Can You Hear Sound in Room Corner by Door?

[RedX]

Say you have a square room with only one door, and the door is as the center of one side of the square. Suppose you put a stereo just outside the door. Can you hear the sound if you are standing in one of the corners of the room that's adjacent to the door?

Fraunhofer diffraction predicts that you can't, since the angular spread would have to be 90 degrees in order to reach that corner.

But obviously that's bogus and you can hear sound from such a corner.

So how is it that you can hear someone talking just outside the door if you are in the near corner?

 

[Matisse]

Reflection from the walls, floor and ceiling.

 

[RedX]

Reflection from the walls, floor and ceiling.

So if the room were infinitely long in the direction across from the door, then you would hear nothing in the corner adjacent to the door?

 

[Matisse]

No, because the sound does not travel horizontally but in all directions radially from the speakers. It travels up and down so it would reflect on the ceiling and floor. (This is not like a laser beam on a slit.)

Sound also travels on the floors, walls and ceiling and each vibrating surface acts like a source.

Other things to consider, the higher the frequency, the shorter the wavelength and the bigger the door compared to the wavelength, so diffraction then would be less and less important.

 

[Matisse]

A few other things do not work in the situation you described.

For diffraction, the Fraunhofer conditions states that the wave should arrive at the slit as a http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/sinint.html#c3". In your case, with the speaker so close to the door, this condition would be far from met.

More importantly, you cannot use as a source something involving many frequencies (like music or someone speaking). If you want to observe a diffraction pattern with sound, you need to use a pure sine wave, preferably with perfectly absorbing walls, floors and ceilings.

Temporal and spatial coherence is crucial for observing diffraction (one wavelength, one phase).

 

[Antiphon]

Energy diffracts 180 degrees anyway.

 

[Matisse]

What do you mean?

 

[RedX]

If the angle of reflection of sound is equal to the angle of incidence, and the room is infinitely long across the door, then sound waves are always moving forward so they will not get reflected back to the corners adjacent to the open door.

Suppose the speakers were placed outside the door instead of directly in front of the door, but still facing the door. Then all the sound rays arriving at the door should be parallel like a laser beam should they not? Just as all the rays from the sun arrive on Earth parallel.

Coherence is required to observe an interference pattern in sound waves. I'm taking the simpler case by only asking the question if you can hear anything at all, so coherence shouldn't matter.

 

[Matisse]

If the angle of reflection of sound is equal to the angle of incidence, and the room is infinitely long across the door, then sound waves are always moving forward so they will not get reflected back to the corners adjacent to the open door.

Yes.

Suppose the speakers were placed outside the door instead of directly in front of the door, but still facing the door. Then all the sound rays arriving at the door should be parallel like a laser beam should they not? Just as all the rays from the sun arrive on Earth parallel.

The speaker (one source) would have to be placed very far away from the door. Straight lines drawn from the source to each side of the door would have to be almost parallel.

Coherence is required to observe an interference pattern in sound waves. I'm taking the simpler case by only asking the question if you can hear anything at all, so coherence shouldn't matter.

Assuming that there are no surface that will vibrate at the frequency of the sound and therefore act as a secondary source or amplifier, I guess that you wouldn't. Which beg the question: How do you construct the door frame, walls, ceiling and floor. Is everything absorbing sound perfectly? Does sound-proofing affect the geometry of the experiment? Does it break the pure lines?

 

[RedX]

I was just curious as to why you can hear sound coming from a door when you are standing in the corner close to the door. So the materials of the frame, ceiling, walls and floor would be typical materials chosen for their strength and cost, and not their acoustic properties.

In reality the wall across the door is not infinitely far away, so reflection of sound at that back wall could be a possibility of why you hear sound at the corner of the front wall.

Or as you say it could be the multi-directional sound coming from the speakers, i.e., the speakers aren't collimated. However, even if this is true, there should be an obliquity factor that dampens the sound - this obliquity factor is usually ignored when considering Fraunhofer diffraction (the obliquity factor is what prevents backwards traveling Huygen wavelets). So really you shouldn't be able to get sound at an angle of pi/2 from the normal to the door, since the obliquity factor prevents that.

 

[Matisse]

I believe that you are thinking mostly in terms of specular reflection when diffuse reflection is very important for sound.

Take a tuning fork for example. Its sound is amplified many folds if you hold it close to a table or a wall, or even better if your let it be in contact with it.

Here is http://digitalcommons.unl.edu/cgi/v...ei-redir=1#search="diffuse+reflection+sound"" that discusses specular reflection in more details.

Sound energy that reflects from a boundary is classified as either specular or
diffuse. A specular reflection, like light reflecting from a mirror, bounces off the
surface with the same angle as it encountered the surface. A diffuse reflection occurs
when the sound energy is scattered into non-specular directions. Three mechanisms
of diffuse reflection are introduced below.

A clear light bulb transmits light directly from the filament through the glass
to objects beyond the bulb. In a frosted light bulb, however, many tiny irregularities
in the glass cause the light to be spread into all directions upon exiting the bulb. In
the same way, the roughness of an acoustical reflecting surface causes acoustical
waves to be spread out in all directions. Surface roughness is the first mechanism of
diffuse reflection.

The second mechanism of diffuse reflection is edge diffraction. Edge
diffraction accounts for a person's ability to hear a neighbor speaking on the other
side of a brick wall even though the neighbor cannot be seen. Edge diffraction can be
visualized as follows: at the boundary between the brick wall and the air at the top of
the wall, new secondary sources are formed that radiate sound spherically into the
space around them (see Figure 1). In this way, a straight-line path from the listener's2
ear to a sound source at the top of the wall is created. In room acoustics, edge
diffraction occurs whenever a sound wave encounters a change in the reflecting
surface. This can be a change in material, such as brick to air, or a change the
orientation of surfaces, such as at the boundary between a wall and the ceiling.
Figure 1. A sound wave undergoing edge diffraction creates a secondary source
at the boundary between the brick wall and the air.

