Why is there a wavelength dependence in diffraction?

[Glenn G]

Hello community,
I'm aware of Huygens construction for helping to explain why you might possibly get diffracted waves if a plane wave passes through a gap (i.e. About the secondary wavelets and superposition bit) but the wavelength dependence bit bothers me because say one wavefront is passing through a gap why should the distance to the trailing wavefront behind the one currently passing through the gap have any impact on how much the curvature of the current wavefront changes?

Just don't get the wavelength dependence to gap. In fact I'd imagine if you sent a single plane water wave to a relatively narrow gap it would diffract on passing through the gap whether or not there was another wavefront lambda metres behind it.

Please help!

Glenn

 

[jambaugh]

Hello community,
I'm aware of Huygens construction for helping to explain why you might possibly get diffracted waves if a plane wave passes through a gap (i.e. About the secondary wavelets and superposition bit) but the wavelength dependence bit bothers me because say one wavefront is passing through a gap why should the distance to the trailing wavefront behind the one currently passing through the gap have any impact on how much the curvature of the current wavefront changes?

As far as the physical/causal "why" keep in mind that the water wave is traveling much slower than the actual speed of sound (pressure waves) in the water. So each bit of water in this process is physically affected by not only nearby disturbances but those farther away. To calculate the diffraction effect in those terms would be problematic although you could code a simulation on a computer and still see the diffaction effects if the model is finely enough resolved.

You should think of Huygen's construction and specifically of the resolution of wave behavior in terms of how sinusoidal plane waves behave as a means to systematically solve for the behavior of the general case. Rather than having to resolve the dynamics of each fluid particle you resolve the bulk motion into normal modes which, mathematically, behave independently of the other modes. Likewise with electromagnetic field propagation and quanta (although the interpretation qualitatively changes in the latter case and one is no longer describing aggregate behavior of many bodies there.)

Just don't get the wavelength dependence to gap. In fact I'd imagine if you sent a single plane water wave to a relatively narrow gap it would diffract on passing through the gap whether or not there was another wavefront lambda metres behind it.

You shouldn't think in terms of "diffracting or not diffracting" both cases will have a diffraction effect. But keep in mind the isolated plane water wavelet is a superposition of many sinusoidal wavelength components (think about its Fourier spectrum). You can calculate the plane wave's diffraction behavior by adding up (via integration) the diffraction of these components each of which will have wavelength dependency.

 

[Glenn G]

Another thing with Huygens and the secondary wavelets what about the part of the wavelets that should be projected backwards? Do we just dismiss this as limitation of the model?

 

[pixel]

I'm aware of Huygens construction for helping to explain why you might possibly get diffracted waves if a plane wave passes through a gap (i.e. About the secondary wavelets and superposition bit) but the wavelength dependence bit bothers me...

Interesting question. Huygen's Principle by itself does not deal with the wavelength of the light so is incomplete as we know there is a strong wavelength dependence to diffraction. So I guess to answer your question, just using Huygen's Principle there should not be an impact of wavelength, but the Principle by itself is not accurate as it does not take into account wavelength. Fresnel later modified the mathematical description of Huygen's Principle to include a factor that depends on wavelength in order to get more accurate results.

 

[ORF]

Hello

Maybe we are referring to different Huygen's principles, but if you are referring to the optical/wave one, I think it deals with the wavelength...
https://en.wikipedia.org/wiki/Huygens–Fresnel_principle#Mathematical_expression_of_the_principle

nteresting question. Huygen's Principle by itself does not deal with the wavelength of the light so is incomplete as we know there is a strong wavelength dependence to diffraction.

Greetings.

 

[lychette]

Interesting question. Huygen's Principle by itself does not deal with the wavelength of the light so is incomplete as we know there is a strong wavelength dependence to diffraction. So I guess to answer your question, just using Huygen's Principle there should not be an impact of wavelength, but the Principle by itself is not accurate as it does not take into account wavelength. Fresnel later modified the mathematical description of Huygen's Principle to include a factor that depends on wavelength in order to get more accurate results.

I think that Fresnel's mathematical modification had more to do with the form of the waves (plane or sperical) rather than the principle of Huygens construction.