Acoustic Phenomena
Syllabus: Content to read
- Sound waves: Pressure amplitude
- Characteristics of sound: Intensity; loudness, quality and pitch
- Doppler's Effects
Acoustic phenomena
Definition:
These waves propagate through a
medium, such as air or water, and can be characterized by their frequency,
wavelength, amplitude, and speed. Acoustic phenomena can also be affected by
factors such as reflection, refraction, and interference.
Sound Wave: Pressure amplitude:
Pressure amplitude refers to the
maximum deviation of a sound wave from its average or equilibrium pressure. It
is a measure of the intensity or loudness of a sound, and is typically reported
in units of decibels (dB).
Derivation:
A longitudinal wave is a type of wave that can be visualized
as a sinusoidal wave, which has a specific frequency, wavelength, and
amplitude. These waves propagate in the x-direction and can be described
mathematically using the appropriate equations:
y=asin(ωt–kx) …… (i)
where, a= amplitude, ω=angular
frequency, k-wave number
In a longitudinal wave, the
displacement of particles is in the same direction as the wave's travel. This
means that the x-axis and y-axis are parallel to each other. The amplitude of
the wave refers to the maximum displacement of a particle in the medium from
its equilibrium position. This displacement amplitude is an important
characteristic of the wave that determines its intensity and energy.
Let ΔPΔ be the
instantaneous pressure fluctuation in a sound wave at a point x at time t. The
absolute pressure at that point is P+ΔP where P is the atmospheric
pressure.
Consider a cylinder of air of
cross-section area A and axis along the direction of wave travel as shown in
the figure. Volume of the cylinder having length Δx at normal condition
is V=A.Δx when there is waves. When a wave is present, left
cross-section of the cylinder displaces through a distance y1 and
the right cross-section through a distance y2. If y2 >
y1, the volume of the cylinder increases and this cause a decrease
of the pressure. If y2<y1, the volume of cylinder
decreases causing an increase in pressure. The pressure fluctuation depends on
the difference between the displacements at neighboring points in the medium.
So, the change in volume,
ΔV= A(y2-y1)
=A. Δy
In limit as Δx → 0, the fractional
change in volume dV/V is
\[\frac{{dV}}{V} = \mathop {\lim }\limits_{x \to 0}
\frac{{A.y}}{{A.x}} = \frac{{\partial y}}{{\partial x}}........(ii)\]
The pressure variation in the
cylinder due to fractional change in volume is given by
\[\begin{array}{l}\Delta P = - B\frac{{\Delta
V}}{V}.....(iii)\\Where\;{\rm{B\; is \;Bulk\; Modulus\; of\; air}}\\From{\rm{\; Equation
(ii)\; and (iii) \;We\; know:}}\\\Delta P = -
B\frac{{\delta y}}{{\delta x}} = Bakcos(\omega t--kx) \ldots (iv)\\{\rm{Let\; }}\Delta {{\rm{P}}_{\rm{m}}}{\rm{ = Bak}}\end{array}\]
The pressure amplitude which is the
maximum increase or decrease in pressure due to wave. The above equation can be
written as
\[\Delta P = \Delta Pmcos(\omega
t--kx) \ldots (v)\]
The compressions where the points
have lowest pressure and density are points of zero displacement and the rarefaction
where the points have lowest pressure and density are points of zero
displacement. Substituting the value B from equation
\[\begin{array}{l}{\rm{v =
}}\sqrt {\frac{B}{\rho }} \\The\;{\rm{ Pressure \; amplitude\; is\; given\; by\;}}\\\Delta
{{\rm{P}}_{\rm{m}}} = {\rm{Bak}} = {{\rm{v}}^2}\rho ka\end{array}\]
Thus, pressure amplitude is
directly proportional to the displacement amplitude and this amplitude is very
small.
Musical Sound and Noise
Musical sound: It refers
to a sound that has a discernible pitch and is usually produced by a musical
instrument or the human voice. It typically consists of a fundamental frequency
and harmonics that give it a distinctive timbre.
Musical sound has several characteristics, including:
1.
Pitch – Sharpness
and shrillness in sound is called Pitch. It is a perceptual attribute of sound
that describes the quality of a sound's highness or lowness, which is
determined by the frequency of the sound wave. High-frequency sound waves are
perceived as high-pitched, while low-frequency sound waves are perceived as
low-pitched.
Pitch | Frequency |
Pitch is a subjective perception of how high or low a
sound is. | Frequency is the objective measurement of the number of vibrations or
cycles per second of a sound wave. |
Pitch is measured in hertz (Hz) and is typically
represented on a musical scale. | Frequency is also measured in hertz (Hz) but is not necessarily
represented on a musical scale. |
Pitch is what we hear and perceive when we listen to
sound. | Frequency is a physical characteristic of sound that can be measured
with scientific instruments. |
The same frequency can be perceived as different
pitches depending on factors such as context, timbre, and individual
differences in hearing. | The relationship between frequency and pitch is not always
straightforward, as the perception of pitch can be influenced by many factors
in addition to frequency. |
2.
