Physics 1240 Notes Tues/Thurs 4/4 - 4/6/06 Wind and brass instruments, pipes Outline • Standing waves in pipes - open at both ends - closed at one end - conical bore • Reed instruments - Tone holes, register keys • Brass instruments - mouth piece/lips as a sound source - flared bell • Flute-type instruments (including recorders, flue organ pipes, etc.) Standing waves in a pipe open at both ends All brass and woodwind instruments involve long tube-like resonant cavities. Let’s look at a few very simple pipe systems. Which we will later see turn out to be simple models (and sometimes fairly accurate simple models) of a variety of instruments. First, the pipe open at both ends. The pipe open at both ends is actually a very good approximation of a flute and we will come back to flutes-type instruments on Thursday. Does this pressure node/anti-node pattern look familiar?The standing waves (or natural modes) in a pipe open at both ends are analogous to the natural modes on a string. Except now, it is a sound wave. Because the pipe ends are open the natural mode has a maximum displacement of the air at the open ends, and a minimum pressure fluctuation at the open ends. This makes sense, since the air is free to move at the open ends and the pressure cannot build up at the open end the way it can in the interior where the air is confined. This gives the node (minimum) and anti-node (maximum) patterns for pressure and displacement (motion) for the sound wave in the open ended pipe. The frequencies of the natural modes are identical to those of a string EXCEPT the wave speed is the speed of sound and not sqrt ( tension/ (mass/length) ). fn = nv/(2L) = n f1 , n = 1,2,3 ... v = 344 m/s. λ = 2L/n, The standing waves for a pipe open at both ends produces the full harmonic series (all integer multiples). Standing waves in a pipe closed at one end All pictures shown are from Hall (1990). At a closed end (fixed boundary) the motion of the air must go to zero. However, the pressure can build up. The open end is the same as before. This leads to the natural modes shown above and these natural modes have the following property fn = nv/(4L) = n f1 , n = 1,3,5 ... where v is the speed of sound as before. The clarinet has a cylindrical bore, up to the bell which is there primarily to increase the radiated sound intensity. The fact that the clarinet has missing even harmonics is important in giving it its unique sound. Conical Bore For a conical bore the pressure is a maximum at the narrow closed end and a minimum at the open end as shown.However, all frequencies are present. This is not obvious, but can be shown by solving the wave equation and is confirmed by experiment. Conical instruments, such as the saxophone and the oboe have a conical bore that causes these instruments to have all integer harmonics present. The larger conical bore on a saxophone is a more efficient radiator and is why the saxophone is a much louder instrument than say a oboe or clarinet. Reed Instruments The reed vibrates in resonance with the natural modes of the attached tube. The vibrating reed causes pressure pulses at rate in resonance with the tube. When the reed swings open the pressure is a maximum and the source of the pressure is the instrumentalist blowing into the mouthpiece. Then the reed swings closed and the pressure is a minimum, again in resonance with the natural modes in the tube. If you disconnect the mouthpiece it vibrates at quite a higher frequency and the frequency is sensitive to embouchure pressure on the reed. When you connect it to the instrument, you lock in on particular frequencies and it is very clear there is feedback controlling the oscillations of the reed between the reed and resonant cavity. For details on why a conical cavity has the natural modes it does see "The physics of Musical Instruments," Fletcher and Rossing (1998). Bore Holes What happens when I open a bore hole or key hole? In woodwind instruments, there are bore holes (or key holes) which effectively truncate the length of the instrument at the location of the hole if they are large enough. Smaller holes cause the frequency to go up, or the effective length of the tube to be shortened to a lesser extent. Register Keys/Holes What happens when I press a register key on a clarinet or saxophone? With a reed instrument, one can accentuate higher harmonics with a tight embouchure and blowing hard. However this is not the optimal way of playing high notes. A small hole placed at the pressure anti-node does a better job. The pressure maximum cannot besupported with the hole and overtones with pressure anti-nodes at the location of the hole are suppressed. In the cylindrical bored clarinet this causes a change in frequency of 3 ( a fifth interval plus an octave). In an conical bored saxophone or oboe it causes a change of frequency of 2. The octave change makes the fingerings of the notes the same. Brass Instruments The buzzing lips in the mouthpiece plays the role of the reed. The lips are more massive than the reed and the instrumentalist has a little more control over what frequencies are excited. This can be shown by comparing what a brass player can do with his mouthpiece versus the control a woodwind player has over only the mouthpiece. When the lips separate, a puff of air or pressure maximum sent into the horn and when the lips close a pressure minimum. It is the feedback of these oscillations with the pressure anti-node that allows the horn player to play particular tones. The brass mouthpiece/lips, like the reed behaves acoustically like a closed end. A horn without a bell will have overtones which are odd integer multiples of the fundamental (pipe closed at one end). Flared Bell Let's talk about a trumpet as a example. If there was not a flared bell the trumpet would have odd integer overtones. The flared bell (and to a lesser extent the mouthpiece) cause the overtones to be harmonic (almost). This is due to many years of artisans tinkering with various shapes and sizes for the bell. The original first natural mode is shifted upward and is not related in a harmonic way with the other normal modes which take on a frequency of 2f, 3f, 4f, ... with the fundamental missing. The missing fundamental can be played by careful control of the lips and heavily relying on the overtones, and is called the "pedal tone". The natural modes of flared bells in
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