Understanding Sound Waves: Properties and Characteristics
Audible Sound Waves
Sound: A longitudinal mechanical wave that propagates through an elastic medium.
A physical disturbance that propagates in an elastic medium such as air.
The sound spectrum is divided into three frequency ranges:
- Audible
- Infrasonic
- Ultrasonic
These intervals are defined as follows: the audible sound corresponds to sound waves in a frequency range of 20 to 20,000 Hz.
Sound waves having frequencies below the audible range are called infrasound.
Sound waves having frequencies above the audible range are called ultrasonic.
Sensory Effects – Physical Properties
- Force: Volume-Intensity
- Tone: Frequency
- Timbre: Quality-Waveform
Sound waves are energy flowing through the area. The intensity of a specific sound wave is a measure of how the energy is spread by a certain volume of space.
Since the amplitude of the current range that the ear is sensitive to, it is more practical to establish a logarithmic scale for sound intensity measurements, which is based on the following rule:
When the intensity I1 of a sound is 10 times greater than the intensity I2 of another, it is said that the intensity ratio is 1 bel (B).
In practice, unit 1 B is too large. For a more useful unit, we define the decibel (dB) sound as:
The logarithmic ratio of two quantities.
Therefore, the response of the example above can also be expressed as 76.8 dB.
The change in intensity varies with the square of the distance from the source. For example, a person placed at twice the distance from a sound source hears the sound at a quarter of the previous intensity, and a person far away three times hears the sound intensity at one-ninth.
If we consider that the sound radiates outward in all directions from a point source, the sound wave looks like a succession of spherical surfaces. Consider the points A and B located at distances r1 and r2 from a source that produces a sound power P.
Tone and Quality
The frequency of a sound determines what the listener hears as the pitch. Musicians designate tone by the letters corresponding to notes of the piano keys.
For example, the notes C, D, and F refer to specific tones or frequencies.
The sounds of the same pitch can be easily distinguished. Suppose that the notes do sound (256 Hz) on a piano, flute, trumpet, and violin. Even if each sound has the same tone, there is a marked difference in the quality or timbre of sound.
A note for each of these instruments has the same frequency (pitch), but all produce sounds very different due to different boundary conditions. The number and relative intensity of the harmonics affect the quality or timbre of sound.
The quality or timbre of a sound is determined by the number and relative intensities of the harmonics.
The difference in quality or timbre between two sounds can be seen objectively by analyzing complex waveforms resulting from each sound.
In general, the more complex the waveform, the greater the number of harmonics that contribute to the complexity.
The Origin of the Musical Scale
The current scale (Western scale) is the result of a long process of learning the notes. The Pythagoreans built a device called a monochord, which consisted of a table, a taut string, and a smaller table that was movable on the larger one.
The Pythagoreans observed that by making a rather long string (moving the movable table), there were different sounds. They chose some sounds that were harmonious with the original sound (whole string).
Most importantly, its simplicity and its importance in building the musical scale are:
- The Octave: When the string measures half the total, the sound is repeated, but sharper. The octave corresponds to a jump of eight white keys of the piano, or rather, one octave is a repetition of a sound with half the rope length, thus producing another harmonious note. Its frequency is twice the first.
- The Fifth: This interval between the notes is obtained with a string of length two-thirds of the initial one. Its frequency is 3/2 of the initial sound. It corresponds to a jump of five white keys on a piano.
- The Fourth: Like its predecessors, this interval between the notes is obtained with a string of length three-quarters of the initial one. Its frequency is four-thirds of the initial note.
The Musical Scale
Thus, from an original sound, we get different notes, which are shown below:
Assuming that the initial note is C, then the octave, fifth, and fourth are the notes corresponding to the fourth, fifth, and eighth grades, respectively, of the diatonic scale (the white keys of the piano). All these relationships between notes are called intervals.
The Construction of the Musical Scale
Ordering the octave of the frequency from smallest to largest:
Relating the frequency of a note to the previous one:
This is called the diatonic scale. It consists of 7 notes; the octave is the same as the previous one, an octave higher. They correspond to the white keys of the piano.
Now, if we used the room to find new notes, we would begin to get the “black keys” on the piano, i.e., sharps and flats. When the scale is complete with 12 notes (black and white keys), that is what is called the chromatic scale.
