Question about force in transverse waves on a string

[kelvin490]

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?

 

[Orodruin]

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

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.

 

[kelvin490]

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.

 

[Orodruin]

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]

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.