Jean-Baptiste le Rond d'Alembert (; French: [ʒɑ̃ batist lə ʁɔ̃ dalɑ̃bɛːʁ]; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the Encyclopédie. D'Alembert's formula for obtaining solutions to the wave equation is named after him. The wave equation is sometimes referred to as d'Alembert's equation, and the fundamental theorem of algebra is named after d'Alembert in French.
I am using Nivaldo Lemos' "Analytical Mechanics" textbook and on section 2.4 (Hamilton's Principle in the Non-Holonomic Case) he uses Hamilton's Principle and Lagrange Multipliers to arrive at the Lagrange Equations for the non-holonomic case.
I don't understand why it is assumed that the...
Q. Calculate the linearised metric of a spherically symmetric body ##\epsilon M## at the origin. The energy momentum tensor is ##T_{ab} = \epsilon M \delta(\mathbf{r}) u_a u_b##. In the harmonic (de Donder) gauge ##\square \bar{h}_{ab} = -16\pi G \epsilon^{-1} T_{ab}## (proved in previous...
Hi,
I was trying to get some practice with the wave equation and am struggling to solve the problem below. I am unsure of how to proceed in this situation.
My attempt:
So we are told that the string is held at rest, so we only need to think about the displacement conditions for the wave...
The solution for the wave equation with initial conditions $$u(x,0)=f(x)$$ and $$u_t(x,0)=g(x)$$
Is given for example on wikipedia : $$u(x,t)=(f(x+ct)+f(x-ct)+1/c*\int_{x-ct}^{x+ct}g(s)ds)/2$$
So a vibrating string, since there is no conditions on ##g## (like ##\sqrt{1-g(x)^2/c^2}##), could...
Well what I did was first use the inverse Fourier transform:
$$u(x,t)=\frac{1}{2\pi }\int_{-\infty }^{\infty }\tilde{u}(\xi ,t)e^{-i\xi x}d\xi$$
I substitute the equation that was given to me by obtaining:$$u(x,t)=\frac{1}{2\pi }\left \{ \int_{-\infty }^{\infty}\tilde{f}(\xi)cos(c\xi...
How were the values of the the regions found in the grid of this solution? I understand that the value should be 0 in every regions that contains the points x = 0, x=4, etc...
I believe the bottom values can be found from the boundary conditions as well, but what about the others?
Was not sure weather to post, this here or in differential geometry, but is related to a GR course, so...
I am having some trouble reproducing a result, I think it is mainly down to being very new to tensor notation and operations.
But, given the metric ##ds^2 = -dudv + \frac{(v-u)^2}{4}...
Hello all,
I understand there are four d'Alembert (fictitious) (non-inertial) forces:
1. Coriolis
2. Centrifugal
3. Linear
4. Angular acceleration.
But then I think about the Gyroscopic Effect (I understand how it arises, so that is not the issue).
I am wondering if one can "classify" these...
I have a few questions about the wave equation and the D'Alambert solution:
0) First of all, I'm a bit confused with the terminology. Wikipedia says that THE wave equation is a PDE of the form: ##\frac{\partial^2 u}{ \partial t^2 } = c^2 \nabla^2 u##, however there are other PDEs that have...
In my textbook there is an explanation of a derivation of D'Alembert equation for pressure waves. (##\frac{\partial^2 y}{\partial x^2}=\frac{\rho}{\beta}\frac{\partial^2 y}{\partial t^2}##)
I put the picture (the only one I found on internet) but I'll call ##y_1 ,y_2## as ##\psi_1,\psi_2## and...
In this video, a man applies an angular acceleration to the base of a rod. While the rod rotates, it bends. Why? What force is there that causes the bending, aside from rod's own weight? It seems to me to be the work of a fictitious inertial force. I was always taught that those forces don't...
If $$\phi(t,x)$$ is a solution to the one dimensional wave equation and if the initial conditions $$\phi(0,x) , \phi_t(0,x)$$ are given, D'Alembert's Formula gives
$$\phi(t,x)= \frac 12[ \phi(0,x-ct)+ \phi(0,x+ct) ]+ \frac1{2c} \int_{x-ct}^{x+ct} \phi_t(0,y)dy . \tag{1}$$
which is...
