This is Part 1 of the presentation that was delivered at the Skunkworks Master Class 2021 Series. The presentation explains the fundamental physics of music and some of the practical aspects of how this applies to playing a musical instrument.
•Sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid;
•Sound waves travel through the conducting medium (e.g. air) as a compression wave (longitudinal wave);
•The sound waves enter our ear and impact on our ear drum and the vibration is transmitted to the cochlea which sends electrical signals to the brain;
•It is interesting to note that it is our brain that interprets these vibrations as sound. Some people suffer from a condition called Synesthesia which is a neurological condition in which information meant to stimulate one of your senses stimulates several of your senses. Synesthetes can often “see” music as colors when they hear it.
A transverse wave (such as a water wave) is where the displacment of the conducting medium (eg water) is perpendicular to the direction of the wave. In this example shown in this video each point on the rope is moving up and down but the wave is travelleing from left to right.
A longitudinal wave (such as a sound wave) is where the displacement of the conducting medium is parallel to the direction of travel of the wave. In this example the compression of the slinky spring represents the compression of, say, air molecules. The air molecules move parallel to the direction of travel of the wave (right to left in this example).
Refering to the diagram above:
From inspection of the formula for frequency above we can see the reason why brass and woodwind instruments get sharp on a hot day. The wave length is constant as it is fixed by the length of the instrument and therefore the increase in the speed of sound causes the frequency/pitch to rise. Conversely on a cold day the pitch will go flat.
Resonance is the phenomenon of a vibration increasing in amplitude when the frequency of a periodically applied force is equal to the natural frequency of the system on which it acts.
The string has a series of modes of vibration/resonance.
Another example of resonance: https://youtu.be/j-zczJXSxnw?t=6
Waves reflect from both a closed end and an open end. This is true for both waves on a string and sound waves in air. When a wave encounters a closed end it reflects with a 180 degree phase shift (i.e. it is inverted). When a wave encounters an open end it reflects with no phase shift.
•When we play a musical instrument we are setting up standing waves in the instrument;
•In the case of a string instrument the input to make the string vibrate is the bow or by plucking the string;
•In string instruments both ends of the string are fixed so that the waves set up on the string will reflect at either end of the string with a phase shift of 180 degress as shown on the previous slide;
•In a wind instrument it is pulses of air blown into the instrument through the lips or a reed.
•It is these reflected waves that set up the standing wave.
•This animation shows how standing waves are created;
•The incident wave (red) travels down the string from left to right;
•The incident wave is reflected off the fixed end but is inverted (blue);
•The incident wave and reflected wave interfere with each other constructively and destructively to form a standing wave (black).
Standing waves on a string can vibrate at many frequencies however all the frequencies of resonsonance are a multiple of the fundamental frequency, mode n=1.
In the same way as a string instrument, standing waves are created in a wind instrument except that the waves are longitudinal (not transverse as for a string);
While some of the energy of the standing wave is radiated from the bell as a normal sound wave, energy is also reflected back into the instrument as a reflected wave thus sustaining the standing wave within the instrument;
The need to constantly blow into the instrument (adding energy to the standing wave) is to replace energy losses due to sound energy radiating from the bell and also losses due to energy being absorbed by the walls of the instrument etc.
Standing waves in a wind instrument can be represented similarly to standing waves on a string except that the wave diagram represents air pressure at a point in the instrument compared with the general surrounding atmospheric pressure;
In this example of a closed tube instrument, e.g. clarinet, The open end (bell) is at atmospheric pressure and is shown as a node;
At the closed end (mouthpiece) the air pressure is constantly changing as you blow into the instrument (regular puffs of air regulated by the reed or lips for brass) and therefore the air pressure changes from high to low to high ..........
Alternatively the standing waves can be represented as the displacment of air. At the closed end of the tube (mouthpiece) the air can not move back and forth longitudinally because the end is closed. At the open end the air is free to move back and forth.
Animations of longitudinal standing waves in a closed tube are shown here: https://www.acs.psu.edu/drussell/demos/standingwaves/standingwaves.html
A flute is a tube that is open at both ends, i.e. the mouthpiece hole is open to the atmosphere;
A flute can resonate on the odd and even harmonics, n=1, 2, 3 ...;
The diagram to the left shows modes n=1, 2 & 3.
The frequency for these modes are:
n=1, f = v/(2L)
n=3, f=(3v)/(2L) where L = length of flute.
A clarinet is a tube closed at one end. This limits the harmonics that can resonate in the instrument to the odd harmonics, n=1, 3, 5, .......
The frequency for these modes are:
n=1, f = v/(4L)
n=5, f=(5v)/(4L) where L = length of clarinet.
Even though they are approximately the same length, the clarinet will sound approximately an octave lower that the flute because it has one end stopped which doubles the wave length of the standing wave (see above). A similar effect is used on organ pipes by stopping one end of the pipe and thus lowering the note by an octave. This video shows that a clarinet can play an octave higher if the mouthpiece end is open (by using a flute mouthpiece) and visa versa for the flute.
Brass instruments are also tubes closed at one end (the mouthpiece) The shape of, say, the trumpet is so designed so that the second and all higher resonances have risen so that they have frequencies in the ratios 2:3:4:5 etc. In other words, the resonances are a complete harmonic series, except for the fundamental. The lowest resonance of the trumpet is not a member of this series. Further, the spectrum of a pedal note has hardly any power at the fundamental frequency. What happens in the pedal note is that the higher resonances (2f, 3f, 4f etc) combine to help the lips establish a nonlinear vibration at the frequency of the missing fundamental f.
Use the app below to hear the shift in harmonics when a bell and mouthpiece are added.