Loudness: Loudness
or intensity refers to the amount of sound energy that reaches the ear per unit
time. It is typically measured in decibels (dB) and is related to the amplitude
of the sound wave. The greater the amplitude, the higher the loudness. However,
the perception of loudness can also be affected by factors such as the
frequency content of the sound, the duration of the sound, and the sensitivity
of the human ear. For example, sounds at lower frequencies may be perceived as
louder than sounds at higher frequencies, even if they have the same amplitude.
Loudness | Intensity |
Perception of the sound's volume by the human ear | Physical measurement of the amount of sound energy per unit area |
Measured in decibels (dB) | Measured in watts per square meter (W/m²) |
Depends on both the intensity and frequency of the sound wave | Depends only on the intensity of the sound wave |
Can be influenced by the listener's age, gender, and hearing ability | Is not influenced by the listener's age, gender, or hearing ability |
Expressed in terms of sound pressure level (SPL) | Expressed in terms of sound power level (SWL) |
Subjective and varies from person to person | Objective and can be measured using instruments |
Used to describe the perceived loudness of musical sound and noise | Used to quantify the strength of sound waves in physical terms |
Noise: Noise refers
to a sound that lacks a discernible pitch and is often perceived as unpleasant
or unwanted. It can be produced by a variety of sources, such as machinery,
traffic, and crowds.
Difference Between Musical Sound and Noise:
Musical Sound | Noise |
Has a specific pitch | Does not have a discernible pitch |
Composed of a fundamental frequency and harmonics | Composed of a mixture of frequencies with no specific pattern |
Often has a rhythmic quality | Does not have a rhythmic quality |
Can be pleasing to the ear | Often considered unpleasant or annoying |
Typically produced by a musical instrument or the human voice | Can be produced by a variety of sources, including industrial and
transportation activities |
Used in music composition and performance | Can interfere with communication and daily activities |
Can convey emotional and artistic expression | Does not convey emotional or artistic expression |
Loudness of sound depends on following factors:
- Amplitude: The greater the amplitude of a sound wave,
the higher its loudness.
- Distance: The loudness of a sound decreases as the
distance from the sound source increases.
- Frequency: The human ear is more sensitive to sounds
in the frequency range of 2,000 to 5,000 Hz, so sounds in this range may
be perceived as louder than sounds of the same amplitude in other
frequency ranges.
- Duration: A sound that lasts longer may be perceived
as louder than a short, intense sound of the same amplitude.
- Acoustics of the environment: The loudness of a sound
can be affected by the acoustic properties of the environment, such as the
presence of echoes, reverberation, and other ambient sounds.
Quality or Timber:
Timber is the characteristic
sound of a particular instrument, voice, or other source of sound. It is what
allows us to distinguish between different sounds even when they have the same
pitch and loudness. Timbre is influenced by a variety of factors, including the
harmonic content, attack and decay characteristics, and other subtle aspects of
a sound wave.
Threshold of Hearing:
The threshold of hearing refers
to the lowest level of sound that can be detected by the human ear.
It is typically defined as a
sound pressure level of 0 decibels (dB), which corresponds to a sound wave with
an amplitude of 20 micropascals (μPa) at a frequency of 1,000 Hz.
However, the threshold of hearing
can vary depending on the individual's age, gender, and hearing sensitivity, as
well as the frequency and duration of the sound.
For example, younger individuals
and those with better hearing sensitivity may have a lower threshold of
hearing.
Intensity of Sound (I)
The intensity of a sound (I) can
be calculated using the following formula:
\[I = \frac{E}{{A \times t}}\]
where E is the energy of the
sound wave, A is the area through which the sound is passing, and t is the time
taken for the sound wave to pass through that area.
Intensity Level:
Intensity level is a measure of
the loudness of a sound, expressed in decibels (dB), which is a logarithmic
unit. It is defined as:
L = klog10(I)
where L is the intensity level in
dB, I is the intensity of the sound in watts per square meter (W/m²), and I0 is
the reference intensity of 10-12 W/m².
The reference intensity of 10-12
W/m² corresponds to the threshold of hearing, which is the minimum sound
intensity that can be detected by the human ear.
Inverse Square Law:
The inverse square law states
that the intensity of the quantity decreases in proportion to the square of the
distance from the source.
For example, if the distance
between a sound source and a listener is doubled, the intensity of the sound
wave will decrease by a factor of four. This is because the same amount of
energy is distributed over a larger area as the sound wave spreads out in three
dimensions.
