Is the sinusoidal pattern on the string longer or shorter with a greater propagation velocity?

Increase in velocity propagation would lead to a longer sinusoidal pattern.

Explanation:

The velocity of propagation has a relationship with the tension in a standing wave and is given by;

v = √(T/μ)

where:

μ is mass per unit length.

T is tension in string

For the sinusoidal pattern on the string to be longer or shorter, it means that the wavelength will be longer or shorter.

Now, relationship between wavelength and velocity and tension is;

v = λ/T

Where V is velocity of propagation, λ is wavelength and T is Tension.

So, λ = vT

From the earlier standing wave equation, we will see that if we increase the velocity, it means the Tension of the spring will increase as well.

Thus, from this wavelength equation, an increase in velocity means an increase in tension and which will in turn mean an increase in wavelength.

Therefore, an increase in velocity propagation means a longer sinusoidal pattern.

in this image, you can see solar radiation going through ocean water.

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