Interference and Pulse
Interference also occurs in the case of longitudinal sound waves, and the superposition principle can also be applied to them.
A common example of interference in sound waves occurs when two tuning forks (or any other sound source of a single frequency) whose frequencies differ slightly, are struck simultaneously.
The sound varies in intensity, alternating between strong and almost silent.
These are known as regular pulse beats.
The vibrato effect obtained in some organs is an application of this principle.
Each note’s vibrato is produced by two tubes tuned at slightly different frequencies.
Consonance and Dissonance
If the frequency difference is very small, then the pulse will be very slow and not perceived as a soft touch but a shell.
For example, if the frequencies are 440.1 and 440 Hz, the difference is 0.1 Hz, i.e., a pulse every 10 seconds.
In this case, since the vast majority of the notes used in music are much shorter than that, a pulse is not completed; it is more of a feeling of singing a more expressive sound.
If the beats are a bit faster, say 1 or 2 Hz, there is an effect called tremolo, like repeated notes. If they are much faster, for example, 5 or 10 Hz to about 50 Hz, the result produces a feeling of agitation commonly called dissonance.
Suppose, for example, a chord made up of two sounds of 220 Hz and 311 Hz (an A and a D#, respectively). It is known that the chord in music is dissonant. If we make the subtraction between the two frequencies, we obtain:
311 Hz – 220 Hz = 91 Hz
That is a touch too fast to cause a sensation of dissonance.
The condition for forming a chord of consonant sounds is that there is no significant interference between harmonics, i.e., intense sound of both.
Thus, we have the most perfect harmony, which is the unison (exactly the same frequency, since in that case, there is absolutely no pulse). Then follows the octave, which is when the sounds are in a frequency ratio of 2:1 (one sound is twice as frequent as the other).
Here, there is no possibility of clashes between sharper harmonics coinciding exactly with the more serious overtones.
Then follows the fifth, which corresponds to a frequency ratio of 3:2 (one sound has a frequency 1.5 times the other).
In this case, take for example the fifth formed by the 220 Hz and 330 Hz. In this case, they differ in successive harmonics by 110 Hz or more.
Sound Propagation
Inverse Square Law: Sets the intensity of sound in free field as inversely proportional to the square of the distance from the source.
Frequency and Amplitude: The number of cycles per second determines what is called the frequency.
Amplitude is the maximum separation or displacement of particles from their equilibrium position.
Auditory amplitude determines the loudness or intensity of sound.
Speed of Sound: When sound travels in the air, which is the medium in which the human ear works, it is usually 341 m/s at a temperature of 20 °C, with a factor of 0.6 m/s per degree Celsius.
Wavelength: Is the distance the sound wave takes to complete its cycle.
For example, for the wavelength of a sound wave of 200 Hz, we perform the following procedure:
Wavelength = 340 m / 200 Hz = 1.7 m.
A less frequent sound has a longer wavelength.
The higher the frequency, the shorter the wavelength.
Period: It is the time it takes for a wave to complete its cycle.
Obtained from the following formula:
Period = 1/Frequency.
A less frequent sound has a longer period.
A more frequent sound has a shorter period.
Phase: Phase is the time relationship of sound waves with respect to the time of initial reference.
With respect to the phase, it is important to emphasize two aspects:
- The polarity in the reproduction of sound waves.
- The delay in the reproduction of sound waves.
Harmonic Content: The characteristic that determines the timbre of a sound source is the harmonic content.
Harmonics are higher frequencies than the original (called the fundamental frequency) and have the characteristic of being multiples thereof.
Surround: This feature describes how the amplitude of sound varies with time since it is generated until it is extinguished.
The envelope consists of four stages: Attack, Decay, Sustain, and Release.
The attack is the time it takes for the sound to emerge until it reaches its peak.
The decay is the time required for the intensity to stabilize.
The sustain is the time when the sound intensity is stable.
The release is the time when the sound intensity decreases until it becomes inaudible.
Decibel: Bel is the relationship between two variables.
Bel = log(mag1/mag2)
dB = 10 log(mag1/mag2)
dB = log(w/Wref)
Calculation of Decibels: The dB indicates the relationship between two quantities expressed in a logarithmic scale (in watts, volts, SPL, etc.).
