Waves Dispersion? can someone help explain?

In summary: In the first situation, v_phi is the frequency (in radians/second) of the phi waves. In the second situation, v_phi is the phase (in degrees) of the phi waves. In summary, the book explains how a beaded string can have a high frequency cutoff, and how the phase of the waves can be determined.
  • #1
belleamie
24
0
Hi I'm studying for a test, and in the suggested reading book review has a few equations that they talk about but I'm not don't really understand how it jumps from one thing to another? the book is very vauge... I've broken the parts i don't understand into A,B,C (I used w = omega)

A)IT shows a graph, explain that the end of a string is given a transverse displacement phi=cosw1t+cosw2t where the two frequencies are almost equal and w1>w2 the resultant motion is a traveling wave of angular frequency (w1+w2)/2, modulated by n envelope which is a traveling wave of (w1-w2)/2 There the speed of this envelop is (w1-w2)/(k1-k2) ...? I don't understand how they got that?

B) A system with dipersion relation w=ak^r...a and r are constants because v(sub g)=xv(sub phi) at all wave frequencies. i duno where then got the other variables v(sub g)? i know that v(sub phi) =c(1+ak^2)^1/2 but i don't understand how they relate?

C) a beaded string above cut off, the dependence of k on frequency is given by w=w(sub c) cosh1/2ka showing a graph, How does k depend on the frequency? i know a beaded string can exhibit high freq cut off and that the part od the system vibrates in anti phase with each other...and k=(pi/a)-ik where k can be found as a function by replacing k=pi/a in w/w(sub c)= sin (1/2 Ka-i1/2ka) but I'm not sure how?
 
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  • #2
A) comes from the trig for cosw1t+cosw2t = (cos at)(cos bt) with
a=(w1+w2)/2 and b=(w1-w2)/2.
 
  • #3
B) v_{phi}=w/k=ak^r/r=ak^{r-1}.
v_g=dw/dk=rak^{r-1}=rv_{phi}.
The x must be a misprint.
 
  • #4
I'm not sure what you're asking. If you have w as a function of k in your first eq.,
can't you just solve that for k? In this and in (B), you may be confusing two different situations.
 

1. What is wave dispersion?

Wave dispersion is the phenomenon in which a wave's speed depends on its wavelength. This results in different parts of the wave traveling at different speeds, leading to the separation or spreading out of the wave.

2. How does wave dispersion occur?

Wave dispersion occurs due to the interaction between the wave and the medium it travels through. The properties of the medium, such as its density and elasticity, affect the speed of the wave and therefore cause dispersion.

3. What are some examples of wave dispersion?

Some examples of wave dispersion include the splitting of white light into different colors by a prism, the breaking of ocean waves on a shore, and the formation of rainbows in the sky.

4. What is the significance of wave dispersion?

Wave dispersion plays a crucial role in many fields, such as optics, acoustics, and oceanography. It allows for the separation and analysis of different wavelengths, which helps in understanding the properties of the medium and the wave itself.

5. Can wave dispersion be controlled or manipulated?

Yes, wave dispersion can be controlled and manipulated through the use of different materials and structures. For example, the design of a prism can alter the dispersion of light passing through it, and the use of waveguides can control the dispersion of electromagnetic waves.

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