Chapter 6: Sound (C9953358)

1 Production of sound

Sound is produced by the oscillation of pressure in solids, liquids or gases. These vibrations can be detected by the organs of hearing.

 Formative learning activity Maps to RK6.1 How is sound produced?

2 Speed of sound

Another example of the speed of a wave (continuing from  list of examples), in a sound wave, the velocity is defined as $v=\sqrt{\dfrac{B}{\rho}}$, where $B$ is the bulk modulus (defined as the tendency of an object to resist deformation when uniformly compressed) (the elastic component), and $\rho$ is the density (the inertial component). For a sound wave in a gas, since $B=\gamma p$, in-substituting, $v=\sqrt{\gamma .\dfrac{p}{\rho}}$, meaning that pressure forms part of the elastic component. Also, since $PV=nRT$ and $\rho=\dfrac{m}{V}$ (introduced ), in-substituting, $v=\sqrt{\gamma .\dfrac{\dfrac{nRT}{V}}{\dfrac{m}{V}}}=\sqrt{\gamma.\dfrac{nRT}{m}}$. Also note that since $n=\dfrac{m}{Mr}$ (introduced ), or reshuffling, $Mr=\dfrac{m}{n}$, in-substituting, $v=\sqrt{\gamma .\dfrac{RT}{Mr}}$, where $\gamma$ is a constant, and $Mr$ is the molar mass of the gas. Note therefore that temperature is related to speed, meaning that sound travels faster in higher temperatures.

 Formative learning activity Maps to RK6.2 What is the speed of sound, and what does this number mean?

3 Intensity of sound

Remember from  that power is the rate at which energy is transferred. Intensity determines loudness of a sound, and is power per unit area, defined by $I=\dfrac{1}{2}\rho \omega^2 A^2 v$, where $\rho$ is density of the medium, $\omega$ is angular frequency $\omega =2\pi f$, $A$ is amplitude, and $v$ is velocity. For a spherical sound source, the intensity from a distance  from the center of the source is $I_{r}=\dfrac{P}{4\pi r^2}$. Note therefore, that intensity is inversely proportional to radius, such that as radius increases, intensity decreases. For example, if radius is doubled, intensity is quartered. Because the human hearing system amplifies low intensity sounds, and reduce high intensity sounds, perceptual differences are better described by the decibel (dB) system, which is defined as $L_{dB}=10.log(\dfrac{I}{I_{0}})$, where $I$ is the actual intensity, and $I_{0}$ is the threshold intensity of human hearing. The threshold of human hearing is the minimum sound level that an average ear of normal hearing can hear. Remember that because of the $log$ present, an increase of decibels by 10 (i.e. $+10$) is equivalent to an increase by intensity by $10^2$ (i.e. $\times 10$).

 Formative learning activity Maps to RK6.3 What does intensity of sound mean?

4 Attenuation

Attenuation is the gradual loss of sound intensity due to its transmission through a medium.

 Formative learning activity Maps to RK6.4 What is attenuation?

5 Doppler effect

Doppler effect is a change in [wave] frequency received by an observer as a result of moving relative to its source. The received frequency is higher when approaching, and lower when going away. When approaching, waves are compressed, thereby decreasing wavelength, and (because there is no change in medium, $\lambda \propto \dfrac{1}{f}$) thus increasing frequency. When moving away, wavelength is lengthened, thus decreasing frequency. The Doppler effect is determined by the formula $f_{r}=f_{s}(\dfrac{c \pm v_{r}}{c \pm v{s}})$, where $f_{r}$ is the frequency received by the receiver, $f_{s}$ is the frequency emitted by the source, $c$ is the velocity of the waves, $v_{s}$ is the velocity of the source (relative to the medium), and $v_{r}$ is the velocity of the receiver (relative to the medium). The key determinant is the signs of $v_{r}$ and $v_{s}$, which can be memorized with the mnemonic, of visualizing the source on the left, and the receiver on the right. Going left can be thought of as positive. Therefore, if the receiver is travelling towards the source, it is positive; and if the source is travelling away from the receiver, it is positive. Because of the complexity of this formula, the approximation (used when the speeds of the source and receiver are small compared to the speed of the wave) can be used, namely, $\dfrac{\Delta f}{f_{s}}=\dfrac{v}{c}$, where $\Delta f=f_{s}-f_{r}$, $v$ is the velocity of the receiver relative to its source ($v=v_{r}-v{s}$), and $c$ is the speed of the wave. An alternative modification is $\dfrac{\Delta \lambda}{\lambda_{s}}=\dfrac{v}{c}$. Relative velocity is positive when the source and receiver are moving towards each other.

 Formative learning activity Maps to RK6.5 What is the Doppler effect? How does it apply to sound?

6 Pitch

Pitch is the perception of sound, and is determined by frequency. It is thus the decibel equivalent [of intensity] for frequency. Higher pitches have higher frequencies, whereas lower pitches have lower frequencies. For example, comparing middle C with the G above, as G is a higher pitch, and has a higher frequency. Hearing range for humans is between 20 to 20,000 Hz, and this range declines with age.

