Violin

Grand Piano

Chapter 21: Musical Sounds

Musical sounds are simply combinations of sound waves with differing frequencies and quality. We list three features:

  • Pitch: the frequency of the sound wave
  • Loudness: the physiological reaction to the intensity of sound. Intensity is given as the square of the amplitude of the sound wave
  • Quality: how the fundamental frequency and its coterie of harmonics blend together

The physics concepts needed for this chapter have been presented already. The sections on Pitch and Loudness are well-presented in the text. Here, I spend most of my time on the last feature Quality, because of its richness in terms of the physics.

Flute

Clarinet

Standing Waves

Two waves moving in opposite directions interfere and set up a standing wave. The wave oscillates but it does not travel. We see that the points where the wave are zero ( Nodes) are stationary. The places where the amplitudes reach maximum and minimum amplitudes are anti-nodes.

How do Standing Waves Arise?

At the wall on the right, the wave's amplitude is forced to zero which causes the wave to flip upon reflection.

At the wall on the right, the wave's amplitude is allowed to float which causes the wave to bounce with the same sense of amplitude upon reflection.

The wave then interferes with itself to set-up the Standing Wave.

Different Modes of Oscillation (different wave frequencies). Again notice that the nodes are stationary. The crests peaks of the combined waves are Antinodes.

Modes of Oscillation (different wave frequencies) on a drumhead. There are now stationary lines where the wave amplitude is zero .

Strings

Strings are held at both ends forcing the amplitudes of propagating waves to go to zero. The top panel shows the lowest frequency mode, the Fundamental mode. The nodes are at the walls and the anti-node in the middle. A violinist can excite the fundamental mode by bowing at the fundamental node's anti-node (the middle of the string). A violinist can preferentially excite different modes by bowing at the anti-nodes of the higher frequency, higher harmonics, of the string (see the lower panel). Guitarists can also exicte different modes on a string as well by applying light pressure on the strings at different points.

In general, when bowing or plucking, the fundamental mode is excited along with the whole set of harmonics. The fundamental has the highest amplitude, but the harmonics add to the richness (the quality) of the sound.

In strings, the waves are supported by the material in the string; the wave speed is

speed = (Tension/mass per unit length)1/2

The Tension is determined by the material make-up of the string.

The wavelength of the mode is set by the length of the string, and so the speed determines the frequency (pitch) of the mode.

Wind Instruments

Wind instruments utilize pipes that can be open at one end and closed at one end, or open at both ends. However, note that a pipe open at both ends is rather simple because we have already met some of its features.

    Open at both Ends:

      The sound waves are pressure waves (representing compressions and rarefactions). So at the ends of the pipe, the air merges into the atmosphere, which is at atmospheric pressure. The pressure waves then reduce to the atmospheric pressure at the ends of the tube and so the amplitude of the pressure must drop to zero at the ends of the tube. This is like the plucked string. This is the situation for flutes. The wave with the longest wavelength is then 1/2 L, where L is the length of the tube. The harmonics series is then frequency = 1:2:3:4:5: ...

    Closed at one End:

      For a pipe closed at one end, the situation is different. At the open end the amplitude must again go to zero, but now, at the closed end, the pressure can be whatever excites the wave. This is the situation for say a clarinet. The wave with the longest wavelength is then 1/4 L. The harmonic series is then frequency = 1:3:5:7: ...

In wind instruments, the waves are supported by the air in the tube; the wave speed is

speed = (1.4 Pressure/density)1/2 = 343 m/s (Temperature in Kelvin/293 Kelvin)1/2

The wave speed is then simply determined by the temperature of the air. This is an issue because this means that the pitch depends fairly strongly on the temperature in the room.

The wavelength of the mode is again set by the length of the pipe, and so the frequency (pitch) of the mode depends mainly on the length of the pipe.

The harmonic content is determine by the boundary conditions on the pipe and so the quality is affected by whether the pipe is open or closed.