Doppler Effects:
The Doppler effect is a
phenomenon in physics that describes the apparent change in frequency of a
wave, such as sound or light, when the source of the wave is moving relative to
an observer.
It can be expressed
mathematically using the following equation:
\[f' = \frac{{v \pm {v_o}}}{{v
\pm {v_s}}}\]
where f is the frequency of the
wave, f' is the perceived frequency, v is the speed of the wave in the medium,
vo is the speed of the observer relative to the medium, and vs
is the speed of the source relative to the medium. The $ \pm $ sign depends
on the direction of the relative motion between the source and observer.
Different Cases in Doppler Effects:
Let ‘v’ be the velocity of sound
‘λ’ be the wavelength of sound wave and ‘f’ be the frequency.
\[f = \frac{v}{\lambda }\]
1.
When source of sound moves towards the
Observer in rest
When the source of the sound
moves towards the static observer, wavelength of sound decreases which results to
increase in frequency of sound.
us= Velocity source
λ’ = changed wavelength
f= frequency of sound
since,
v>v- us
i.e. f’>f
So, frequency increases when source wave is
towards the observer in rest.
2.
When source of sound moves away from the
Observer in rest
When source of sound moves away from the static observer,
the wavelength of sound wave increases. Therefore, apparent change in wavelength
is given by:
\[\begin{array}{l}\lambda ' = \frac{{v + {u_s}}}{f}\\If{\rm{ f' be\; the\; apparent \;change \;in\; frequency\; }}\\{\rm{then,}}\\{\rm{f' = }}\frac{v}{{\lambda '}} = \frac{v}{{v + {u_s}}} \times f\\{\mathop{\rm Sin}\nolimits} ce,\\v < v + {u_s}\\i.e.\;{\rm{f' > f}}\end{array}\]
So, frequency decreases when source moves away from the observer in rest.
3.
When observer moves towards the source in
rest
When observer towards the source
in stationary then relative velocity of sound wave to the observer is v +uo.
So, frequency increases when
observer moves towards the source in rest.
4.
When observer moves away from the source
in rest
When observer moves away from the
source in rest then relative velocity of sound wave to the observer is v + uo.
So, frequency decreases when
observer moves away from the source in rest.
5.
When source and observer move towards
each other
When the source and observer are
approaching towards each other with the velocity us and uo respectively,
then
So, frequency increases when
source and observer towards each other
6.
When source and observer move away from
each other
When the source and observer move
away from each other with the velocity us and uo respectively,
then
So, frequency decreases when
source and observer move away from each other.
7.
When source leads the observer
When the source and observer move
in same direction and the source is leads the observer, then
So the frequency will change
depending on uo and us.
Limitations of Doppler effect:
The Doppler Effect is not
applicable in following conditions:
a. If the velocity of sound of
the source is greater than that of the sound because the wave gets
distorted due to which no change in frequency will be observed.
b. If the velocity of the sound
of the observer is greater than that of the sound.
Applications of Doppler effects
- Medical ultrasonography: In medical imaging, the
Doppler effect is used to measure blood flow and velocity within the body.
This technique is commonly used in obstetrics to monitor the fetal heart
rate and diagnose various conditions.
- Radar technology: Doppler radar is used to detect and
track the motion of objects, such as aircraft, ships, and weather
patterns. The radar sends out pulses of radio waves and measures the
Doppler shift of the reflected waves to determine the velocity and
location of the object.
- Astronomy: The Doppler effect is used to study the
motion of celestial objects, such as stars, galaxies, and planets. By
analyzing the Doppler shift of the light emitted by these objects,
astronomers can determine their velocity, distance, and other properties.
- Sound engineering: The Doppler effect is used in
sound engineering to create special sound effects, such as the sound of a
moving vehicle or a passing train, in movies, TV shows, and video games.
Infrasonic and ultrasonic sound
Infrasonic sound is below the
lower limit of human hearing (<20 Hz) and can travel long distances and
penetrate solid objects easily.
Ultrasonic sound is above the
upper limit of human hearing (>20 kHz) and has practical applications in
various fields such as medical imaging, cleaning, and material testing.
Application of Ultrasonic:
- Medical imaging: Ultrasound is commonly used for
diagnostic imaging in medicine, including obstetrics, gynecology,
cardiology, and radiology. It can be used to visualize internal organs,
monitor fetal development, and detect abnormalities.
- Cleaning: Ultrasound can be used for cleaning a wide
range of objects and surfaces, including jewelry, surgical instruments,
electronic components, and automotive parts. The high-frequency vibrations
can dislodge dirt and debris from hard-to-reach areas without causing
damage.
- Non-destructive testing: Ultrasound is often used for non-destructive testing in industry to detect flaws or defects in materials such as metals, plastics, and composites.