Understanding Waves Dispersion: A, B, & C

In summary, the conversation discusses equations and concepts related to wave motion and group velocity. The first part (A) talks about a graph showing a transverse displacement and the resultant motion of a traveling wave. The speed of the envelope is also mentioned, but the person does not understand how it was derived. The second part (B) mentions a system with a dispersion relation and the variables involved, but the person is confused about how they relate. Finally, part (C) discusses the dependence of k on frequency in a beaded string and how it exhibits high frequency cut off. The person is unsure of how to find k as a function using the given equations and mentions a helpful simulation for further understanding.
  • #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
These problems are about group velocity [tex]v_g = {\rm d}\omega/{\rm d}k[/tex]. You need to write the superposition as a product of average and beat frequency, using those relations there are for sin a + sin b.

Here is some help, with a nice simulation:
http://webphysics.davidson.edu/faculty/dmb/bernstein/qmwave/section2b.html
 
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  • #3


Hi there,

Understanding wave dispersion can be tricky, but I'll try my best to explain it to you.

A) In this part, the book is discussing a string that has a transverse displacement, meaning that the wave is moving perpendicular to the direction of the string. The equation given is phi=cosw1t+cosw2t, which means that the string is vibrating at two different frequencies, w1 and w2. When these frequencies are close to each other and w1 is greater than w2, the resulting motion is a traveling wave with an angular frequency of (w1+w2)/2. The envelope of this wave is also a traveling wave, but with a frequency of (w1-w2)/2. The speed of this envelope can be found by taking the difference between the frequencies and dividing it by the difference between the wave numbers (k1-k2). This is because the speed of a wave is equal to its frequency divided by its wave number (v=w/k). I hope this helps to clarify how they got to that equation.

B) This part is discussing a system with a dispersion relation of w=ak^r, where a and r are constants. The book is saying that the group velocity (vg) is equal to the phase velocity (vphi) at all wave frequencies. The phase velocity can be found using the equation vphi=c(1+ak^2)^1/2, where c is a constant. To relate this to the dispersion relation, we can set vphi=vg and solve for k. This will give us the equation k=(w/a)^1/r, which shows how the wave number (k) is related to the frequency (w). The other variables, vphi and vg, are both related to the speed of the wave, but they are looking at it from different perspectives (phase velocity vs. group velocity).

C) In this part, the book is discussing a beaded string that has a high frequency cut-off. This means that there is a certain frequency at which the string can no longer vibrate. The equation given is w=w(sub c)cosh(1/2ka), where w(sub c) is a constant and a is the distance between the beads on the string. This equation shows how the frequency (w) is related to the wave number (k). To find k as a function of w, you can substitute k=pi/a into the equation
 

Related to Understanding Waves Dispersion: A, B, & C

1. What is wave dispersion?

Wave dispersion refers to the phenomenon in which waves of different frequencies travel at different speeds through a medium. This causes the waves to separate or spread out over time, leading to a change in the shape or composition of the wave.

2. How does wave dispersion occur?

Wave dispersion occurs due to the interaction of waves with the medium they are traveling through. The properties of the medium, such as its density and elasticity, affect how the waves behave, causing them to disperse or spread out over time.

3. What is the significance of understanding wave dispersion?

Understanding wave dispersion is crucial in many fields of science and engineering, including optics, acoustics, and seismology. It allows us to predict and control the behavior of waves, which is essential for developing new technologies and solving real-world problems.

4. What are the different types of wave dispersion?

There are two main types of wave dispersion: temporal and spatial. Temporal dispersion occurs when waves of different frequencies travel at different speeds over time. Spatial dispersion, on the other hand, refers to the separation of waves in space due to their different frequencies.

5. How can we measure or study wave dispersion?

There are various techniques and instruments for measuring and studying wave dispersion, depending on the type of wave and the medium it is traveling through. Some common methods include spectroscopy, interferometry, and seismic surveys. Computer simulations and mathematical models are also used to understand and predict wave dispersion.

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