How Do You Calculate Max and Min Transverse Speeds in a Wave on a String?

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To calculate the maximum and minimum transverse speeds of particles on a string given the wave equation D=0.40sin(7.0x+38t), one must derive the speed from the displacement formula. The wave speed is determined to be 5.4 m/s, but this value does not directly provide the transverse speeds of the particles. The maximum transverse speed can be found by differentiating the displacement equation with respect to time, yielding a sinusoidal function for particle speed. This approach reveals that the maximum speed corresponds to the amplitude and angular frequency of the wave. Understanding this relationship is crucial for accurately determining the transverse speeds of the string's particles.
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Homework Statement


You have a long string, and you oscillate one end, causing a (transverse) wave to propagate along it. The wave's formula is D=0.40sin(7.0x+38t) where displacement D and position x are given in meters and time t is given in seconds.

Determine the maximum and minimum transverse speeds of particles of the string.

Homework Equations


v= omega/k

The Attempt at a Solution


I found the wave speed, which is equal to 5.4 m/s, but I am not sure how to find the maximum and minimum speeds from this.
 
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The given wave equation for the particles on the string is sinusoidal. Note that this equation is for the motion of the individual particles on the string, and so how would you obtain the equation describing the speed of the individual particles (from the displacement equation given to you)?

Hint: It's not v = ω/k
 

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