# In Mechanical Wave v = w/k. EM wave w/k = c. How Equated ?

• morrobay
In summary: The velocity of propagation for a transverse wave is not related to the amplitude of the wave at all.
morrobay
Gold Member
With ω/k = 2π/T / 2π/λ = velocity for both transverse mechanical waves and EM waves.
I can understand velocity as distance over time in mechanical wave. But how is the ratio Em/Bm = ω/k = c.
That is the maximum amplitudes of the E and B fields in the y and z planes corresponding to c in x direction ?
Is it correct to say that the " leading edges" of the E and B fields in y and z planes are at c ?

http://www.santarosa.edu/~lwillia2/42/WaveEquationDerivation.pdf

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morrobay said:
With ω/k = 2π/T / 2π/λ = velocity for both transverse mechanical waves and EM waves.
I can understand velocity as distance over time in mechanical wave. But how is the ratio Em/Bm = ω/k = c.
That is the maximum amplitudes of the E and B fields in the y and z planes corresponding to c in x direction ?
Is it correct to say that the " leading edges" of the E and B fields in y and z planes are at c ?

http://www.santarosa.edu/~lwillia2/42/WaveEquationDerivation.pdf

Too late edit : With dE/dx and t constant, dB/dt and x constant --> ∂E/∂x = - ∂B/∂t ... then to ω/k = Em/Bm = c. Does answer the change in x/change in t question. Again still not sure how this is related to amplitudes ?

morrobay said:
Too late edit : With dE/dx and t constant, dB/dt and x constant --> ∂E/∂x = - ∂B/∂t ... then to ω/k = Em/Bm = c. Does answer the change in x/change in t question. Again still not sure how this is related to amplitudes ?
The velocity of propagation for a transverse wave is not related to the amplitude of the wave at all.

SteamKing said:
The velocity of propagation for a transverse wave is not related to the amplitude of the wave at all.
Well that's my question , with ω/k = Emax/Bmax = c. (from Halliday-Resnick) The propagation velocity is related to amplitude. Unless I am misinterpreting Em/Bm

This is from a page in the text : kEm cos (kx-ωt) = ωBm cos (kx-ωt).
ω/k = Em/Bm = c
Thus the speed of wave c is the ratio of the amplitude of the electric and magnetic components of the wave.
So can someone show how Em/Bm = ω/k ?

morrobay said:
So can someone show how Em/Bm = ω/k ?

Consider the following plane wave as an example: $$\vec E = \hat x E_m \cos (kz - \omega t) \\ \vec B = \hat y B_m \cos (kz - \omega t)$$ where ##\hat x## and ##\hat y## are unit vectors in the x and y directions. That is, ##\vec E## and ##\vec B## are in the x and y directions respectively, and the wave propagates in the z direction. Substitute these into the third Maxwell equation in free space: $$\vec \nabla \times \vec E = - \frac {\partial \vec B}{\partial t}$$ and you will get the desired result.

[added: I originally had ##E_m## instead of ##B_m## in my equation for ##\vec B## above. I've fixed this.]

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## 1. What does the equation v = w/k represent in mechanical waves?

The equation v = w/k represents the speed of a mechanical wave, where v is the wave's velocity, w is its angular frequency, and k is its wavenumber.

## 2. How is the speed of a mechanical wave related to its frequency and wavelength?

The speed of a mechanical wave is directly proportional to its frequency and inversely proportional to its wavelength. This means that as the frequency increases, the speed of the wave also increases, while a longer wavelength results in a slower wave speed.

## 3. What is the significance of the angular frequency in the equation v = w/k?

The angular frequency, w, represents how many radians an oscillating particle goes through in one second. It is related to the wave's frequency, f, by the equation w = 2πf. In the context of the v = w/k equation, the angular frequency helps determine the speed of the wave.

## 4. How is the equation v = w/k different for electromagnetic waves?

The equation v = w/k is not applicable for electromagnetic waves, as they do not require a medium to propagate and therefore do not have a wavenumber. Instead, the speed of an electromagnetic wave, c, is determined by the properties of the medium it is traveling through.

## 5. Can the equation v = w/k be used to calculate the speed of any type of mechanical wave?

No, the equation v = w/k is only valid for transverse waves in a medium. Longitudinal waves, such as sound waves, have a different equation for calculating their speed. Additionally, the medium through which the wave is traveling must have a linear relationship between the applied force and the resulting displacement for the equation to apply.

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