Question

State wave equation in terms of angular velocity & wave number.​

Answer 1

Answer:

This question can be solved by substituting the expression for frequency and wavelength in the wave velocity equation. The expression of frequency can be obtained from the angular frequency equation and the wavelength expression from the angular wave number equation. These expressions then when substituted in the wave velocity equation will give the relation between angular frequency, angular wave number and wave velocity.

Explanation:

Formula used: ω=2ππT,k=2πλ,v=υλ

Complete step-by-step solution:

Angular frequency is given by the equation ω=2πT=2πυ

Where, ω is the angular frequency, T is the time period and υ is the frequency

Let’s write the above equation in terms of υ, υ=ω2π……………… (1)

Angular wave number is given from the equation k=2πλ

Where k is the wave number and λ is the wavelength

Writing the above equation in terms of λ=2πk………………. (2)

From the wave equation,v=υλ ……………… (3)

Substituting the values of υ and λ from equation (1) and equation (2) respectively in equation (3)

Therefore equation (3) becomes, v=υλ=ω2π×2πk

Simplifying the equation we get, v=ωk

Hence, this is the relation between angular velocity, angular wave number and angular frequency.

Note: This problem can also be solved by taking the ratio between angular frequency and angular wave number.

Angular frequency,ω=2πυ and wave number, k=2πλ

Taking the ratio between angular frequency and angular wave number

We get, ωk=2πυ2πλ

Simplifying the expression we get, ωk=υλ

Using the relationv=υλ, we get v=ωk

Using this method also we can derive a relation between angular frequency, angular wave number and wave velocity.