What is Wave equation: Definition and 594 Discussions
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Due to the fact that the second order wave equation describes the superposition of an incoming and outgoing wave (i.e. rather a standing wave field) it is also called "Two-way wave equation" (in contrast, the 1st order One-way wave equation describes a single wave with predefined wave propagation direction and is much easier to solve due to the 1st order derivatives).
Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.
Homework Statement
A stretched string occupies the semi-infinite interval -\infty<x\leq0.
y(x,t) := f(x-ct) + f(-x-ct) is a solution of the wave equation.
What boundary condition does y satisfy at x=0?
Describe what is going on in terms of incident and reflected waves.
Homework...
In relativity, the scalar wave equation in the coordinate system (x,y,z,ict)
is
\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}+\frac{\partial^2\phi}{\partial (ict)^2}=0
In 3D classical mechanics, the Laplace equation is:{when...
Homework Statement
A string is stretched to a tension T, its ends x=0 and x=L are attached to rings with mass M which are able to slide on parallel smooth wires perpendicular to the string. Show:
1) The transverse displacement must satisfy Mytt = Tyx at x=0, Mytt = -Tyx at x=0
2) The...
Homework Statement
Show that the wave equation becomes
\left(1-\frac{V^{2}}{c^{2}}\right)\frac{\partial^{2}\psi'}{\partial x'^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}\psi'}{\partial t'^{2}}+\frac{2V}{c^{2}}\frac{\partial^{2}\psi'}{\partial t' \partial x'} = 0
under a Galilean transform if the...
As a computer scientist and applied math guy, I've recently taken interest in learning to simulate fluids. I've done simple wave equation and KDV simulations before, but after reading the paper
"Rapid, Stable Fluid Dynamics for Computer Graphics" by Kass and Miller 1990.
I thought I'd like...
I'm not understanding something here. Maxwell's wave equation is:
Laplacian of E = (1/c^2) * second partial of E
(sorry, I don't know how to write symbols)
But the second partial derivative is the Laplacian. So how can you scale the laplacian of E by a number and get the laplacian of E as...
1. Solve the wave equation u_(tt) = 4u_(xx) on the interval [0, π] subject to the
conditions
u(x, 0) = cos x, u_t(x, 0) = 1, u(0, t) = 0 = u(π, t).
Homework Equations
3. Hello. This appears to be a common separation of variables question. Only problem is after using...
Homework Statement
According to the website, the statement is as follows:
Write a program which will calculate the evolution of the Wave Equation,
in the case of a bound string. Test this program on the base eigenfunctions,
i.e. the sinusoids, and on more interesting combinations. You...
Homework Statement
Solve, u_{t} = u_{xx}c^{2}
given the following boundary and initial conditions
u_{x}(0,t) = 0, u(L,t) = 0
u(x,0) = f(x) , u_{t}(x,0) = g(x)Homework Equations
u(x,t) = F(x)G(t)
The Attempt at a Solution
I solved it, I am just not sure if it is right.
u(x,t) =...
Homework Statement
I am trying to calculate the angular momenta for
\psi(x,y,z) = A(ar^2 + bz^2)
A is given as a constant.
Homework Equations
The Attempt at a Solution
I know that z=r\sqrt{4\pi/3} * Y_0^1
What I have so far is:-
\psi(x,y,z) = r^2Aa +...
Hello,
I want to use Galerkin method to solve 3-D wave equation \nabla^2 u+k^2 u=0, with the following boundary conditions: at z=z_1 plane, u=g, and when x,y,z go to the infinity, u becomes 0.
My question is how to choose the basis function \phi_n for u: u=\sum \lambda_n \phi_n. As my...
Hi,
I have this electrodynamics problem sheet on the paraxial approximation, and I am not getting very far with it. It starts off talking about a laser beam traveling in the z-direction, and says that a scalar wave has the form F(r,w)eiwt.
The first part of the question ends with me proving...
a wave equation is given as A = A cos (kx - ωt)
so why if someone describes the wave equation to be A = A sin (ωt - kx) , the argument of the sin function changes by a minus sign?
and is there a meaning to it?
also i still don't really understand why the minus sign in the first equation...
I need help solving 3Utt+10Uxt+3Uxx=sin(x+t)
I have found the homogeneous part, which is U(x,t)=f(3x-t) +g(x-3t), but I don't know where to go from there. Any help would be much appreciated!
I am unable to determine the relationship between x and t in the following equation.
y\left(x,t\right)=A\sin\left( kx-\omega t \right)\\
If \nu=\frac{x}{{t}} then the numbers within the bracket goes to zero; because kx=\omega t
for all points on y(x,t).
