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5 Acoustics in Biology and Medicine
of the wave. This speed can be found by dividing how far the wave moves by the
time taken to move this distance, which is its wavelength λ divided by the wave
period T , so v = λ/T . A surfer on such a wave can measure both the wavelength
(distance to the next bump) and period (time the water takes to move up and down
at each location), and so can easily get the speed of the wave he or she is riding.
The frequency f of a simple wave is the number of cycles per unit time, i.e. the
inverse of its period (which is time per cycle), making f = 1/T . 7 This means we
can calculate the speed of any simple wave by
v = f λ .
(5.19)
Because this relation comes from the definitions of the speed of a wave, its
characteristic wavelength, and its period, it is universally true for all types of waves,
including light.
In materials, coherent vibrations are meaningful only if the wavelengths are
longer than twice the separation of the atoms or molecules. For liquid water,
molecules are separated by about 3 Å. The speed of sound in water is about
1500 m/s, so sound frequencies in water above about 2.5 × 10 12 Hz are not possible.
Molecules in air at STP, standard temperature and pressure, 8 are about ten times
further apart than in liquid water, and the speed of sound about four and a half times
slower. This makes the maximum frequency in air about 6 × 10 10 Hz. As cells and
tissue contain a high percentage of water, the upper limit for sound frequency in soft
tissue is about 2 × 10 12 Hz.
Corresponding to frequency as the number of cycles per unit time, the ‘spatial
repeatance’ κ is defined as the number of cycles per unit length, which is the inverse
of the wavelength (length per cycle): κ = 1/λ. 9
This gives an alternate way of expressing Eq. (5.19), namely
f = v κ .
(5.20)
Sometimes, the ‘angular frequency’ ω = 2πf and angular wave number k = 2π κ
are introduced so that the sinusoidal wave can be more simply expressed as
ξ(x, t) = A sin(kx ∓ ωt + φ).
7 The physical unit for frequency is inverse seconds, but cycles per second is commonly used to
remind the listener what is being talked about. The SI unit for one cycle per second is a Hertz,
after the first sentient being to detect radio waves, almost 20 years after Maxwell predicted them.
A move away from unit clarity still Hertz!
8 Standard temperature and pressure (STP) are taken (since 1982) as 273.15 K and 1.00×10 5 N/m 2 .
Normal temperature and pressure (NTP), are T = 293.15 K and p = 1.01325 × 10 5 N/m 2 .
9 The spatial repeatance, κ = 1/λ, is commonly referred as the wave-number, even though this
terminology ambiguates with the angular wave number k = 2π/λ.
5 Acoustics in Biology and Medicine
of the wave. This speed can be found by dividing how far the wave moves by the
time taken to move this distance, which is its wavelength λ divided by the wave
period T , so v = λ/T . A surfer on such a wave can measure both the wavelength
(distance to the next bump) and period (time the water takes to move up and down
at each location), and so can easily get the speed of the wave he or she is riding.
The frequency f of a simple wave is the number of cycles per unit time, i.e. the
inverse of its period (which is time per cycle), making f = 1/T . 7 This means we
can calculate the speed of any simple wave by
v = f λ .
(5.19)
Because this relation comes from the definitions of the speed of a wave, its
characteristic wavelength, and its period, it is universally true for all types of waves,
including light.
In materials, coherent vibrations are meaningful only if the wavelengths are
longer than twice the separation of the atoms or molecules. For liquid water,
molecules are separated by about 3 Å. The speed of sound in water is about
1500 m/s, so sound frequencies in water above about 2.5 × 10 12 Hz are not possible.
Molecules in air at STP, standard temperature and pressure, 8 are about ten times
further apart than in liquid water, and the speed of sound about four and a half times
slower. This makes the maximum frequency in air about 6 × 10 10 Hz. As cells and
tissue contain a high percentage of water, the upper limit for sound frequency in soft
tissue is about 2 × 10 12 Hz.
Corresponding to frequency as the number of cycles per unit time, the ‘spatial
repeatance’ κ is defined as the number of cycles per unit length, which is the inverse
of the wavelength (length per cycle): κ = 1/λ. 9
This gives an alternate way of expressing Eq. (5.19), namely
f = v κ .
(5.20)
Sometimes, the ‘angular frequency’ ω = 2πf and angular wave number k = 2π κ
are introduced so that the sinusoidal wave can be more simply expressed as
ξ(x, t) = A sin(kx ∓ ωt + φ).
7 The physical unit for frequency is inverse seconds, but cycles per second is commonly used to
remind the listener what is being talked about. The SI unit for one cycle per second is a Hertz,
after the first sentient being to detect radio waves, almost 20 years after Maxwell predicted them.
A move away from unit clarity still Hertz!
8 Standard temperature and pressure (STP) are taken (since 1982) as 273.15 K and 1.00×10 5 N/m 2 .
Normal temperature and pressure (NTP), are T = 293.15 K and p = 1.01325 × 10 5 N/m 2 .
9 The spatial repeatance, κ = 1/λ, is commonly referred as the wave-number, even though this
terminology ambiguates with the angular wave number k = 2π/λ.