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Four waves are described by the following equations, where distances are measured in meters and times in seconds. I) y = 0.12 cos(3x - 21t) II) y = 0.15 sin(6x + 42t) III) y = 0.13 cos(6x + 21t) IV) y = -0.23 sin(3x - 42t) Which of these waves have the same period?

A) I and III, and also II and IV

B) I and IV, and also II and III

C) I and II, and also III and IV

D) All of them have the same period

E) They all have different periods

Respuesta :

IamJerry

Answer:

A) I and III, and also II and IV

Explanation:

Generally, the equation of a progressive wave is given by:

y = Asin(kx - ωt)

or

y = Acos(kx - ωt)

depending or the waveform

where;

A is the amplitude of the wave in meters

y(x,t) is the displacement in meters

k = 2π/λ is the propagation constant

ω [tex]=\frac{2\pi }{T}[/tex] is the angular frequency in radian per seconds

[tex]T = \frac{2\pi }{w}[/tex]

T is the period in seconds

+ve and -ve signs show the direction of the wave.

I) Comparing y = 0.12cos(3x - 21t) with the general equation;

           ωt = 21t

           ω = 21

         [tex]\frac{2\pi }{T}[/tex] = 21

         [tex]T = \frac{2\pi }{w}[/tex]

         [tex]T = \frac{2\pi }{21}[/tex] seconds

II) Comparing y = 0.15 sin(6x + 42t)  with the general equation;

         ωt = 42t

         ω = 42

         [tex]\frac{2\pi }{T}[/tex] = 42

         [tex]T = \frac{2\pi }{w}[/tex]

         [tex]T = \frac{2\pi }{42}[/tex]

         [tex]T = \frac{\pi }{21}[/tex] seconds

III) Comparing y = 0.13 cos(6x + 21t) with the general equation;

         ωt = 21t

         ω = 21

         [tex]\frac{2\pi }{T}[/tex] = 21

         [tex]T = \frac{2\pi }{w}[/tex]

         [tex]T = \frac{2\pi }{21}[/tex] seconds

IV) Comparing y = -0.23 sin(3x - 42t) with the general equation;

         ωt = 42t

         ω = 42

         [tex]\frac{2\pi }{T}[/tex] = 42

         [tex]T = \frac{2\pi }{w}[/tex]

         [tex]T = \frac{2\pi }{42}[/tex]

         [tex]T = \frac{\pi }{21}[/tex] seconds    

 Therefore I and III have the same period, II and IV have the same period.

A wave's period is the amount of time it takes a particle on a medium to complete one full vibrational cycle. Option A is correct. Wave I and III, and also II and IV have the same period.

What is the time period of the wave?

A wave's period is the amount of time it takes a particle on a medium to complete one full vibrational cycle. A period is a unit of time that is measured in seconds, hours, days, or years.

The Earth's orbit around the Sun lasts around 365 days; it takes 365 days for the Earth to complete one cycle.

The equation of wave is given as;

[tex]\rm y = Acos(kx - \omega t)[/tex]

A is the amplitude of the wave in meters

y(x,t) is the displacement in meters

k = 2π/λ is the propagation constant

ω is the angular frequency in radian per seconds

T is the period in seconds

We have to analyze the different cases;

I) y = 0.12 cos(3x - 21t)

[tex]\rm \omega t= 21 t \\\\ \omega =21 \\\\ \frac{2\pi }{T} = 21 \\\\ T= \frac{2 \pi }{\omega} \\\\ \rm T = \frac{2 \pi}{21}[/tex]

II) y = 0.15 sin(6x + 42t)

[tex]\rm \omega t= 42 t \\\\ \omega =42 \\\\ \frac{2\pi }{T} = 42 \\\\ T= \frac{2 \pi }{\omega} \\\\ \rm T = \frac{2 \pi}{42}\\\\ \rm T = \frac{2 \pi}{21}[/tex]

III) y = 0.13 cos(6x + 21t)

[tex]\rm \omega t= 21 t \\\\ \omega =21 \\\\ \frac{2\pi }{T} = 21 \\\\ T= \frac{2 \pi }{\omega} \\\\ \rm T = \frac{2 \pi}{21}[/tex]

(IV) y = -0.23 sin(3x - 42t)

[tex]\rm \omega t= 42 t \\\\ \omega =42 \\\\ \frac{2\pi }{T} = 42 \\\\ T= \frac{2 \pi }{\omega} \\\\ \rm T = \frac{2 \pi}{42}\\\\ \rm T = \frac{2 \pi}{21}[/tex]

Hence option A is correct. Wave I and III, and also II and IV have the same period.

To learn more about the time period of the wave refer to the link;

https://brainly.com/question/467475

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