Sound lens
I.N. Galidakis
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] (*)
Let's do it now parametrically.
A sound wave front before focusing:
x(t,Theta)=v_s*t*cos(Theta);
y(t,Theta)=RL+v_s*t*sin(Theta);
After focusing:
xp(t,Theta)=(t_c-t)*v_s*cos(Theta);
yp(t,Theta)=(t_c-t)*v_s*sin(Theta);
Therefore the lens shape must be the shape of the curve:
[X(t_0,Theta,-Theta),Y(t_0,Theta,-Theta)], with Theta\in [0,Pi] (**)
with:
X(t,Theta,Phi)=xp(t,Theta)-x(t,Phi)
Y(t,Theta,Phi)=yp(t,Theta)-y(t,Phi);
with v_s = 343m/s, R_L = 3, R_1 = 1, R_2 = 2, I am getting:
(*) h(x,t_0)=sqrt(4-x^2)-3+sqrt(1-x^2), x\in [-1..1]
so the lens is described by the vector
[x, sqrt(4-x^2)-3+sqrt(1-x^2)], x\in [-1..1]
while from the second method I get:
(**) [cos(Theta), 3*sin(Theta)-3], Theta\in [0..Pi]
(*) and (**) are not the same curves. Can anyone explain the discrepancy?
Many thanks,
--
Ioannis
Mensanator
Wouldn't it simply be a parabolic dish?
> --
> Ioannis
I.N. Galidakis
[snip]
> Wouldn't it simply be a parabolic dish?
Yes, for a reflective lens. A refractive lens is a different beast.
--
Ioannis
gudi
> 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.
Narasimham
Ken Pledger
"I.N. Galidakis" <morp...@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....
IIRC the expression "Cartesian oval" may be worth looking up.
Ken Pledger.
Peter Webb
"I.N. Galidakis" <morp...@olympus.mons> wrote in message
news:1258654768.128962@athprx04...
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.
Virgil
"Peter Webb" <webbf...@DIESPAMDIEoptusnet.com.au> wrote:
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".
Mensanator
How are you going to make a refractive lens for sound?
> --
> Ioannis
Mensanator
> 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?
Virgil
<2e5a6ffb-1771-47ed...@g27g2000yqn.googlegroups.com>,
Mensanator <mensa...@aol.com> wrote:
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.
>
> > --
> > Ioannis
Virgil
<de13b0a7-7ae5-44d7...@m16g2000yqc.googlegroups.com>,
Mensanator <mensa...@aol.com> wrote:
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.
I.N. Galidakis
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
jbriggs444
<webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
> "I.N. Galidakis" <morph...@olympus.mons> wrote in message
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.
Peter Webb
"Virgil" <Vir...@home.esc> wrote in message
news:Virgil-A40EAC....@bignews.usenetmonster.com...
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).
I.N. Galidakis
[snip]
> 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).
--
Ioannis
I.N. Galidakis
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.
My guess is that the hyperbola's "reflection principle" (See Geometrical
Properties)
http://en.wikipedia.org/wiki/Hyperbola
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".
> Narasimham
--
Ioannis