WAVES       MUSICAL INSTRUMENTS              STRINGS

How do we make musical sounds?

 To make a sound, we need something that vibrates. If we want to make musical notes, you usually need the vibration to have an almost constant frequency, that means stable pitch. Many musical instruments make use of the vibrations of strings to produce the notes.   Fig. 1.   Various string instruments. What do all these instruments have in common?  What factors determine the pitch of the sound made by plucking, bowing or striking a string?   The physics of the stringed musical instruments is very simple. The notes played depend upon the string which is disturbed. The string can vary in length, its tension and its linear density (mass / length).   Fig.2.   Sketch of a string instrument. String parameters: length , tension  and linear density .   ·       Length : different strings may have different lengths or the length of a string can be changed by using the fingers. ·       Linear density :  different strings have differing linear density ·       String tension : a string can be tuned by altering the string tension by using the tuning knobs.   It is only these three factors and how the string is disturbed that determines the vibrations of the string and the notes that it plays.   A string is disturbed and this sets up transverse waves travelling backward and forwards along the string due to reflections at the terminations of the string. The terminations act as nodes where the displacement of the string is always zero.   Only for a set of discrete frequencies, (natural or resonance frequencies of the string) can large amplitude standing waves be formed on the string to produce the required notes.  The frequency  of vibration of the string depends upon the wavelength  of the wave and its speed  of propagation          (1)                             vibration frequency of string   The speed  of propagation of the waves along the string depends upon the string tension  and the strings linear density  whereas the wavelength is determined by the length  of the string and the mode of vibration of the standing wave setup on the string.           (2)                 speed of a wave on a string   A steel piano string has a length of 0.400 m and a mass of 3.00x10-3 kg. A piano tuner adjusts the string tension to 800 N. What is the speed of the wave on the string?   Fig. 3.   Variation in the propagation speed with the string tension for a constant linear density.  When the piano tuner adjusts the frequency, they are making small changes in the propagation speed.   Fig. 4.   Variation in the propagation speed with the linear density for a constant string tension. The smaller the value of the linear density, the greater the speed.

 Fig. 5.   Propagation of pulses along two strings with different linear densities but under the same tensions.   What is the ratio of the linear densities for the two strings? Answer:                                 N.B. the different time scales for the blue and red pulses

 The Estonia-Minion, a small grand piano has 88 strings arranged over a sound board (figure 6). To excite a string to vibrate, it is struck with a hammer. The interaction when the string is struck with the hammer plays a very important role for the tone quality of the notes played. The low notes (bass) are played with the long strings and the high frequencies (treble) are played using the shorter strings.   Fig. 6.   Estonia-Minion Piano:  Position of strings over the soundboard. There are 88 strings. The longest string L1 plays a note of 27.5 Hz and the shortest string L88 plays the note 4186 Hz.   String L1         f = 27.5 Hz     L = 1239 mm     FT = 1350 N     = 0.2906  kg.m-3 String L26      f = 130.8 Hz   L = 831.2 mm     FT =  625 N     = 0.0167  kg.m-3

Reflection of waves at boundaries and standing waves

 The reflections of the waves at the boundaries of the string are a very important aspect in the music that is played by musical instruments.     Fig. 7.   Reflection of a pulse from a fixed end (node) and a free end (anitinode).               Fig. 8.   Reflection of a pulse:                        x = 0   fixed end (node) and  x = 100 m   fixed end (node).        Fig. 9.   Reflection of a pulse:                      x = 0   fixed end (node) and  x = 100 m   free end (antinode).     Fig. 10.   Colliding pulses. The superposition determines the resulting shape of the waveform of the two colliding pulses.  Note how the two pulses pass through each other unaltered. Node at x = 0 and an antinode at x = 100 m.       For the waves on strings, the boundary conditions are always fixed ends, therefore, upon reflection the wave is always inverted. The ends of the string correspond to nodes.  The initial disturbance of the string sets up waves that travel along the string and are reflected. The resultant waveform is determined by the superposition of the multiple waves travelling backward and forward along the string. The resulting oscillation can form standing waves. The positions where the oscillations reach their maximum values are known as antinodes. At points where the amplitude of the oscillation is zero are called nodes  these points do not oscillate.  Figure 11 shows the resultant standing wave produced by two waves travelling in opposite direction that have the same frequency and same amplitude.   Fig. 11.   Standing wave due to the superposition waves two that travel in opposite direction and have the same frequency and same amplitude.  What is the wavelength of the travelling waves? What is the distance between adjacent nodes? What is the distance between adjacent antinodes?   For a standing wave, the distance between adjacent nodes or adjacent antinodes is .