The mathematical equation of dB (in watts):
dB = 10 log(power1/power2)
Decibel Scale:
- Rule No. 1: (In power, watts) 3 dB indicates a 2 to 1.
- Rule No. 2: (In power, watts) 10 dB indicates a 10 to 1.
- Rule No. 3: (For pressure, volts, or SPL) 6 dB indicates a 2 to 1.
- Rule No. 4: (For pressure, volts, or SPL) 20 dB indicates a 10 to 1.
Decibel Scale Units:
- dBm
- dBW
- dBSPL
- dBu
- dBV
The dBm is related to electrical units with 1 milliwatt as reference, namely:
0 dBm = 0.001 watt (1 milliwatt).
This unit became standard in 1940 to be applied in telephone lines where resistance is 600 ohms.
This value (0.001 watts) is the power dissipated when 0.775 volts are inserted on a line with 600 ohms of resistance.
dBm = 10 log(power/0.001)
Its utility meters focus on electronic components (mixers, equalizers, crossovers, etc.).
dBW: The dBW relates units with 1 watt power reference, namely:
0 dBW = 1 watt.
This is implemented to reduce values in high-power specifications.
dBW on the scale when the meter reads 0 means that the unit produces 1 watt of power.
dBW = 10 log(power/1)
dBW scale is commonly used in power amplifiers meters.
dBu: 0 dBu = 0.775 volts.
dBu = 20 log(voltage/0.775)
dBu scale is used in dBm meters and electronic components (mixers, equalizers, crossovers, etc.).
dBV: The electrical unit is related to 1 volt reference.
0 dBV = 1 volt.
dBV = 20 log(voltage/1)
dBV scale is used in measuring dBu in electronic components (mixers, equalizers, crossovers, etc.).
dBSPL: Unlike previous scales (measuring electrical signals), the dBSPL scale measures sound waves.
dBSPL measures the intensity of sound waves and is governed under rule number 3 and rule 4.
0 dB SPL = 20 µPascal.
dB SPL = 20 log(pressure/20 µPascal).
The wide range of perception of the human ear is dBSPL, the scale between 0 and 120 dB SPL.
Examples:
While 100 volts is 10 times 10 volts, when we refer to the values of power which comes from the dB, we find that this represents a power ratio of 20 dB.
This is why the voltages are 2 times the multiplier before the equation log dB.
Absolute Levels vs. Relative Levels:
Example: “The maximum output level of the console is +20 dB.” This statement makes no sense because the zero reference for the dB is not specified. It’s like telling a stranger, “I can do only 20” without providing a clue what the 20 describes.
Example B: “The maximum output level of the console is 20 dB above 1 milliwatt.” This tells us that the console is capable of delivering 100 milliwatts (0.1 watt) to some load. How do we know that it can deliver 100 milliwatts? From the 20 dB expressed, representing the first 10 dB as a tenfold increase in power (1 mW to 10 mW), and the following 10 dB as another tenfold increase (from 10 mW to 100 mW).
Example C: “The maximum output level of the console is 20 dBm.” This example tells us exactly the same as Example B, but in other words. Instead of saying, “the maximum output level is 100 milliwatts,” we say +20 dBm.
Example D: “The maximum output level of the console is +20 dBm at 600 ohms.” This example tells us that the output is virtually the same as expressed in examples B and C, but gives us the information that the load is 600 ohms. This allows us to calculate the maximum output voltage at the load is 7.75 volts RMS, even when the output voltage is not given in the specification.
Example E: “The maximum output level is +20 dBu.” This example tells us that the maximum output voltage of the console is 7.75 volts, and we calculate for example D, but there is a significant difference.
The output in example D is at 600 ohms, while in example E, it specifies a minimum load impedance of 10,000 ohms. If this console was connected to a termination of 600 ohms, its output would probably fall in voltage and probably increase in distortion.
Example F: “The nominal output level is +4 dBV.” The two statements appear to be identical, but with a more detailed notice, the ancient use of the tiny “v” and the letter ‘v’ after the ‘dB’.
This means that the first specified output gives a nominal output of 1.23 volts RMS, while the second mixer specified level will deliver a nominal output of 1.6 volts RMS.
Example G: “The nominal output level is +4 dBV.” Both statements 1 and 2 are identical, although the latter is currently used. Both indicate that the nominal output level is 1.23 volts.