Can anyone...
Homework Statement
a) Assuming the presence of sources (J flux density) and (p charge density) , write out Maxwell’s equations in the time domain in terms of and only for a lossless, but inhomogenous medium in which
ε = ε(r) , μ = μ(r).
b) Derive the vector differential...
Homework Statement
\Psi(x) = \frac{C}{a^2 + x^2}
Homework Equations
I know to do this I need to solve for:
\int_{-\infty}^{\infty} \left|\Psi(x)\right|^2 = 1
The Attempt at a Solution
I'm not sure how to do it for this function. I've tried various methods to solve
C^2...
Homework Statement
Show that Y(x,t) = cos(kx)exp(-iwt) is a solution to the time-dependent Schrodinger wave equation.
where k is the wavenumber and w is the angular frequency
Homework Equations
Hamiltonian of Y(x,t) = ihbar d/dt Y(x,t)
The Attempt at a Solution
When I plug...
Problem:
Applied Partial Differential Equations (Richard Heberman) 4ed.
#12.3.6
Consider the three dimensional wave equation
\partial^{2}u/\partial t^2 = c^2\nabla^2 u
Assume the solution is spherically symetric, so that
\nabla^2 u =...
[b]1. Homework Statement [/b
Determine whether the function D=A sin kx cos \omegat
is a solution of the wave equation.
Homework Equations
D=Asin (kx-\omegat)
The Attempt at a Solution
sorry completely lost please help
This comes from Jackson's Classical Electrodynamics 3rd edition, page 613. He finds the Green's function for the covariant form of the wave equation as:
D(z) = -1/(2\pi)^{4}\int d^{4}k\: \frac{e^{-ik\cdot z}}{k\cdot k}
Where z = x - x' the 4 vector difference, k\cdot z = k_0z_0 -...
The Shroedinger equation defines the time evolution of the wave function. If we observe a region of large gravitational fields where observed time has slowed, the wave function will be observed to evolve slowly. In the limit of a Black Hole it will stop evolving altogether.
Still quantum...
Hello, I have a question about the following problem:
Given a wave equation \Psi(n,t) where t is the time, and n is an integer. What is the Fourier transform?
I'm trying to reproduce this paper: One-dimensional Quantum Walks by Ambainis et al...
I was interrested in the general solutions to the wave equation depending on only one spatial coordinate.
For one linear coordinate, the general solution is:
a f(x-ct) + b g(x+ct)
For one radial spherical coordinate, the general solution is:
a f(r-ct)/r + b g(r+ct)/r
I thought that...
Hi there,
i would like everyone to evaluate my working here,
this is my attempt to wave equation / vibrating system using vector approach.
please correct me if i had made some mistakes.
Your help is much appreciated,
Thanks, and have a nice day. :smile:
Regards,
Daniel.
1D wave equation -- bizarre problem!
I am trying to write a solver for a 1D wave equation in python, and I have run into a bizarre problem that I just can't find a way out of.
I start with the wave equation, and then discretise it, to arrive at the following,
phi(i,j+1) = deltat2/deltax2...
Hey Everyone,
So I've been working on some very basic QM mathematics. Basically I've worked out the wave equation for a particle in one dimension (briefly) like so:
-\frac{\hbar 2}{2m}\psi"(x) + V(x)\psi(x) = E\psi(x)
V = 0 for 0 < x < L ; (L = "Length" of the Boundary)
=>...
Homework Statement
The displacement of the wave traveling in + x direction is: Y(x, t) = 0.35 (m) Sin (6x- 30t); where x is in meter and t is in second.
If the wave reaches its maximum displacement after 0.04 sec,
what is the value of x corresponding to y (max).
Homework Equations...
Homework Statement
Prove that, for any solution to the wave equation, the sum of the values of phi at the points (x0,ct0 +- a) is equal to the sum of the values of phi at the points (x0 +- a,ct0). Prove this directly from the wave equation, not from the general solution.
Hint: change...
Homework Statement
How can I find out if a function is a solution of a wave equation such as:
(a) xt
(b) log(xt)
(c) x² + c²t²
The Attempt at a Solution
Is it simply differentiating the funtion with respect to 'x' twice and equating this to the product of 1/c² and...
I am trying to derive the wave equation for 'Second Sound in superfluid Helium-4 using the basic tenets of the two-fluid model. I am following the derivation in a book which has intermediate steps along the way - I am trying to fill in the gaps. I am almost there - there is only one step that I...
