify partial differential equations. This involves assuming the solution can be written as a product of functions, each depending on a single variable. For the wave equation describing the transverse vibrations of a stretched membrane, we assume the solution in the form:
ewline \( u(x, y, t) = X(x)Y(y)T(t) \).
ewline This assumption allows the complex problem to be broken down into simpler, solvable ordinary differential equations (ODEs) for each component function. By substituting this form into the original wave equation and then separating variables, we can treat and solve each spatial and temporal part individually. This process also makes it easier to apply boundary conditions and find specific solutions.
Boundary Conditions
Boundary conditions are constraints necessary for solving differential equations completely. They are usually defined by the physical setup of the problem. For a membrane stretched on a frame of length \(a\) and width \(b\):
ewline \[ X(0) = 0, \quad X(a) = 0, \quad Y(0) = 0, \quad Y(b) = 0 \].
ewline These conditions imply that the membrane is fixed at its edges. Applying these to the general solutions \(X(x) = A \sin(\beta x) + B \cos(\beta x)\) and \(Y(y) = C \sin(\gamma y) + D \cos(\gamma y)\) simplifies the equations and specifies the allowable values of \(\beta\) and \( \gamma \):
ewline
  • \( \beta a = n\pi\) implying \(\beta = \(n\pi/a\)\) \
  • \( \gamma b = m\pi\) implying \(\gamma = m\pi/b\) \
where \(n\) and \(m\) are positive integers. These boundary conditions help in arriving at unique solutions for the spatial parts of the wave equation.
Natural Angular Frequencies
Natural angular frequencies characterize the inherent vibration modes of the system. For the membrane described by the wave equation, once we solve for the spatial part as \(X(x) = \sin(\frac{n\pi x}{a})\) and \(Y(y) = \sin(\frac{m\pi y}{b})\), we consider the time-dependent part. The time-dependent ODE takes the form:
ewline \[ \frac{\text{d}^2 T}{\text{d} t^2} + \omega^2 T = 0 \].
ewline Solving this leads to expressions that determine the natural angular frequencies \( \omega \):
ewline \[ \omega^2 = \frac{T\pi^2}{\rho}\left( \frac{n^2}{a^2} + \frac{m^2}{b^2} \right) \].
ewline These frequencies are crucial in understanding how the membrane vibrates naturally. The values of \(n\) and \(m\) determine the specific modes of vibration, showcasing how the geometry and tension of the membrane influence its dynamic behavior.
Transverse Vibrations
Transverse vibrations refer to motions where particles of the medium (in this case, the membrane) move perpendicular to the direction of the wave propagation. For the membrane described by the wave equation, the vibrations are governed by the wave equation:
ewline \[ T\left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) = \rho \frac{\partial^2 u}{\partial t^2} \].
ewline Here, \(u(x, y, t)\) represents the displacement of the membrane in the direction perpendicular to its surface. The transverse nature means that when waves travel, for example, along the x-axis, the displacement occurs along the z-axis (out of the plane).
ewline This sort of vibration is critical in analyzing the behavior of stretched membranes, such as drumheads, and can be visually and physically observed as peaks and troughs perpendicular to the surface.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) Find the form of the solution of Laplace's equation in plane polar coordinates \(\rho, \phi\) that takes the value \(+1\) for \(0<\phi<\pi\) and the value \(-1\) for \(-\pi<\phi<0\) when \(\rho=a\) (b) For a point \((x, y)\) on or inside the circle \(x^{2}+y^{2}=a^{2}\), identify the angles \(\alpha\) and \(\beta\) defined by $$ \alpha=\tan ^{-1} \frac{y}{a+x} \quad \text { and } \quad \beta=\tan ^{-1} \frac{y}{a-x} $$ Show that \(u(x, y)=(2 / \pi)(\alpha+\beta)\) is a solution of Laplace's equation that satisfies the boundary conditions given in (a). (c) Deduce a Fourier series expansion for the function $$ \tan ^{-1} \frac{\sin \phi}{1+\cos \phi}+\tan ^{-1} \frac{\sin \phi}{1-\cos \phi} $$.

