Stationary waves in composite strings

In summary, the conversation discusses the concept of stationary waves in a composite string ABC made up of two strings AB and BC with different mass per unit length and a joint at B. It is mentioned that a stationary wave will set up at a particular frequency with joint B as a node. The doubts brought up include the possibility of a perfect stationary wave in string AB due to partial reflection, the reason for a node at the junction, and the difference in amplitude between the stationary waves in AB and BC. The experts clarify that the existence of exact standing waves with perfect nodes is only an approximation, and depends on the damping of the system. They also mention that the orientation of the principal axes of the vibrating points can vary at different positions on the object
  • #1
arvindsharma
21
0
Dear All,

I was reading the concept of stationary waves in composite string ABC made up of joining two strings AB AND BC with different mass per unit length and a joint at 'B'.the two ends of the composite string are clamped at 'A' and 'C' and a transverse wave is set up by an external source at one of the clamp say clamp 'A'.it was written in the book that stationary wave will set up in the composite string ABC at a particular frequency of external source with joint 'B' as a node.following are my doubts
1)the stationary wave in AB must be due to incident wave and reflected wave at junction B but at junction 'B' there is some reflection and some transmission(so that stationary wave in B can also be set up).due to partial reflection the amplitude of incident wave and reflected wave must be different so a perfect stationary wave is not possible in string AB.however wave in string BC is perfectly stationary(assuming no energy loss at clamp C).is my reasoning correct in this sense?
2)Why there is a node at junction?does this always happen in composite string that joint is a node?
3)the stationary wave in string AB and BC will have different amplitude because the amplitude of reflected and transmitted wave is not same?
 
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  • #2
1. because of the partial refection at B you won't get opposite traveling waves to interfere to produce a non-zero amplitude there and get standing waves. You can check by solving the wave equation for standing waves or just do the experiment.
2. see above. yes - for standing waves.
3. yes

Merry Xmas.
 
  • #3
THinking about a mode of vibration as a "stationary wave" made from "traveling waves" in opposite directions is useful for a simple structure like a uniform string, but less useful for more complicated structures.

You will certainly get "stationary" modes of vibration of the whole structure, but the traveling waves are complicated because the speed will be different in the two parts of the string, and there will be partial reflection and partial transmission at B, for the traveling waves going in both directions. The "shape" of the traveling waves will not be the same as the shape of the stationary vibration mode, and the "shape" of the stationary wave willl not be a simple sine wave like a uniform string

In general, point B will not be a node. If it is a node in your example, there must be something special about the problem, for example some relation between the mass per unit length of the two strings, or the lengths of parts AB and BC. Or, the book is wrong!
 
  • #4
not yet satisfied.please clear my doubts
 
  • #5
You need to tell us EXACTLY what the book says about the system.

it should be obvious that if AB and BC were the same length, and the mass of BC was 1.01 times the mass of AB, the vibration modes of the system would be very close to a uniform string, and there would be standing waves that did NOT have a node at B.

If the system in the book has a node at B, it is because of the particular masses, lengths, etc, of that system, not because every possible set of two strings has a node there.
 
  • #6
AlephZero said:
You need to tell us EXACTLY what the book says about the system.

it should be obvious that if AB and BC were the same length, and the mass of BC was 1.01 times the mass of AB, the vibration modes of the system would be very close to a uniform string, and there would be standing waves that did NOT have a node at B.

If the system in the book has a node at B, it is because of the particular masses, lengths, etc, of that system, not because every possible set of two strings has a node there.

Of course. To get a standing wave, you must have solutions to the wave equations on both sections that are satisfied by the boundary conditions. I have a feeling that the bandwidth of the system (i.e. the losses) may allow some latitude but I can't be sure.
 
  • #7
sophiecentaur said:
To get a standing wave, you must have solutions to the wave equations on both sections that are satisfied by the boundary conditions. I have a feeling that the bandwidth of the system (i.e. the losses) may allow some latitude but I can't be sure.

You are right, the existence of "exact" standing waves with "perfect" nodes (zero motion at all times) is only an approximation, and depends on the way the system is damped.

Making models of damping based on physics is notoriously difficult, especially for lightly damped systems like vibrating strings etc. The usual technique is to make a model that is mathematically convenient, and fits reasonably well to measurements close to the resonant frequency of each mode. What happens away the resonances often isn't important from an engineering point of view, because the amplitudes are small compared with the resonance peaks.

In real life, different positions on a vibrating object don't necessarily vibrate in phase with each other, and don't necessarily even vibrate back and forth through some equilibrium position. For example it a rotating object is vibrating, the coriolis forces make every point move in an ellipse around the equilibrium point (as viewed by somebody rotating at the same speed as the object), not back and forth along a straight line. And the orientations of the principal axes of the ellipses vary at different positions on the object.

But all this is probably irrelevant to the OP's question.
 
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  • #8
i got the point. thank you all for your contribution
 

1. What are stationary waves in composite strings?

Stationary waves in composite strings refer to a type of vibrational pattern that occurs when two or more strings of different materials are joined together. These waves are characterized by points of no motion, known as nodes, and points of maximum motion, known as antinodes.

2. How do stationary waves form in composite strings?

Stationary waves in composite strings form when two waves of the same frequency traveling in opposite directions interfere with each other. This interference creates a standing wave pattern, where the amplitude of the wave remains constant over time.

3. What properties of the composite strings affect the formation of stationary waves?

The properties of the composite strings such as their mass, tension, and the material they are made of can affect the formation of stationary waves. The different properties of the strings can lead to differences in their wave velocities, which in turn affect the interference and formation of the stationary waves.

4. How are the wavelengths of stationary waves in composite strings determined?

The wavelengths of stationary waves in composite strings are determined by the length and properties of the individual strings. The wavelength is equal to twice the length of the string divided by the number of nodes present in the wave pattern.

5. What are some practical applications of stationary waves in composite strings?

Stationary waves in composite strings have several practical applications, including musical instruments such as guitars and pianos. They can also be used in imaging techniques such as ultrasound, where the interference patterns can be used to create detailed images of internal body structures.

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