Converting dBu to dBV (dBm or through a line of 600 Ω):
Provided in the case of voltage (not power), you can convert dBV to dBu (or dBm through 600 Ω) by adding 2.2 dB to any value in dBV required. To convert dBu (dBm) to dBV, simply do the opposite operation, i.e., subtract the value 2.2 dB from dBu.
Exercise: Construct the following table:
Logarithmic and Exponential Forms:
Diffraction generates sound, which normally travels in a straight line, to deviate in other directions.
The wave fronts and rays travel in straight lines.
Sound rays traveling at right angles relative to the wave fronts except when something comes in their way.
The obstacles can cause the sound to change the direction of their original path.
The process by which this change of direction takes place is called diffraction.
For smaller wavelengths (higher frequencies), the phenomenon of diffraction is less noticeable.
An excellent example is when the waves pass by a small spring in which they have no problem passing it; however, trying to move an island, the waves will be returned.
Diffraction and Wavelength:
The effectiveness of a barrier to refract sound is determined by the acoustic size of the obstacle.
The acoustic size is measured in terms of the wavelength of sound.
One obstacle A may have the same physical size as another obstacle B, but the frequency of the sound system at a tenth of the frequency of sound B.
Diffraction of Sound Through Openings:
When the wave fronts of sound hit a heavy barrier, part is reflected and part continues through the opening. Arrows indicate some of the energy in the main bar is deflected.
What mechanism will accomplish these deviations?
The answer is the Huygens principle, which states:
Each point of the sound wave fronts that have passed through an opening or past the diffraction limit is considered a point source that radiates energy into the shadow zone (area behind obstacles).
The sound energy at any point in the shaded area can be obtained by summing the contributions of all point sources of wave fronts.
Diffraction of Sound by Obstacles:
Each wave front passing the obstacle becomes a line of new point sources that radiate sound into the shadow zone.
Diffraction of Sound Slot:
In this arrangement, the source slot, the radiometer measurement was at a distance of 8 meters. The slot width was 11.5 cm, and the wavelength of the sound measured was 1.45 cm.
The dimension B indicates the geometric boundaries of the beam.
Anything wider than B is caused by diffraction of the beam due to the slot.
A narrower slit diffraction generates greater and greater beam width.
The graph shows the sound intensity against the angle of deviation.
Diffraction Due to the Zone Plate:
The zone plate can be considered an acoustic lens.
It consists of a circular plate with concentric grooves cleverly planned with a radius.
If the focal point is at a distance R from the plate, the next longest path must be R + λ, where λ is the wavelength of the sound arriving on the plate from the source.
Significant lengths of paths are given by R + R λ + 3/2 + 2 R λ and these lengths differ by λ/2.
This means that the sound through all the slots arrives at the focal point in phase, which also means they will add constructively to intensify the sound.
Diffraction Around the Human Head:
This diffraction pattern due to the human head, as well as reflections and diffractions from the shoulders to the upper torso, affect human perception of sound.
In general, sounds of 1-6 kHz frequency coming from in front, the diffraction caused by the head tends to increase the sound pressure in front and decrease it behind the head. For frequencies in the lower range, the directional pattern tends to be circular.
Diffraction Due to the Edges of the Cabinets of the Speakers:
The sound that reaches the observation point is the combination of direct sound over the edge diffraction.
Fluctuations due to diffraction from the edge for this particular situation are about 5 dB.
This effect can be controlled by increasing the area in front of the baffle.
It is also possible to round the edges using a sponge or deletion.
Diffraction Due to Multiple Objects:
Old sound level meters (the edges of the boxes of sound level meters affecting the calibration of microphones).
Material mounted acoustic panels in the measurement of their absorption coefficient.
Small slits in recording studios that can destroy the insulation because the sound that emerges from the other side of the slot is spread in all directions by diffraction.
Refraction of Sound:
Refraction changes the direction of travel of sound due to differences in the speed of propagation.
An example of refraction of light is approaching or when inserting a rod into the water and depends on changes in the environment.
Refraction of Sound in Solids:
When two wave fronts arrive at a parallel surface, the sound rays are refracted at the interface of the two having different sound velocities, so they are no longer parallel.
Refraction of Sound in the Atmosphere:
In the absence of thermal gradients, a sound beam can be propagated in a straight line.
