2 sign questions related to the wave equation

[Clara Chung]

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.

 

[olivermsun]

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.

 

[Clara Chung]

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.