“In high school, kids would put on that irritatingly high frequency sound that only kids can hear, and the teacher couldn’t,” Mandy said, “SO annoying!”

Remembering from  that since velocity dependent on the medium’s physical properties, wave velocity will thus be constant in a given medium. Beat is interference between two sounds of slightly different frequencies [and therefore wavelengths], which causes total constructive interference at some point (demonstrated by increased amplitude, and since $\propto A^2$ , therefore heard as increased intensity or loudness), and total destructive interference at some other point (demonstrated by, for vice versa reasons, softness), because of the differences in wavelengths. It sounds like periodic variations in volume, with a beat frequency equal to the difference between the two frequencies.

Piano tuners don’t just use the intuition of pitch to tune, but beats. Piano tuners play a note on the tuning fork and piano together, and hear for a beat. As stated , because the beat frequency is the difference between the two frequencies, the smaller the beat frequency, the closer the two notes. From  since $T=\dfrac{1}{f}$, the less the beat frequency, the greater the beat period. Therefore, pianos cannot really be perfectly tuned, because a period of infinity would be required (i.e. wait forever).

 Formative learning activity Maps to RK6.6 What is pitch?

7 Resonance

Phase difference is important only when considered in relation to a reference plane, in particular, another wave. Interference is where two waves superimpose to form a resultant wave, determined by the superposition of the vertical displacements of the individual waves, which is addition of the constituent waves at every point. Where there is a phase difference of $180^\circ$ out of phase, note that there is total destructive interference, meaning that the vertical displacement is smaller at every point. Where the constituent waves are in phase, there is total constructive interference, meaning that the vertical displacement is greater at every point.

Note that in spite of medium change, for example, a thicker rope connected to a thinner rope, a wave in order to remain continuous (strings connected), must have the same frequency. Therefore, in medium change, frequency remains the same, and only wavelength and speed change. Additionally, there may be reflection and/or refraction. Reflection is the change in direction of a wave at an interface between different media, such that the wave returns into the medium from which it originated. Note that a wave reflecting off a string clamped on the opposing side, on the end of the string, will try to cause a force in the direction the amplitude is directed. However, because it is clamped down, it cannot occur. However, by Newton’s 3rd law, an equal but opposite force is created to the amplitude, thereby creating a wave pulse similar to the initial one, but with opposite polarity (upside down). In contrast, if it is not clamped down, the wave simply reflects with the same polarity. Analogously, if moving to a denser medium (similar to a hard boundary), the wave will be inverted ($180^{circ}$ out of phase). In contrast, if moving to a less dense medium, the wave will be upright (in phase). Refraction is the change in direction of a wave due to change in medium.

Standing wave is a wave that remains in a constant position. It is caused by two waves of the same frequency, wavelength and amplitude, travelling in opposite directions. The effect is a series of nodes (total destructive interference) and anti-nodes (total constructive interference) at fixed points along the line. Where the waves meet each other (in the middle) at $\dfrac{\lambda}{2}$, there is a node. Note that by definition, because the starting ($0$) and ending points ($n\lambda$) are fixed, they are also nodes. On either side of the middle, at $\dfrac{\lambda}{4}$ and $\dfrac{3\lambda}{4}$, there are anti-nodes.

Because the start and end points of a string are fixed and thus nodes, standing waves must be periodic at these nodes. Note there are essentially infinitely small division of waves that are periodic at the start and end nodes. The wavelength from the start and end node is the fundamental frequency, which is the lowest frequency [and since velocity is constant in any medium see  $f\propto \dfrac{1}{\lambda}$, the longest wavelength]. The waves that are periodic at the fundamental frequency is known as resonance frequencies, which are frequencies that are an integer multiple of a fundamental frequency. So this would include 2f, 3f, 4f, etc.

 Formative learning activity Maps to RK6.7 What is resonance?

8 Harmonics

The corresponding wavelengths to the frequencies are known as harmonics. In-substituting into $\lambda=\dfrac{c}{f}$, the equivalent periods would be $\dfrac{c}{2f}, \dfrac{c}{3f}, \dfrac{c}{4f}$ or otherwise, $\dfrac{1}{2}\lambda, \dfrac{1}{3}\lambda, \dfrac{1}{4}\lambda$, etc. If both ends are nodes or antinodes, the harmonics can be found by $\lambda=\dfrac{2L}{n}$, where $L$ is the length, $n$ is any positive integer, and $\lambda$ is the wavelength of the harmonic. For example, the 1st harmonic ($n=1$) is $\lambda=2L$, the 2nd harmonic ($n=2$) is $\lambda=L$, the 3rd harmonic ($n=3$) is $\lambda=\dfrac{2}{3}L$, etc. If one end is a node, and the other end is an antinode, the harmonics can be found by $\lambda=\dfrac{4L}{n}$, where $n=1, 3, 5, etc$.

 Formative learning activity Maps to RK6.8 What are harmonics?

9 Ultrasound

Ultrasound are sounds that have a frequency greater than the upper limit of human hearing, which is approximately 20kHz+.

 Formative learning activity Maps to RK6.9 What is ultrasound?

# Assessment e-submission

(Formative assessments are not assessed for marks. Assessments are made on the unit level.

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