Schr枚dinger's equation for a non-relativistic particle in a constant potential region can be taken as $$ -\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}\right)=i \hbar \frac{\partial u}{\partial t} $$ (a) Find a solution, separable in the four independent variables, that can be written in the form of a plane wave, $$ \psi(x, y, z, t)=A \exp [i(\mathbf{k} \cdot \mathbf{r}-\omega t)] $$ Using the relationships associated with de Broglie \((\mathbf{p}=\hbar \mathbf{k})\) and Einstein \((E=\hbar \omega)\), show that the separation constants must be such that $$ p_{x}^{2}+p_{y}^{2}+p_{z}^{2}=2 m E $$ (b) Obtain a different separable solution describing a particle confined to a box of side \(a(\psi\) must vanish at the walls of the box). Show that the energy of the particle can only take the quantised values $$ E=\frac{\hbar^{2} \pi^{2}}{2 m a^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right) $$ where \(n_{x}, n_{y}, n_{z}\) are integers.

Consider the PDE \(\mathcal{L} u(\mathbf{r})=\rho(\mathbf{r})\), for which the differential operator \(\mathcal{L}\) is given by $$ \mathcal{L}=\nabla \cdot[p(\mathbf{r}) \nabla]+q(\mathbf{r}) $$ where \(p(\mathbf{r})\) and \(q(\mathbf{r})\) are functions of position. By proving the generalised form of Green's theorem, $$ \int_{V}(\phi \mathcal{L} \psi-\psi \mathcal{L} \phi) d V=\oint_{S} p(\phi \nabla \psi-\psi \nabla \phi) \cdot \hat{\mathbf{n}} d S $$ show that the solution of the PDE is given by $$ u\left(\mathbf{r}_{0}\right)=\int_{V} G\left(\mathbf{r}, \mathbf{r}_{0}\right) \rho(\mathbf{r}) d V(\mathbf{r})+\oint_{S} p(\mathbf{r})\left[u(\mathbf{r}) \frac{\partial G\left(\mathbf{r}, \mathbf{r}_{0}\right)}{\partial n}-G\left(\mathbf{r}, \mathbf{r}_{0}\right) \frac{\partial u(\mathbf{r})}{\partial n}\right] d S(\mathbf{r}) $$ where \(G\left(\mathbf{r}, \mathbf{r}_{0}\right)\) is the Green's function satisfying \(\mathcal{L} G\left(\mathbf{r}, \mathbf{r}_{0}\right)=\delta\left(\mathbf{r}-\mathbf{r}_{0}\right)\).

Electrostatic charge is distributed in a sphere of radius \(R\) centred on the origin. Determine the form of the resultant potential \(\phi(\mathbf{r})\) at distances much greater than \(R\), as follows. (a) express in the form of an integral over all space the solution of $$ \nabla^{2} \phi=-\frac{\rho(\mathbf{r})}{\epsilon_{0}} $$ (b) show that, for \(r \gg r^{\prime}\), $$ \left|\mathbf{r}-\mathbf{r}^{\prime}\right|=r-\frac{\mathbf{r} \cdot \mathbf{r}^{\prime}}{r}+\mathrm{O}\left(\frac{1}{r}\right) $$ (c) use results (a) and (b) to show that \(\phi(\mathbf{r})\) has the form $$ \phi(\mathbf{r})=\frac{M}{r}+\frac{\mathbf{d} \cdot \mathbf{r}}{r^{3}}+\mathrm{O}\left(\frac{1}{r^{3}}\right) $$ Find expressions for \(M\) and \(\mathbf{d}\), and identify them physically.

(a) Show that the gravitational potential due to a uniform disc of radius \(a\) and mass \(M\), centred at the origin, is given for \(ra\) by $$ \frac{G M}{r}\left[1-\frac{1}{4}\left(\frac{a}{r}\right)^{2} P_{2}(\cos \theta)+\frac{1}{8}\left(\frac{a}{r}\right)^{4} P_{4}(\cos \theta)-\cdots\right] $$ where the polar axis is normal to the plane of the disc. (b) Reconcile the presence of a term \(P_{1}(\cos \theta)\), which is odd under \(\theta \rightarrow \pi-\theta\), with the symmetry with respect to the plane of the disc of the physical system. (c) Deduce that the gravitational field near an infinite sheet of matter of constant density \(\rho\) per unit area is \(2 \pi G \rho\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.