ACOUSTICS OF HORNS
You've seen the strange shapes of old gramophone speakers. You've seen the cacophonous musical instruments used by noisy marriage bands. You've heard unconvincing election campaigns blowing out of loudspeakers, policemen bellowing instructions over megaphones. But have you ever wondered why each instrument is shaped the way it is? Does its shape affect the way it sounds?
All the instruments mentioned above have one thing in common that is a circular tube of increasing cross sectional area called a HORN. However, there are two distinct functions that are performed by these horns, depending on where they are used: effective radiation over a range of frequencies in loudspeaker horns, and reinforcement of characteristic frequencies in musical horns.
Loudspeaker Horns
Two basic shapes used in designing loudspeaker horns are:
1.Exponential,
2. Hyperbolic,
where R0 is the radius of the small end of the horn and m is the flare constant which is a measure of how rapidly the radius of the horn increases. These horns are driven directly by a vibrating diaphragm at the small end.
Musical Horns
Wind musical instruments with horns can be broadly classified into the brasses, which include the French horn, the trumpet, the trombone; and the woodwinds, like the oboe, the piccolo and the clarinet. Such instruments consist of a mouthpiece, a main cylindrical bore and a flaring horn at their ends, where the sound is radiated into the surrounding space.
Mathematically, musical horns are best described as bessel horns, in which
Here b and x0 are constants which are determined by the minimum and maximum radii of the horn, and a is the flare constant.
The Horn Equation
The Webster Horn Equation describing plane waves propagating in a horn is
where x is the horn axis coordinate, c is the velocity of the sound wave in air, p is the pressure fluctuation, and S is the geometrical area of cross-section of the horn at any point. Replacing S by pR2, the differential equation for a simple harmonic pressure wave (p(x,t) = P(x) e-iwt) can be written as
To understand the propagation processes in a horn, we transform the Webster equation by defining a new variable Y= PS1/2, and we get
where U = R'' / R, which is a measure of the rate of flare of the horn wall. Now U, which is called the Horn Function, plays the same role in horn theory as does the potential energy function in quantum theory (Schrödinger Equation). In other words, U acts as a barrier to the propagating wave. The above equation has oscillatory (propagating type) solutions for k2 >U, and exponentially decaying (non-propagating type) solutions for k2
If U is not constant along the horn, then different frequencies get reflected at different points, as shown for musical horns. This means that the cylinder-horn combination behaves as a cylinder of different effective length for different frequencies.
The source of vibration in both loudspeakers and wind musical instruments with reeds corresponds to a constant velocity fluctuation. The response of the horn, i.e. the pressure fluctuation, is proportional to the acoustic impedance at the input. Thus by studying the variation of input impedance with frequency, we can determine the acoustic behaviour of the horn. (See box below.) We have considered the following simplified model of the horn for all our calculations. For musical wind instruments we have a cylindrical tube that is attached to a flaring horn.
|
Acoustic Impedance Any medium through which waves travel offers impedance (or opposition) to those waves. For sound waves, the Acoustic Impedance (Z) offered by an elastic medium is defined as Pressure due to wave Z = ----------------- Particle velocity This quantity is analogous to electrical impedance, which opposes the flow of current in an electrical circuit. When a sound wave reaches a boundary separating two regions of different acoustic impedance, it is partially reflected and partially transmitted. The same thing occurs for light and in general for any kind of wave. In horns, sound waves travelling from the small end get reflected near the larger open end. |
This terminates in a large cone that simulates the room in which the instrument is "played". The cylindrical section is omitted for loudspeaker horns.
Arbitrary initial conditions are set up and the horn equation, slightly modified by the inclusion of a damping term, is numerically integrated. Input Impedance and Power Reflection Coefficient (which determines how much of the incident energy is reflected) are calculated for a range of frequencies. The results are shown in the next panel.
|
Resonance In general, whenever a system capable of oscillating is acted on by a periodic series of impulses having a frequency equal or nearly equal to one of the natural frequencies of oscillation of the system, the system oscillates with a relatively large amplitude. This phenomenon is called resonance, and the system is said to resonate with the applied impulses. This phenomenon is seen in daily life too. A play-swing pushed at just the right frequency oscillates with ever larger amplitudes, much to the delight of its occupant. In musical horns, resonance occurs when the reflected wave combines constructively with the incident wave. Frequencies for which this occurs are known as resonant frequencies, and maximum build up of sound wave amplitude takes place inside the horn. (See computer display.) |
The difference between musical and loudspeaker horns is evident from the input impedance graphs. The presence of large peaks for the musical horn illustrates the fact that energy build up takes place only at certain frequencies. This happens because for certain frequencies, the wave reflected back into the cylinder on reaching the input end combines constructively with the wave sent in by the source. For other frequencies, this kind of reinforcement doesn't take place, and they don't build up. (See box on resonance.) The resonant frequencies of the air column can be identified by the peaks of the input impedance curve. Also, the amount of energy leaking out gradually increases with frequency (as can be seen from the corresponding decrease in the power reflection coefficient), because the barrier becomes thinner. This results in weaker resonances, as is evident from the decrease in height of the impedance peaks. The loudspeaker horn shows a markedly different behaviour. The input impedance curve illustrates that its response is almost uniform for all frequencies above a certain cut-off. Very little power is radiated at frequencies below this; but at those above the cut-off, the power reflection coefficient drops quickly to zero. This implies that almost all of the energy is transmitted, which is what a loudspeaker is supposed to do.
|
The Musicality of Horns The resonant frequencies of a cylinder closed at one end are odd multiples of some fundamental frequency. In a musical horn, the input of the cylindrical portion behaves like a closed end, because the pressure is maximum there (i.e. it is an antinode). But due to the fact that the effective length of the instrument increases with frequency, the higher harmonics fall below the odd number sequence. So we find that all the harmonics except the fundamental are present in the spectrum, resulting in rich musical sound. |
|
|
|
Copyright © 2022 ICICI Centre for Mathematical Sciences
All rights reserved. Send us your suggestions at