Homework Statement
Mechanical crane raises 225kg at a rate of 0.031m/s^2. from a rest to a speed of 0.5m/s over a distance of 4m. Frictional resistance is 112N.
m1=225kg
a1=0.031m/s^2
u1=0m/s
v1=0.5m/s
s=4m
Fr=112N
A. Work input from the motor
B. Tension in the lifting cable
C. Max power...
When considering the Wave equation subject to initial conditions as follows…
…then D'Alembert's solution is given by (where c is wave speed):
I'd like to understand physically how this formula allows us to know the value of u (where u is the height of the wave, say) at some point (x0,t0)...
The D'Alembert equation for the mechanical waves was written in 1750. It is not invariant under a Galilean transformation.
Why nobody was shocked about this at the time? Why we had to wait more than a hundred years (Maxwell's equations) to discover that Galilean transformations are wrong...
I can't find a derivation of d'Alembert principle. Wikipédia says there is no general proof of it. Same with stackexchange. I find it surprising so I thought I'd come here to check with you guys. D'Alembert principle has indeed no proof ?
Hi, I'm in the masters year of a theoretical physics course which begins this September. I'm reading the classical mechanics notes ahead of time, and I came across the idea of holonomic and non-holonomic constraints. I understand that in the case of a holonomic system, you can use the...
1. A car accelerates 1800m down an incline of 1 in 4 at 0.4ms^2. The car has a mass of 4,000kg and the resistance to motion is 400N
Determine:
a) The Tractive effort required by using D'Alemberts principle
b) The Tractive effort required by using the conservation of energyHomework Equations...
Homework Statement
i have a question on D'alemberts principle in which it asks me to find the velocity of a hammer immediately before impact with a pile
the information i have been given is as follows:
mass of hammer;300kg
height of hammer;3.5m
gravity to be taken as;9.81
mass of pile ;500kg...
Homework Statement
Graph snapshots of the solution in the x-u plane for various times t if
\begin{align*}
f(x) =
\begin{cases}
& 3, \text{if } -4 \leq x \leq 0 \\
& 2, \text{if } 4 \leq x \leq 8 \\
& 0, \text{otherwise}
\end{cases}
\end{align*}
Homework Equations
Assuming that c=1 and g(x)...
Homework Statement
Hi all, I'm currently reviewing for a final and would like some help understanding a certain part of this particular problem: Determine the retarded Green's Function for the D'Alembertian operator ##D = \partial_s^2 - \Delta##, where ##\Delta \equiv \nabla \cdot \nabla## ...
hello everybody,
I have a question regarding the fictitious force (d'alembert force) we usually add to an examined body in a noninertial reference system. As I understood from reading and leraning about this topic, this force is artificially added only to compensate for exploring this body in...
Can someone please help me in plotting a D'Alembert wave equation solution in MATLAB? I am so confused as how to plot it in MATLAB
I need to plot a graph like the one below
Homework Statement
Find the solution of the wave equation using d'Alembert solution.
Homework Equations
u(0,t)=0[/B] and u(x,0)=0
u_t(x,0)=\frac{x^2}{1+x^3}, \, x\geq0
u_t(x,0)=0, \, x<0
The Attempt at a Solution
For a semi infinite string we have the solution
u(x,t)=\frac{1}{2}\left(...
Hey guys,
The expression \partial_{\mu}\partial^{\nu}\phi is equal to \Box \phi when \mu = \nu. However when they are not equal, is this operator 0?
I'm just curious cos this sort of thing has turned up in a calculation of mine...if its 0 I'd be a very happy boy
I understand why this is a good method, but in one of the problems I am trying I yield 4 unknown parameters in a second order differential equation. I believe I should only have 2.
Let f(x) be a part of a homogeneous solution and and u(x) be some unknown function in x. Then a particular...
Hi there,
This is a problem concerning hyperbolic type partial differential equations. Currently I am studying the book of S. J. Farlow "Partial differential equations for scientists and engineers". The attached pages show my problems. Fig. 18.4 from case two (which starts in the lower part...