 Exercise  REFRACTION A pulse travels along a string and meets a discontinuity where there is a change in the linear density of the string. The string tension is uniform along the string. Carefully observe the animation. What is the value of x for the discontinuity? Estimate the velociites for the incident, reflected and refracted pulses? What is the change in phase of the reflected and refracted pulses?  At the discontinuity, does the linear density increase or decrease? Explain your answer? Estimate the ratio of the linear densities  Fig. 12.   Refraction of a pulse at a discontinuity due to a change in linear density of the string.   Answers ·       discontinuity at x = 60 m ·         ·       Reflected pulse:  rad change in phase (inverted) ·       Refraction pulse: 0 rad change in phase (upright) ·       Since the reflected pulse is inverted, discontinuity acts like a fixed end, therefore, the linear density must increase. ·                     from equation 2

Oscillations of a string

 When transverse oscillations are produced in a stretched string fasten at both ends, standing waves are setup. There must be nodes at the positions where the string is fastened. Hence, only oscillations are produced with appreciable amplitude when an integral number of half-wavelengths fit into the length of the string. This gives the condition          (3)                    condition for standing waves   Therefore, from equation 1, the frequencies of the standing waves that can vibrate with appreciable amplitudes are          (1)                                           (4a)            natural frequencies of vibration   The velocity  of the waves on the string depend only on the tension of the string  and linear density          (2)                              The frequencies  are called the natural frequency of a string.          (4b)           natural frequencies of vibration   The fundamental frequency  (n = 1) is the lowest frequency of oscillation          (5)                                   fundamental frequency   The natural frequencies are integral multiples of the fundamental frequency          (6)                                                          natural frequencies of vibration   The natural frequencies are also called the harmonics: ·       1st harmonic   n = 1    fundamental   ·       2nd harmonic   n = 2   ·       nth harmonic         The number n is called the mode number, and the value of n gives the natural or normal mode of the oscillation.   The oscillations of the string are remarkable in the respect that according to classical physics, we get discrete values of one of the quantities characterizing the oscillations  the frequency. Such a discrete nature is an exception for classical physics. For quantum processes, it is the rule rather than the exception.     Fig. 13.   Normal modes of vibration of a string fixed at both ends. The standing waves pattern has a series of nodes at fixed positions that are separated by the distance . The antinodes occur at positions of maximum displacement of the string, where each part of the string moves with simple harmonic motion. The period of oscillation is the time for complete oscillation in the shape of the waveform. The harmonic series of allowed frequencies is: .   A musical tone is a steady periodic sound. A musical tone is characterized by its duration, pitch (frequency), intensity (loudness), and timbre or quality.  Timbre or quality describes all the aspects of a musical sound other than pitch or loudness. Timbre is a subjective quantity and related to richness or perfection  music maybe heavy, light, murky, thin, smooth, clear, etc. For example, a note played by a violin has a brighter sound than the deeper sound from a viola when playing the same note (figure 14). A simple tone or pure tone, has a sinusoidal waveform. A complex tone is a combination of two or more pure tones that have a periodic pattern of repetition.   When the string of a musical instrument is struck, bowed or hammered, many of the harmonics are excited simultaneously.  The resulting sound is a superposition of the many tones differing in frequency. The fundamental (lowest frequency) determines the pitch of the sound. Therefore, we have no difficultly in distinguishing the tone of a violin and the tone from a viola of the same pitch  a different combinations of harmonic frequencies are excited when the violin and the viola the play same note (figure 14).   Fig. 14.   The sound recordings for a violin and viola playing the same note at a pitch of 440 Hz. The sounds from the two instruments have a different frequency spectrum. The violin has a richer sound because many more higher harmonics are excited.   The French mathematician Joseph Fourier discovered a mathematical regularity in periodic wave forms. He found that even the most complex periodic wavefunction can be dissembled into a series of sine wave components. The components correspond to sine functions of different frequencies and amplitudes and when added together reproduce the original wavefunction. The mathematical process of finding the components is called Fourier Analysis. Figure 14 shows the component frequencies for the sound recordings of the violin and viola.   Fourier synthesis is a method of electronically constructing a signal with a specific and desired periodic waveform from a set of sine functions of different amplitudes and that have a harmonic sequence of frequencies. Fourier synthesis is used in electronic music applications to generate waveforms that mimic the sounds of familiar musical instruments. Fig.  7. Fourier synthesis and an electronic music synthesizer.

