I am trying to determine the shape of a so called "sound-lens", which is a device which can focus sound waves. I am using two methods, and they give different results, hence one (or both) must be wrong.
Imagine a sound source at (0,R_L). The object is to focus the emanating waves at (0,0). For this, denote the lens transition time t_0 = R_1/v_s, with R_1+R_2=R_L and t_c=R_L/v_s.
A sound wave front before focusing as a function of time t and x:
y_1(x,t)=RL-sqrt((v_s*t)^2-x^2).
A sound wave front after focusing:
y_2(x,t)=sqrt((R_2-v_s*(t-t_0))^2-x^2);
It follows that if the lens shape is h(x,t), then this shape must be the shape of the curve:
h(x,t_0) = y_2(x,t_0) - y_1(x,t_0), x\in some [-A, A] (*)
> I am trying to determine the shape of a so called "sound-lens", which is a > device which can focus sound waves. I am using two methods, and they give > different results, hence one (or both) must be wrong.
> Imagine a sound source at (0,R_L). The object is to focus the emanating waves at > (0,0). For this, denote the lens transition time t_0 = R_1/v_s, with R_1+R_2=R_L > and t_c=R_L/v_s.
> A sound wave front before focusing as a function of time t and x:
> y_1(x,t)=RL-sqrt((v_s*t)^2-x^2).
> A sound wave front after focusing:
> y_2(x,t)=sqrt((R_2-v_s*(t-t_0))^2-x^2);
> It follows that if the lens shape is h(x,t), then this shape must be the shape > of the curve:
On Nov 19, 9:12 pm, "I.N. Galidakis" <morph...@olympus.mons> wrote:
> I am trying to determine the shape of a so called "sound-lens", which is a > device which can focus sound waves. I am using two methods, and they give > different results, hence one (or both) must be wrong.
---
Pl. see, fwiw for light rays,earlier I derived a hyperbola lens section whose eccentricity equals refractive index ( > 1 ) that brings rays to a focus, with constant equal weighted path lengths.
> > Yes, for a reflective lens. A refractive lens is a different beast. > > -- > > Ioannis
> I don't understand why this isn't exactly the same maths (Snell's Law) as > for an optical lens, which AFAIK also determines a parabolic curve.
For sound waves, a reflecting or refracting medium is too often of a similar order of magnitude as the wave lengths, and Snell's law only works well when the wave lengths are much smaller that the reflecting or refracting media, as in light "waves".
> In article <4b05d9fd$0$5425$afc38...@news.optusnet.com.au>, > "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
> > "I.N. Galidakis" <morph...@olympus.mons> wrote in message > >news:1258654768.128962@athprx04... > > > Mensanator wrote: > > > [snip]
> > >> Wouldn't it simply be a parabolic dish?
> > > Yes, for a reflective lens. A refractive lens is a different beast. > > > -- > > > Ioannis
> > I don't understand why this isn't exactly the same maths (Snell's Law) as > > for an optical lens, which AFAIK also determines a parabolic curve.
> For sound waves, a reflecting or refracting medium is too often of a > similar order of magnitude as the wave lengths, and Snell's law only > works well when the wave lengths are much smaller that the reflecting or > refracting media, as in light "waves".
Then how do those parabolic dishes I see at football games work?
In article <2e5a6ffb-1771-47ed-8510-fa4d5f046...@g27g2000yqn.googlegroups.com>,
Mensanator <mensana...@aol.com> wrote: > On Nov 19, 12:19 pm, "I.N. Galidakis" <morph...@olympus.mons> wrote: > > Mensanator wrote:
> > [snip]
> > > Wouldn't it simply be a parabolic dish?
> > Yes, for a reflective lens. A refractive lens is a different beast.
> How are you going to make a refractive lens for sound?
A suitably shaped balloon containing a gas whose speed of sound differs sufficiently from that of the ambient air, because of density or temperature differences, would work.
> > > > Yes, for a reflective lens. A refractive lens is a different beast. > > > > -- > > > > Ioannis
> > > I don't understand why this isn't exactly the same maths (Snell's Law) as > > > for an optical lens, which AFAIK also determines a parabolic curve.
> > For sound waves, a reflecting or refracting medium is too often of a > > similar order of magnitude as the wave lengths, and Snell's law only > > works well when the wave lengths are much smaller that the reflecting or > > refracting media, as in light "waves".
> Then how do those parabolic dishes I see at football games work?