A third mechanism of diffuse reflections is a wall treatment called a numerical
diffuser. The mechanism of diffuse reflections for numerical diffusers is neither
surface roughness, nor edge diffraction. A numerical diffuser is comprised of wells
of equal width with varying depth. Incident sound travels down the wells and
reemerges from each well with a different phase. The well openings then become
individual sound sources, which combine with one another to produce reflections in
nonspecular directions (see Figure 2). This is the third mechanism of diffuse
reflections.

 

[RedX]

Diffuse reflection sounds good, particularly surface roughness. I don't believe edge diffraction works for the case I have because of the obliquity/inclination factor. Not sure I understand a numerical diffuser. But diffuse reflection from rough surfaces, along with specular reflection from the back wall (and any objects in the room) sounds like good reasons why you can hear sound in the corner next to the door. Thanks.

 

[Matisse]

My pleasure. :)

 

[Antiphon]

What do you mean?

Energy will diffract into all directions at all times from say any sharp corner or obstruction. Fraunhofer is an approximate to the full diffraction phenomenon.

If you have an infinite half-plane and launch a semi-infinite wave moving along the upper part of the half plane, energy will diffract at the place where the half plane ends. This energy will propagate outwards in all directions including in the backwards direction (180 degree change of direction from the incident wave.)

So the sound would make it around the corner even if the angle were more than 90 degrees.

 

[RedX]

Energy will diffract into all directions at all times from say any sharp corner or obstruction. Fraunhofer is an approximate to the full diffraction phenomenon.

If you have an infinite half-plane and launch a semi-infinite wave moving along the upper part of the half plane, energy will diffract at the place where the half plane ends. This energy will propagate outwards in all directions including in the backwards direction (180 degree change of direction from the incident wave.)

So the sound would make it around the corner even if the angle were more than 90 degrees.

In the Fresnel-Kirchoff formula, there is an obliquity term:

\hat{n}\cdot(\hat{s}+\hat{r})

where n is the normal to the aperture, r is the distance from the source to the point you want to calculate the field, and s is the distance from the slit to the point you want to calculate the field. When s and r are colinear but in opposite directions, you get zero, so there should be no waves going backwards at 180 degrees.

You also get zero if n is at 90 degrees to s and r.

s is definitely perpendicular to n at the corner.

r might not be though if the source is placed far from the slit. If the stereo is placed close to the door, then r would make about 90 degrees with n.

In any case even if r is far away from the door, the edge effect ought to be really small. I don't know the size of it, but shadows from light shining on a half-plane are almost geometric. So I don't know how far sound can reach into the corner. Does anyone know of how far the shadow reaches into the geometric shadow for half-plane diffraction? Is it a wavelength or two?

 

[Matisse]

Energy will diffract into all directions at all times from say any sharp corner or obstruction. Fraunhofer is an approximate to the full diffraction phenomenon.

If you have an infinite half-plane and launch a semi-infinite wave moving along the upper part of the half plane, energy will diffract at the place where the half plane ends. This energy will propagate outwards in all directions including in the backwards direction (180 degree change of direction from the incident wave.)

So the sound would make it around the corner even if the angle were more than 90 degrees.

I have not done physics at a high level in a long time, but I do not recall being familiar with the expression. "energy diffracts". To be honest, it kinda shocks my sensibilities. I understand diffraction as a phenomenon happening when waves will interfere constructively or destructively and I can't imagine that happening to energy, a positive scalar(except fot potential energy which is another animal). Energy just adds up and dissipates.

What you say though reminds of edge diffraction, that I talked about earlier, where the boudary on a half-planes emits Huygen's wavelets in all directions that will combine to create spherical wavefront.

Attachment taken from the link above.

 

[Antiphon]

The Fresnel-Kirchoff approximation yields erroneous results in cases like this. The zero field propagation in the rear hemisphere is part of the approximation which is correct for the main beam (zero order diffraction) but not for the higher order terms (edge diffraction).

Sound will reach all the way around and propagate backwards.

Take a cement pipe and put a radio in it. You will hear it (but very quiet) when you stand behind the pipe.

 

[Antiphon]

Matisse: you're confusing interference with diffraction. Diffraction is what happens when energy scatters in a way which is not reflection, refraction or absorption.

I use the term energy to underscore the idea that this doesn't merely apply to sound or light but to all wave phenomena; water waves, etc. With the exception of QM, these diffracting fields all involve propagating energy.

Your attached diagram shows diffracted fields propagating in all directions from the edge which is correct.

 

[Matisse]

Antiphon -

I think I use the term diffaction correctly, at least I hope so since I teach introductory physical optics. Diffraction shows minima given by \delta_{}min = a sin \theta = m \lambda. If one is at the exact position of the minima the intensity should be zero (or very close to it).

One can prove the formula by imagining 12 rays passing through the slit at regular intervals and interacting deconstructively at a point on a screen (much larger than the slit) when \delta = m \lambda.

You said:

"Diffraction is what happens when energy scatters in a way which is not reflection, refraction or absorption. "

Thanks for clarifying. I knew you meant scattering and to me it's a much better word when applied to energy than diffraction.PS - I have problems with subscripts in Latex at home and here. Bizarre, I click the same button I used to click when I did my Masters with LaTex (the one with the black rectangle at the correct spot next to the empty box). It doesn't work and the indices do not come out the right size. Anyone know what is the problem. Sorry for being non-topical.