# Relation between the direction of propogation of sound and loudspeaker's diameter

Sound waves from a conical loudspeaker spread nearly uniformly in all directions if the wavelength of the sound is much larger than the diameter of the loudspeaker.

Sound is essentially transmitted in the forward direction if the wavelength is much shorter than the diameter of the speaker.

Q. What is the explanation behind this phenomenon and why the diameter is significant for the direction of propagation of wave?

Research:

This seems to be a result of diffraction. Diffraction: the bending of waves around small* obstacles and the spreading out of waves beyond small openings.(*small compared to the wavelength)

Q.But where does the bending occur?

Q.And what is the obstacle here?

I guess that bending occurs at the edges of loudspeaker

Q.What does the bending has to do with the loudspeaker's diameter?

If I hold my previous assumption correct,the thickness of edges should only matter and should be responsible for diffraction.

• You will basically have to solve the acoustic wave equation with the motion of the membrane as a boundary condition. The "diffraction" occurs on the entire shape of the membrane and speaker, if you will. – CuriousOne Feb 16 '16 at 7:48
• Could you please explain at high school level? – user106015 Feb 16 '16 at 8:06
• Imagine that every point on the membrane is the source of a spherical wave and all these waves spread out and are interfering with each other. Where the waves hit a reflecting surface they create new spherical waves, which are also adding to the interference. If, in the most ideal scenario, you could make a perfectly spherical membrane, then the sound field would be spherical. The second useful shape would be a cylindrical membrane with infinite length to make a cylindrical sound field. Neither is technically possible, so we get unwanted interference from the actual shapes. – CuriousOne Feb 16 '16 at 8:58

It is a diffraction effect.
Working out the wavelengths you get 16.5 cm for the 2000 Hz source and 330 cm for the 100 Hz source.
The amount of diffraction becomes significant when the aperture size becomes comparable to the wavelength which is very much the case in your experiment.

If you want to discuss some theory then you might consider using a simple derivation to explain the reason for more spreading if the wavelength is larger given the same aperture size.

Here you have a vibrating membrane $XY$ producing sound waves.
Every part of the membrane produces sound waves.
Splitting the membrane in half $YZ$ and $YZ$ and considering the waves produces by a point on the membrane in the upper half $A$ and the waves from a corresponding point in the lower half $A'$,
The waves leave the membrane in phase.
At a certain angle $\theta_1$ the waves from these two point sources at your ear $BB'$ exactly out of phase and so there is no sound in that direction. The sound waves are concentrated within directions $\theta_1$ and $-\theta_1$.
For a larger wavelength (right hand diagram) the angular spread is larger.

The second animation in this link might get the idea of the range of directions of the waves being larger as the wavelength decreases.

The effect of loudspeaker size can be analysed in a similar way.

People spend a lot of money buying loudspeakers and this has meant that a lot of research has been done on making them better.

The HyperPhysics site has a nice introduction to loudspeakers and there you will find some nice nice directional pattern plots.