The Wave Equation and Traveling Waves

[harambe]

The wave equation in one space dimension can be written as follows:

.A traveling wave which is confined to one plane in space and varies sinusoidally in both space and time can be expressed as combinations of

What is the difference between these two wave equations?? And is traveling wave always sinusoidially vary

 

[Charles Link]

The wave equation in one space dimension can be written as follows:

.A traveling wave which is confined to one plane in space and varies sinusoidally in both space and time can be expressed as combinations of

View attachment 222954What is the difference between these two wave equations?? And is traveling wave always sinusoidially vary

The general solution of the wave traveling to the right is $y=A \cos(kx-\omega t)+B \sin(kx-\omega t)=C \cos(kx-\omega t -\phi)$ where $C=\sqrt{A^2+B^2}$ and $\phi=\tan^{-1}(\frac{B}{A})$. Factoring out $\sqrt{A^2+B^2}$ and using the trigonometric identity for $\cos(\theta-\phi)$ will show this result.

 

[harambe]

The general solution of the wave traveling to the right is y=Acos(kx−ωt)+Bsin(kx−ωt)=Ccos(kx−ωt−ϕ)y=Acos⁡(kx−ωt)+Bsin⁡(kx−ωt)=Ccos⁡(kx−ωt−ϕ) y=A \cos(kx-\omega t)+B \sin(kx-\omega t)=C \cos(kx-\omega t -\phi) where C=√A2+B2C=A2+B2 C=\sqrt{A^2+B^2} and ϕ=tan−1(BA)ϕ=tan−1⁡(BA) \phi=\tan^{-1}(\frac{B}{A}) . Factoring out √A2+B2A2+B2 \sqrt{A^2+B^2} and using the trigonometric identity for cos(θ−ϕ)cos⁡(θ−ϕ) \cos(\theta-\phi) will show this result.

Is this the general form for a traveling wave varying sinysodially in one plane...what about the partial differential equation?

 

[hilbert2]

The traveling wave can have any continuous and differentiable shape, as can be shown by checking that the general function $f(x-ct)$ satisfies the wave equation (as does the one moving to opposite direction, $f(x+ct)$).

The sine and cosine solutions are useful just because a wave of any shape can be constructed by summing an appropriate set of them together, as in Fourier series/integral.

 

[harambe]

So the sine and cos function waves are solutions of the wave equation..right?

 

[hilbert2]

Yes, they represent one kind of a solution, but not the only one. When it's a partial differential equation, it's not possible to write a simple formula that represents every possible solution, as is possible with an ordinary differential equation like

$\frac{dy}{dx}=ky$,

for which all solutions are of the form $y(x)=Ae^{kx}$ with $A$ a constant number.

 

[harambe]

I got that .Thanks

Like traveling waves ,are there any other special category of waves which satisfy the wave motion. Wave motion should apply to every type of wave,right

 

[hilbert2]

If you define an initial condition like $u(x,0)=Ae^{-kx^2}$ and $\left.\frac{\partial u}{\partial t}\right|_{t=0} = 0$, you will get a solution that is not a traveling wave.

 

[harambe]

The wave f(x-ct).Can this wave travel in any plane

 

[hilbert2]

The wave f(x-ct).Can this wave travel in any plane

If you have a 3D scalar wave equation $\frac{\partial^2 u(x,y,z,t)}{\partial t^2} = c^2 \nabla^2 u(x,y,z,t)$, you can make traveling wave solutions like

$u(x,y,z,t) = A\sin(\mathbf{k}\cdot \mathbf{x} - ct)$,

where $\mathbf{k}$ is a wavevector with any direction and $\mathbf{x} = (x,y,z)$ is a vector position coordinate. So in 3D the traveling waves can move to any direction. If it's not a scalar wave and the function $u$ is also a vector, then the constant $A$ in the solution is a vector too and does not necessarily have the same direction as the wave vector $\mathbf{k}$

 

[olivermsun]

I got that .Thanks

Like traveling waves ,are there any other special category of waves which satisfy the wave motion. Wave motion should apply to every type of wave,right

Several common types of waves do not (even approximately) follow the behavior described by solutions like $\frac{1}{2}f(x-ct) + \frac{1}{2}f(x+ct)$. For example, think of an ocean wave that gradually steepens before it breaks in the surf zone.