At a nearly perpendicular incidence to a hard surface, reflection of sound waves works fairly well even for waves fairly large with respect to the extent of that reflecting surface, but I'll bet those dishes pick up piccolos better than Sousaphones.
Mensanator wrote: > On Nov 19, 12:19 pm, "I.N. Galidakis" <morph...@olympus.mons> wrote: >> Mensanator wrote:
>> [snip]
>>> Wouldn't it simply be a parabolic dish?
>> Yes, for a reflective lens. A refractive lens is a different beast.
> How are you going to make a refractive lens for sound?
A refractive sound lens can be as Virgil says, any partial solid delimited by revolution and by two surfaces, a bottom radial surface and a top surface equal to the shape I derive placed at the right distance from the source.
Can we now address my question which is, why the two methods give different results? -- Ioannis
> > Yes, for a reflective lens. A refractive lens is a different beast. > > -- > > Ioannis
> I don't understand why this isn't exactly the same maths (Snell's Law) as > for an optical lens, which AFAIK also determines a parabolic curve.
The situation with a refractive lens is complicated by the fact that there are two surfaces.
I worked out a few cases in high school and if your lens has an index of refraction greater than one and you cancel out one surface by making it spherical centered at the source then the other surface becomes an ellipsoid. [Unless I f'd up that calculation many years ago]
For a source at infinity, this reduces to a flat surface toward the source and a paraboloid surface toward the receiver, so your conjecture fits that case correctly.
For a thin lens, the surfaces are always approximately spherical. Well, barring a Fresnel lens anyway.
>> > > > Yes, for a reflective lens. A refractive lens is a different beast. >> > > > -- >> > > > Ioannis
>> > > I don't understand why this isn't exactly the same maths (Snell's >> > > Law) as >> > > for an optical lens, which AFAIK also determines a parabolic curve.
>> > For sound waves, a reflecting or refracting medium is too often of a >> > similar order of magnitude as the wave lengths, and Snell's law only >> > works well when the wave lengths are much smaller that the reflecting >> > or >> > refracting media, as in light "waves".
>> Then how do those parabolic dishes I see at football games work?
> At a nearly perpendicular incidence to a hard surface, reflection of > sound waves works fairly well even for waves fairly large with respect > to the extent of that reflecting surface, but I'll bet those dishes pick > up piccolos better than Sousaphones.
Yes, because the "gain" of the "antenna" is proportional to the square of the aperture divided by the wavelength.
However, exactly the same considerations apply to lenses.
Snell's law does not have a term in it which relates to frequency. The derivation that I recall from 40 years ago was based upon velocity considerations, and not wavelength.
Despite the claims to the contrary in this thread, I can't see that sound waves are much different to EM waves with respect to considerations such as reflection, refraction, coherence and diffraction. Unless somebody can prove to me otherwise, I still believe that a sound lens would require the same profile as an EM lens, ie parabolic. (Whether such a device could actually be built which gives a worthwhile gain after the losses deriving from reflection off the various surfaces is a completely different question).
> Snell's law does not have a term in it which relates to frequency. The > derivation that I recall from 40 years ago was based upon velocity > considerations, and not wavelength.
Can you recall the derivation?
> Despite the claims to the contrary in this thread, I can't see that sound > waves are much different to EM waves with respect to considerations such as > reflection, refraction, coherence and diffraction. Unless somebody can prove > to me otherwise, I still believe that a sound lens would require the same > profile as an EM lens, ie parabolic. (Whether such a device could actually > be built which gives a worthwhile gain after the losses deriving from > reflection off the various surfaces is a completely different question).
gudi wrote: > On Nov 19, 9:12 pm, "I.N. Galidakis" <morph...@olympus.mons> wrote: >> I am trying to determine the shape of a so called "sound-lens", which is a >> device which can focus sound waves. I am using two methods, and they give >> different results, hence one (or both) must be wrong. > ---
> Pl. see, fwiw for light rays,earlier I derived a hyperbola lens > section whose eccentricity equals refractive index ( > 1 ) that brings > rays to a focus, with constant equal weighted path lengths.
The solution seems indeed to be a hyperboloid of revolution on the upper surface, facing the one focus and a radial concave surface on the lower surface, facing the second focus.
is the refractive analogue of the reflection principle for an ellipse. For an ellipse, if the the source coincides with one focus, the waves are brought together (by reflection) to the other focus.
For the hyperbola, its reflection principle does the same, albeit in a "refractive analogue".
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