Homework Statement
Find a formal solution to the vibrating string problem..
alpha=4, 0<x<pi t>0
u(0,t)=u(pi,t)=0 t>0
f(x)= x^2(pi-x)
g(x)=0
Homework Equations
u(x,t) = sum[a cos(alpha*n*t/L + b sin(alpha*n*t/L)*sin(n pi x / L)
Fourier series for sine
The Attempt at a Solution
a =...
Hi,
While trying to simulate wave a mixing scenario I wrote below.
But it seems that matlab's matrix manipulation can easily unrol the marked loop (but i cannot find how). Can somebody help
% speed of light
c = 1;
% matrix containing wave parameters in each row for number of waves...
Let u be a solution of the wave equation utt-uxx=0 on the whole plane. Suppose that ux(x,t) is a constant on the line x=1+t. Assume that u(x,0)=1 for all x in R and u(1,1,)=3. Find such a solution u.
I need help trying to incorporate the ux(x,t) is a constant on the line x=1+t
A transverse wave on a rope is given by y(x, t)=
(0.750\; {\rm cm})\, \cos ( \, \pi [(0.400\;{\rm cm}^{ - 1})x+(250\; {\rm s}^{ - 1})t])
Find the period.
This should be simple, but I keep getting the wrong answer in Mastering Physics. I can't find any explanation in my book, and it's...
Homework Statement
the transverse displacement y(x,t) (assumed small) of a string stretched between the points x=0 and x=a satisfies the equation
d2y/c2dt2=d2y/dx2
find the solution satisfying the following (t=0) initial conditions
(i) y(x,0)= Lsin(\pix/a), (dy/dt)(x,0)=0
all derivatives are...
I'm reposting this because there was a problem with the title/LaTeX last time.
Homework Statement
Solve the wave equation (1) on the region 0<x<2 subject to the boundary conditions (2) and the initial condition (3) by separation of variables.Homework Equations
(1) \frac{\partial^2...
Hello, I have some trouble seeing why the solution of the wave equation in 2 dimensions exist at all later times once it passes an initial disturbance...
For example, take a simple case where the initial position is zero, and the initial velocity equals some function inside some circle domain...
Hi,
I was wondering if anybody could help me understand a derivation connected to the double-slit experiment that I came across within an introduction to quantum theory paper. I was interested in understanding this approach because it seems to provide a useful correlation of the meaning of the...
How do you use separation of variables to solve the damped wave equation
y_tt + 2y_t = y_xx
where y(0,t) = y(pi,t) = 0
y(x,0) = f(x)
y_t (x,0) = 0
---
These are partial derivatives where y = X(x)T(t)
So rewriting the equation I get
X(x)T''(t) + 2X(x)T'(t) = X''(x)T(t)
which...
Homework Statement
I must use the wave equation to to find the speed of a wave.
y(x,t) = (3.0mm) sin [(4.00mm^-1)x - (7.00 s^-1)t]
Homework Equations
Here's the wave equation. It has strange symbols.
(∂^2 y) / (∂ x^2) = (1 / v^2) ((∂^2 y)/ (∂^2 t)
The Attempt at a Solution...
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
Really stuck with this, i can't work out how to apply the boundary conditions to generate the simultaneous equations to find the specific solution. Can't find any similar examples either.
Help appreciated.
Hey there!
I'm faced with this problem:
http://img7.imageshack.us/img7/4381/25686658nz9.png
It's a 1D nonhomogeneous wave equation with a "right hand side" equaling to the dirac delta function in x * a sinusoidal function in t. I have to find its general solution with the constraints...
Homework Statement
A steel guitar string with a diameter of .3 mm and 65 cm long has a tension of 100 N. Find the frequencies of the first five modes of vibration and sketch a graph of the associated eigenfunctions. The density of steel,7700 kg/m^3 is needed to find \mu.
Homework...
Homework Statement
For which values of the constants a and b does u(x,t) = sin(ax)sin(bt) satisfy the wave equation Utt = C2uxx?
Homework Equations
The Attempt at a Solution
I've taken the partial differentials:
Ux = acos(ax)sin(bt)
Uxx = -a2sin(ax)sin(bt)
Ut = bcos(bt)sin(ax)...
Write the BVP for the small vertical vibrations of a homogeneous string. Assume that the wave's speed is c.
Problem: take L = 2[Pi] , c= 2,
SUppose that the right end is fixed while the left end is allowed to move vertically. At the left end, the tangent line at any time t is horizontal...
This isn't a homework question, but something which has been bugging me. I can't figure it out. Maybe it's late, but it's probably a very stupid question
If light shines onto a denser medium, the light slows down in this medium right? If the light slows down, then by the wave equation, v =...