2 sign questions related to the wave equation

In summary, the conversation discusses the driving force and potential energy per unit length of a point in a wave, with one question regarding the mechanics of the driving force and another question about the signs of the potential energy and work done against tension. The expert summarizes that the driving force must counteract the tension in the string, and the potential energy stored has the opposite sign of the work done against tension.
  • #1
Clara Chung
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In the picture about, I gave 1. a picture of a segment of string for reference, 2. a description of the driving force and 3. a description about the potential energy per unit length of a point in the wave.

I have two questions here.

1. Why does the driving mechanism produce a force to balance the transverse component of the tension in the string at x=0? Intuitively, shouldn't the driving mechanism produces the transverse component of the tension to initiate and support the wave motion?

2. Isn't the force against the tension -T? Therefore, why isn't the work done against the tension per unit length U= -1/2 T (Ψ/∂x)^2 ?

I am having a difficult time on figuring out what are the signs. Thank you so much for your help.
 

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  • #2
If I understand your questions correctly, then I would respond this way:

1. The string is already stretched, so it is exerting a tension force (balanced by its stationary endpoints). To move the end of the string slightly, thereby forcing the wave motion, the driver also has to counteract this tension, i.e., exert a force/do work against the tension. It's like raising an object against the force of gravity, which is (always) acting on the object.

2. The potential energy stored (i.e., the change in potential energy) has, by convention, the opposite sign of the work. To continue the previous analogy, if you do work against gravity to raise an object, you increase its potential energy.
 
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  • #3
olivermsun said:
If I understand your questions correctly, then I would respond this way:

1. The string is already stretched, so it is exerting a tension force (balanced by its stationary endpoints). To move the end of the string slightly, thereby forcing the wave motion, the driver also has to counteract this tension, i.e., exert a force/do work against the tension. It's like raising an object against the force of gravity, which is (always) acting on the object.

2. The potential energy stored (i.e., the change in potential energy) has, by convention, the opposite sign of the work. To continue the previous analogy, if you do work against gravity to raise an object, you increase its potential energy.
Thank you.
 

1. What is the wave equation?

The wave equation is a mathematical equation that describes the behavior of waves in a variety of physical systems, such as water waves, sound waves, and electromagnetic waves. It is often written as: d2y/dx2 = (1/v2) d2y/dt2, where y is the displacement of the wave, x is the position, t is time, and v is the speed of the wave.

2. What is the significance of the wave equation?

The wave equation is significant because it allows scientists and engineers to understand and predict the behavior of waves in various systems. It is used in fields such as acoustics, optics, and electromagnetics to design and analyze devices and systems that utilize waves, such as microphones, telescopes, and antennas.

3. What are the two sign questions related to the wave equation?

The two sign questions related to the wave equation are: What is the sign of the second derivative term? and What is the sign of the constant term?

4. How do the signs in the wave equation affect the behavior of waves?

The signs in the wave equation determine the nature of the wave's behavior. The sign of the second derivative term determines whether the wave is propagating in the positive or negative direction. The sign of the constant term affects the amplitude of the wave, with positive values resulting in an increasing wave and negative values resulting in a decreasing wave.

5. Can the wave equation be solved analytically?

Yes, the wave equation can be solved analytically for simple cases, such as a one-dimensional wave on a string or in a uniform medium. However, for more complex systems, numerical methods are often used to approximate the solution.

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