Wave Velocity

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Wave

A wave is a result of external perturbation in a plane surface. We can define a wave as - a wave is a disturbance propagating in space with transportation of energy and momentum from one point to another without transfer of the matter. The most commonly used examples for waves are the ripples in a pond, Sound that reaches us propagates through wave motion, TV signals, etc.  The waves are classified into different types depending upon the type of medium, propagation energy, dimensions, and the vibration of particles. 


What is Wave Velocity?

Now, we are constantly talking about the term wave velocity. To understand the wave velocity first, let us look at the meaning and define wave velocity. 

  • The wave velocity definition is given as the velocity associated with the disturbance propagating in the given medium or in other words, wave velocity is the distance travelled by period motion per unit time. 

  • The wave velocity depends upon the nature of the medium used.

  • The wave velocity is also known as phase velocity because all the particles at the crust and trough are in phase with each other, we can say that the phase travels with wave velocity.

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Now formula of wave velocity is given as follows. The wave velocity formula says it is the product of wavelength and the frequency of the wave. I.e.,

Wave velocity (v) mathematically is given by,

⇒ v = \[\frac{w}{k}\] ……….(1)

Where,

w - The angular velocity

k - the wavenumber

We know that, 

The value of the angular velocity = w = 2πv; where v - Frequency of the wave

The value of the wavenumber = k = \[\frac{2 \pi}{\lambda}\]; where \[\lambda\] - The wavelength 

Substituting these value in equation (1) we get,

⇒ v = \[\frac{2\pi v}{\frac{2\pi}{\lambda}}\] = \[\lambda\]v

Therefore, we have,

⇒ v = \[\lambda\]v …….(2)

Where,

\[\lambda\] - The wavelength 

v - Frequency of the wave

Equation (2) is known as the equation of wave velocity or wave velocity formula.


Wave Velocity Formula:

In wave motion, the perturbations travel through the medium due to repeated periodic oscillations of the particles. The velocity of the wave will be different from the velocity of the particles with which they vibrate about their mean positions. The wave velocity will always be constant but the particle velocity will be changing with time periods.


Properties of Wave Velocity:

  • The wave velocity in a given medium is always constant.

  • The wave velocity is independent of the time and source of the wave, but the wave velocity depends on the wavelength of the propagating wave in a given medium.

  • The wave velocity depends on the medium used.

Examples:

1. How to Calculate Wave Velocity for a Given Periodic Wave with a Wavelength of 3m Has a Frequency 6Hz?

Sol:

Given,

The wavelength of the periodic wave = λ = 3m

The frequency of the given periodic wave = v = 6Hz

We have to calculate the wave velocity of the given periodic wave. From the equation of wave velocity we have,

⇒ v = λv

Where,

λ - The wavelength 

v - Frequency of the wave

Substituting the corresponding values in equation (1) we get,

⇒ v = (3)(6) = 18 m/s

Therefore, the wave velocity of a given periodic wave is 18 m/s.


2. How Do You Find the Velocity of a Wave with a Wavelength of 20m has a Frequency 70Hz?

Sol:

Given,

The wavelength of the periodic wave = λ = 20m

The frequency of the given periodic wave = v = 70Hz

We have to calculate the wave velocity of the given periodic wave. From the equation of wave velocity we have,

⇒ v = λv

Where,

λ - The wavelength 

v - Frequency of the wave

Substituting the corresponding values in equation (1) we get,

⇒ v = (20)(70) = 1400 m/s

Therefore, the wave velocity of a given periodic wave is 1400 m/s.


3. The Velocity of Wave 70m/s. If the Wavelength of the Wave is 1m then Calculate the Frequency of the Given Wave.

Sol:

The wavelength of the wave = λ = 1m

The wave velocity of the given wave = v = 70m/s

We have to calculate the Frequency of the given wave. From the equation of wave velocity we have,

⇒ v = λv …….(1)

Where,

λ - The wavelength 

v - Frequency of the wave 

On rearranging the equation (1) for the frequency of the wave,

⇒ v = \[\frac{v}{\lambda}\]……(2)

Substituting the given values,

⇒ v = \[\frac{70}{1}\] = 70Hz

Therefore, the frequency of the given wave is 70Hz.


4. A Wave with a Frequency 450Hz is Travelling at a Speed of 200m/s. Then Calculate the Wavelength of the Wave.

Sol:

The frequency of the wave = v = 450Hz

The wave velocity of the given wave = v = 200m/s

We have to calculate the wavelength of the given wave. From the equation of wave velocity we have,

⇒ v = λv …….(1)

Where,

λ - The wavelength 

v - Frequency of the wave 

On rearranging the equation (1) for the wavelength of the wave,

⇒ λ = \[\frac{v}{v}\]  ………(2)

Substituting the corresponding values in (2) we get,

⇒ λ = \[\frac{200}{450}\] = 0.44m

Therefore, the wavelength of the given wave is 0.44m.

FAQ (Frequently Asked Questions)

Q1. How to Calculate the Velocity of a Wave?

Ans: The velocity of the wave can be calculated by knowing the wavelength of the wave and the frequency of the wave. Mathematically it is given by,

⇒ v = λv

Where,

λ - The wavelength 

v - Frequency of the wave

Q2. What are the Factors that Affect the Wave Velocity?

Ans: The wave velocity depends on the following factors:

  • The type of medium used.

  • The wavelength of the propagating wave in a given medium.