Question about force in transverse waves on a string

• kelvin490
In summary, the force on a point of a sinusoidal wave transmitted by a flexible string is zero if the slope at that point is zero. However, if the slope is nonzero, the transverse force is proportional to the second derivative of the string shape.
kelvin490
Gold Member
In deriving wave equation or power transmission of wave transmitted by a string, it is usually stated (with some assumptions) that the transverse force on a point of the string is proportional to the slope at that point. An example is given in p.20 of this notes: http://www.people.fas.harvard.edu/~djmorin/waves/transverse.pdf

If the slope is zero the transverse force is also zero. It can also be seen in the way that if some portion of the string is horizontal the tensions on both side are also horizontal and thus cancel out, therefore no transverse force.

However, in the case that the wave is sinusoidal, the points at the amplitude of the wave should have greatest acceleration and should experience the greatest force because every point is performing SHM. There seems like a contradiction here. Why?

The assumption is that the string is flexible and thus cannot transmit any forces orthogonal to its direction.

kelvin490 said:
However, in the case that the wave is sinusoidal, the points at the amplitude of the wave should have greatest acceleration and should experience the greatest force because every point is performing SHM. There seems like a contradiction here. Why?

If I understand you correctly, you think it surprising that the points currently at the maximum amplitude should experience the largest force? Why do you find this surprising? It is true of any harmonic motion that the force is greatest at the turning points.

Orodruin said:
The assumption is that the string is flexible and thus cannot transmit any forces orthogonal to its direction.
If I understand you correctly, you think it surprising that the points currently at the maximum amplitude should experience the largest force? Why do you find this surprising? It is true of any harmonic motion that the force is greatest at the turning points.

I have no doubt that the force is greatest at turning points. It is the SHM model. However, my concern is if we consider the common mathematical description of wave we know that the transverse force acting on it is proportional to the slope of that portion of string. At maximum position of the sinusoidal wave the slope is zero and thus transverse force is zero. It looks like there is a contradiction to the SHM model.

kelvin490 said:
However, my concern is if we consider the common mathematical description of wave we know that the transverse force acting on it is proportional to the slope of that portion of string. At maximum position of the sinusoidal wave the slope is zero and thus transverse force is zero. It looks like there is a contradiction to the SHM model.

No, this is not correct. If you just have a slope, the force on one side of a small string element cancels the force on the other side. What is important is the change of the slope, i.e., the second derivative of the string shape, which describes the difference of the forces on each side of a small string element.

kelvin490
Orodruin said:
No, this is not correct. If you just have a slope, the force on one side of a small string element cancels the force on the other side. What is important is the change of the slope, i.e., the second derivative of the string shape, which describes the difference of the forces on each side of a small string element.

I see your point. Thank you.

1. What is the definition of force in transverse waves on a string?

The force in transverse waves on a string is the external force that is applied to the string causing it to vibrate in a perpendicular direction to the direction of the force. This force is responsible for the propagation of the wave along the string.

2. How does the amplitude of the wave affect the force in transverse waves on a string?

The amplitude of the wave does not affect the force in transverse waves on a string. The force remains constant regardless of the amplitude of the wave. However, a larger amplitude wave may result in a larger displacement of the string, which in turn may require a larger force to maintain the wave's motion.

3. What is the relationship between frequency and force in transverse waves on a string?

There is no direct relationship between frequency and force in transverse waves on a string. The force is dependent on the tension of the string and the amplitude of the wave, while the frequency is determined by the speed of the wave and the length of the string.

4. Can the direction of the force change in transverse waves on a string?

Yes, the direction of the force in transverse waves on a string can change. As the wave travels along the string, the force acts in a perpendicular direction to the direction of the wave's motion. However, if the string is bent or curved, the direction of the force may also change to align with the new direction of the string.

5. How does the tension of the string affect the force in transverse waves on a string?

The tension of the string has a direct impact on the force in transverse waves on a string. A higher tension in the string will result in a greater force being required to maintain the wave's motion. This is because a higher tension creates a greater resistance to the wave's motion, requiring a stronger force to overcome it.

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