Question

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

Answer 1

The sinusoidal pattern on a string will have a longer wavelength with a greater propagation velocity, assuming the frequency remains constant. The time-averaged power of the wave is not directly proportional to the linear density but increases with higher amplitude and angular frequency and is also proportional to the wave speed.

Step-by-step explanation:

The sinusoidal pattern on a string represents a wave, and the pattern's wavelength is related to the wave's propagation velocity. According to the wave equation v = f λ, where v is the propagation velocity, f is the frequency, and λ is the wavelength, we can deduce that with a greater propagation velocity and a constant frequency, the wavelength will be longer. Hence, the sinusoidal pattern would be longer with a greater propagation velocity.

If the propagation velocity of a wave on a string increases, the time-averaged power of the wave also increases, but it is not directly proportional to the linear density of the string. The time-averaged power depends on the square of the amplitude and the square of the angular frequency. It is also proportional to the speed of the wave on the string, meaning that if the wave speed doubles due to an increase in tension, the power doubles as well.

Answer 2

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

Step-by-step 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.