Homework Statement
I have a general wave equation on the half line
utt-c2uxx=0
u(x,0)=α(x)
ut(x,0)=β(x)
and the boundary condition;
ut(0,t)=cηux
where α is α extended as an odd function to the real line (and same for β)
I have to find the d'alembert solution for x>=0; and show that in...
I need someone helps me to solve this problem using method of virtual works by d'alembert.
The datas are:
m1=2kg;m3=4kg;m=0,5kg;M=1.5kg;R=0.6;r=0,2m;\alpha=30°.
One of the requests is to calculate the acceleration of the mass m1, the solution is: 3.90m/s^2.
The wheel is formed by two coassial...
[b]1.
For the 1-D wave equation, the d’Alembert solution is u(t, x) = f (x + ct) + g(x − ct) where f , g are each a function of 1 variable.
Suppose c = 1 and we know f (x) = x^2 and g(x) = cos 2x for x > 0.
Find u(t, x) for al l t, x ≥ 0 if you are also given the BC: u ≡ 1 at x = 0...
In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein arrives at the expression (equation 1.46) \mathbf{v}_i = \frac{d\mathbf{r}_i}{dt} = \sum_k \frac{\partial \mathbf{r}_i}{\partial q_k} \dot{q}_k + \frac{\partial \mathbf{r}_i}{\partial t}
where \mathbf{r}_i...
Homework Statement
The question is here:
http://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/assignments/probwave1solns.pdf
It's a long question and I figured attaching the link here would be better.
I need help with the question on page 4.
when...
Hi I have been trying to figure out how this method works.
Searched the forum but got no results. Have attached an example I am trying to do. If someone could show me it would be great.
Or even show me a similar example worked out.
Thanks in advance
One dimensional wave equation:
\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}
Where c is the vertical velocity of the vibrating string.
This will give D'Alembert solution of u(x,t) = \frac{1}{2}[f(x+ct) + G(x+ct)] + \frac{1}{2}[f(x+ct) + G(x+ct)]
Where...
Hi this is just a general question about using the ratio test for convergence.
If I have to carry out the test to find out if something converges (and I don't need to find out if its absolutely converges, but just convergence), then can my answer to the test be negative?
Or does the...
Homework Statement
Ok so hope someone will be able to help...
I've used the D'Alembert method to solve the wave equation and have got that the general form should be
y(x,t) = f(x+ct) + g(x-ct)
Now I am also told that the time dependence at x=0 is sinusoidal..
that is, y(x,0) =...
Homework Statement
This is an example I copy from the book. The book showed the steps of solving and provide the answer. I don't understand the book at all. Below I show the question and the solution from the book. Then I am going to ask my question at the bottom.
Question
Use...
In am studying PDE and I have question about D'Alembert solution for one dimension wave equation.
I am going to reference Wolfram:
http://mathworld.wolfram.com/dAlembertsSolution.html
1) I want to verify the step of \frac{\partial y_0}{\partial t} of step (14) of the page...
D'Alembert problem for semi infinite string on R-
utt=c2 uxx ( -\infty<x<0)
Initial condition:
u(x,0)=f(x)
ut(x,0)=g(x)
Boundary condition:
u(0,t)=0
please help me to solve it
Well,most of You know this principle,
I just know 1 part of his work, which is that there is a force in an accelerated frame on a mass that equals MA and is in the opposite direction.
so I have a question, is the force(fictitious or not) used in the accelerated frame ,conservative or not...
1. Is there anywhere where I can find a derivation for the generalized inhomogeneous wave equation? I found a derivation of the 1D wave equation for an infinitesimal region of an elastic string here:
http://www.math.ubc.ca/~feldman/apps/wave.pdf
But I am looking for proof of the...
Homework Statement
I am looking at the derivation of the D'alembert equation, and I'm having trouble with understanding where the limits of integration come in.
Homework Equations
Given the 1-d wave equation:
u_{tt} = c^2u_{xx} , with the general solution u(x,t)= \theta(x-ct) +...