In a system where hot air is high and cold air is near the ground, and because sound travels faster in warm air than cold air, the top of the wave fronts travel faster than the low wavefront, causing the sound to propagate over longer distances.
In the event that the present system has warm air near the ground and cold air above, the lower parts of the wave fronts travel faster relative to the upper parts, which causes diffraction sound up, so the sound will travel shorter distances.
The sound travels straight up from the source S; the temperature gradient enters at right angles and is refracted. All sound except the vertical rays will be refracted downward. Near the vertical rays, the refracted rays are much less than those more or less parallel to the surface of the earth.
In the case where cold air is above, shaded areas are created. Again, the vertical beam is the only one that escapes from the refractive effects.
Against the wind creates a shady area, and downwind creates a refraction. Therefore, it is said that one experiences a better perception of sound in favor of the wind than against the wind.
The wind speed is usually lower near the surface of the earth at high altitudes.
Plane waves that travel downwind generated by a distant source sound into the earth will curve.
Plane waves traveling against the wind will be sent up.
Refraction of Sound in the Ocean:
In the 1960s, some oceanographers devised an ambitious plan to see how far they could detect underwater sound.
They detonated 600 pounds deep in the Australian waters.
The sounds of these discharges were detected near Bermuda.
Still thinking that the sound in water travels 4.3 times faster than in air, it took 3.71 hours for the sound to travel.
The distance is close to 12,000 miles, about half the circumference of the earth.
At the edge of the ocean, the speed of sound decreases with depth due to temperature changes.
Reflection of Sound in the Ocean:
The following figure illustrates the reflection of waves from a sound source of a rigid, flat wall. Spherical wave fronts (solid lines) hit the wall, and reflected wave fronts (dashed lines) are returned to the source.
Reflections from Flat Surfaces:
Analogy as the light/mirror, reflected wave fronts behave as if they originated from a sound image.
The image source is located the same distance behind the wall as if the real source was in front of the wall.
This is the simple case of a single reflecting surface.
In a rectangular enclosure, there are six surfaces, and the source has an image in the six sending energy back to the receiver.
Doubling the Pressure Reflection:
The sound pressure on a surface normal to the incident waves is equal to the energy density of radiation across the surface.
If the surface is a perfect absorber, the pressure equals the energy density of the incident radiation.
If the surface is a perfect reflector, the pressure equals the energy density of the incident and reflected radiation.
Therefore, the pressure in the face of a perfectly reflecting surface is twice that of a perfectly absorbing surface.
Reflections on Convex Surfaces:
Spherical wave fronts generated by a point source tend to become plane waves at the greatest distance from the source.
For this reason, the sound incident on the areas used will be considered as plane wave fronts.
The reflection of plane wavefronts of sound coming from a strong convex surface tends to scatter the sound energy in many directions. This is equivalent to the spread of incident sound.
The plane wavefront of a sound hitting a concave surface tends to be focused to a point as shown in the figure below:
The precision with which the sound is focused to a point is determined by the shape of the concave surface.
Spherical concave surfaces are common because they are easily manufactured.
They are often used to make a highly directional microphone by placing the focal point.
These microphones are often used to collect sounds from outdoor sporting events or recording the songs of birds or other animals in the wild.
The concave surfaces in churches or auditoriums can be the source of serious problems because they produce sound levels in direct opposition to the goal of achieving an even distribution of sound.
The effectiveness of the reflectors for microphones depends on the size of the reflector with respect to the wavelength of the sound.
A spherical reflector 3 ft in diameter gives good directivity at 1 kHz (wavelength of 1 ft), but is virtually non-directional at 200 Hz (wavelength of approximately 5.5 ft).
A parabola has the feature to focus the sound exactly to one point.
This is generated by the simple equation 
deep dish surface as shown, which exhibits directional properties much better than shallow.
Reflections in a Cylinder:
In this case, the source and receiver are both within a cylindrical enclosure with a massive rigid surface.
At the source, a whisper directed tangentially to the surface is clearly heard on the receiver side. The phenomenon is supported by the fact that the dome-shaped walls.
The Corner Reflector:
The corner reflector in the following figure receives sound from the source S, and it sends a direct reflection.
If the angles of incidence and reflection are carefully reviewed, a source in B also sends a direct reflection from the two surfaces.