Questions about Waves - Answers and Explanations

[Tosh5457]

Hi, I got a couple of questions which I couldn't find the answer for when I studied. I hope some of you can answer them or provide links explaining :

1) When a wave is written in the complex form, such as

u(x,t)=A(x,t)e^{i(ωt-kx)}

what's the physical meaning of u(x,t)? I imagine u doesn't have a physical meaning, but what about its real and imaginary parts (Re(u) and Im(u))?

2) Is there a difference between wave packet, gaussian packet and a wave?

3) In a propagation wave, is the phase velocity the velocity of the wavefront?

4) How do you know what's the wavefront from the wave function?

5) Do ω (temporal frequency) < 0 and k (spatial frequency) < 0 have a physical meaning? Why are negative values considered?

6) Why is the intensity of a wave u, ||u||^2? The only argument I saw was one of dimensional analysis...

7) The solution of the wave's equation in one dimension can be approximated by: u(x,t)=A(x,t)e^{i(ωt-kx)}

The general solution is a Fourier series/Fourier transform if I'm not mistaken. When is this approximation valid?

8) How is the spatial frequency vector k related to the Poyinting vector?

9) How to show that the Poyinting vector points to the direction of the wave's propagation? (just the basic idea behind that proof would be appreciated).

Thanks.

 

[schaefera]

To answer the first two: because of Euler's identity, u(x,t) has the same physics meaning as whatever the wave equation has in the none-imaginary representation-- or at least the real part does!

And a wave packet is essentially an "envelope" in which a little bit of a wave that increases then decreases- and envelope that has within it a packet of the wave! The wave itself is the inside thing.

 

[Tosh5457]

Thanks for your answer. Nobody else can help? :(

 

[PhilDSP]

1) When a wave is written in the complex form, such as

u(x,t)=A(x,t)e^{i(ωt-kx)}

what's the physical meaning of u(x,t)? I imagine u doesn't have a physical meaning, but what about its real and imaginary parts (Re(u) and Im(u))?

The real part contains all of the frequency components which are non-zero when an inner product is taken against the cosine function. The imaginary part contains all of the frequency components which are non-zero when an inner product is taken against the sine function. (The real and imaginary parts are orthogonal to each other)

3) In a propagation wave, is the phase velocity the velocity of the wavefront?

Sometimes. For that to be the case the phase velocity vector must be oriented in the same direction as the wave propagation vector (group velocity vector) and the medium cannot be dispersive. There are other stipulations.

4) How do you know what's the wavefront from the wave function?

The wavefront should be the gradient of the wave function solution I believe.

5) Do ω (temporal frequency) < 0 and k (spatial frequency) < 0 have a physical meaning? Why are negative values considered?

They might be more visualizable as having physical meaning if you convert them to frequency and wavelength. (ω is angular frequency). Negative values mean the wave component travels in the reverse direction.

7) The solution of the wave's equation in one dimension can be approximated by: u(x,t)=A(x,t)e^{i(ωt-kx)}

The general solution is a Fourier series/Fourier transform if I'm not mistaken. When is this approximation valid?

I believe the FS or FT is exact, not an approximation. The Dirichlet conditions specify when a FS is not valid.