FUNDAMENTALS of
ACOUSTICS
2.1 FREQUENCY AND WAVELENGTH
Frequency
A steady sound is produced by the repeated back and forth movement of an object at regular intervals. The time interval over which the motion recurs is called the period. For example
if our hearts beat 72 times per minute, the period is the total time 60 seconds divided by the number of beats 72, which is 0.83 seconds per beat. We can invert the period to obtain
the number of complete cycles of motion in one time interval, which is called the frequency.
f = 1
T 2.1
where f = frequency cycles per second or Hz
T = time period per cycle s The frequency is expressed in units of cycles per second, or Hertz Hz, in honor of the
physicist Heinrich Hertz 1857–1894.
Wavelength
Among the earliest sources of musical sounds were instruments made using stretched strings. When a string is plucked it vibrates back and forth and the initial displacement travels in
each direction along the string at a given velocity. The time required for the displacement to travel twice the length of the string is
T = 2 L
c 2.2
38 Architectural Acoustics
Figure 2.1
Harmonics of a Stretched String Pierce, 1983
where T = time period s
L = length of the string m c = velocity of the wave m s
Since the string is fixed at its end points, the only motion patterns allowed are those that have zero amplitude at the ends. This constraint called a boundary condition sets the
frequencies of vibration that the string will sustain to a fundamental and integer multiples of this frequency, 2f , 3f , 4f , . . . , called harmonics. Figure 2.1 shows these vibration patterns.
f = c
2 L 2.3
As the string displacement reflects from the terminations, it repeats its motion every two lengths. The distance over which the motion repeats is called the wavelength, and is
given the Greek symbol lambda, λ, which for the fundamental frequency in a string is 2 L. This leads us to the general relation between the wavelength and the frequency
λ = c
f 2.4
where λ = wavelength m
c = velocity of wave propagation m s f = frequency Hz
When notes are played on a piano the strings vibrate at specific frequencies, which depend on their length, mass, and tension. Figure 2.2 shows the fundamental frequencies
associated with each note. The lowest note has a fundamental frequency of about 27 Hz, while the highest fundamental is 4186 Hz. The frequency ranges spanned by other musical
Fundamentals
of Acoustics
39 Figure
2.2 Frequency Range of a Piano Pierce, 1983
Kategori