Exercise

A guitar string is 900 mm long and has a mass of 3.6 g.  The distance from the bridge to the support post is 600 mm and the string is under a tension of 520 N.

1    Sketch the shape of the wave for the fundamental mode of vibration.

2    Calculate the frequency of the fundamental.

3    Sketch the shape of the string for the sixth harmonic and

calculate its frequency.

4    Sketch the shape of the string for the third overtone (fourth harmonic) and

calculate its frequency.

 f1 = 300 Hz     f6 = 1.8΄103 Hz     f4 = 1.2΄103 Hz

 Exercise A violin string plays at a frequency of 440 Hz. If the tension is increased by 8.0%, what is the new frequency?   Answer:  457 Hz

 Exercise A string has a mass per unit length of 2.50 g.m-1 and is put under a tension of 25.0 N as it is stretched taut along the x-axis. The free end is attached to a tuning fork that vibrates at 50.0 Hz, setting up a transverse wave on the string having an amplitude of 5.00 mm. Determine the speed, angular frequency, period, and wavelength of the disturbance.   Answers:  100 m.s-1, 3.14x102 rad.s-1, 2.00x10-2 s, 2.00 m

Stringed Instruments

 The natural frequencies of vibration of a string are given by          (4a)           natural frequencies of vibration   A stringed instrument is tuned by adjusting the tension of the string. This changes the speed of the transverse waves travelling along the string, hence changing the frequency of vibration. When a finger is placed on a violin string, the effective length of the string is shorter. So, its fundamental frequency (pitch) is higher since the wavelength of the fundamental is also shorter. The strings on a violin are all the same length. They sound at different pitches because the strings have different linear densities (mass / length), which affects the speed and hence frequency. The greater the linear density of a string, the smaller the speed and so the pitch is lower for the same length of string. In piano and harps, the strings are different lengths. For the lower notes, the strings are not only longer, but also heavier as well.

 Example Estonia-Minion Piano  the highest note played on the piano is 4186 Hz and the lowest note is 27.5 Hz.  If the string for the highest note is 45 mm long, how long would the string be for the lowest note if the two strings have the same linear density and under the same tension.   Solution The linear density and the tensions in the two strings are the same and for the fundamental n = 1            Therefore           where the subscripts L and H refer to the low note and the high note respectively. The answer of 6.8 m is ridiculously long for a piano  the longer strings for the low notes must be made heavier, so that on even grand pianos, the strings are less than 3 m long.

 If you have any feedback, comments, suggestions or corrections please email: Ian Cooper   School of Physics   University of Sydney ian.cooper@sydney.edu.au