Sound waves

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Introduction

In this post I tried to give some basic background about sound waves (a ‘disturbance’ traveling through a medium such as air; the medium itself does not travel with it) resulting in the derivation of the wave function, the velocity of the wave through a medium, velocity of particles inside a medium, and finally wave energy, power, and intensity.

I mainly copied text and figures from University Physics Volume 1 (Moebs et al, 2021; see References below) but also from some few other websites that I refer to below. A such this page does not contain any new information and, in fact, there there are many other excellent (and better) resources about this topic. Nevertheless, by putting some details together one a single page I hope this helps in understanding some concepts of sound waves. I didn’t try to be complete and left out many details.

Sound waves

In our context we consider a speaker (or vocal cords) producing sound waves by oscillating a cone, causing vibrations of air molecules. In Figure 1, a speaker vibrates at a constant frequency and amplitude, producing vibrations in the surrounding air molecules. As the speaker oscillates back and forth, it transfers energy to the air resulting into compressing and expanding the surrounding air, creating slightly higher (HP) and lower (LP) local pressures. These compressions (high-pressure regions) and rarefactions (low-pressure regions) move out as longitudinal pressure waves having the same frequency as the speaker. Sound waves in air and most fluids are longitudinal because fluids have almost no shear strength. In solids, sound waves can be both transverse and longitudinal)

Thus, although we usually visualize sound waves are sinus-like waves (in programs like Cubase, Wavelab, etc) these are actually three dimensional pressure waves.

Figure 1(a) shows the compressions and rarefactions, and also shows a graph of gauge pressure versus distance from a speaker (assuming no damping). As the speaker moves in the positive x-direction, it pushes air molecules, displacing them from their equilibrium positions. As the speaker moves in the negative x-direction, the air molecules move back toward their equilibrium positions due to a restoring force. The air molecules oscillate in simple harmonic motion (explained below) about their equilibrium positions, as shown in part (b). Note that sound waves in air are longitudinal, and in the figure, the wave propagates in the positive x-direction and the molecules oscillate parallel to the direction in which the wave propagates. See also animation in Figure 2 and the Youtube videos below.

Figure 1 (a) A vibrating cone of a speaker, moving in the positive x-direction, compresses the air in front of it and expands the air behind it. As the speaker oscillates, it creates another compression and rarefaction as those on the right move away from the speaker. After many vibrations, a series of compressions and rarefactions moves out from the speaker as a sound wave. The red graph shows the gauge pressure (P) of the air versus the distance from the speaker. Pressures vary only slightly from atmospheric pressure for ordinary sounds. Note that gauge pressure is modeled with a sine function, where the crests of the function line up with the compressions and the troughs line up with the rarefactions. (b) Sound waves can also be modeled using the displacement (S) of the air molecules. The blue graph shows the displacement of the air molecules versus the position from the speaker and is modeled with a cosine function. Notice that the displacement is zero for the molecules in their equilibrium position and are centered at the compressions and rarefactions. Compressions are formed when molecules on either side of the equilibrium molecules are displaced toward the equilibrium position. Rarefactions are formed when the molecules are displaced away from the equilibrium position.

Figure 2. This figure is copied from the Institute of Sound and Vibration Research (ISVR; Southampton, UK). This animation clearly demonstrate the nature of a acoustic longitudinal wave.

Below, I give a more mathematical description of sound waves starting with the definition of angular frequency.

Angular frequency

In physics, angular frequency ( $\omega$ ) (also referred to as angular velocity) is defined as $\omega = \frac{2 \pi}{T} = 2 \pi f$ since $f=1/T$ , and has units of $s^{-1}$ . Here T is the period of the sinusoid in seconds and $f$ is the ‘ordinary’ frequency in Hertz (or $s^{-1}$ . Figure 3 and 4 shows the relation between the angular frequency and a sinusoid wave, which follows from the well-known rule $sin(\omega) = \frac{opposite}{hypotenuse}$ where opposite is the position of the green dot. Thus, $opposite = hypotenuse \times sin(\omega)$ . Thus, the hypotenuse can be considered to be the amplitude of the sinus.

Assume that the red dot on the circle moves counter clock-wise with a constant velocity. The green dot is the projection of the red dot, which moves up/down if the red dot moves around the circle. One revolution implies that $\omega$ moves from 0 to $2\pi$ radians ( $=360^{\circ}$ ). Plotting the movement of the green dot on a linear scale from 0 to $2\pi$ gives a sinusoidal oscillation. The number of cycles through which the green dot goes up and down in a second is the oscillation frequency of the sinusoid. Thus, angular velocity of the red dot can be measured in radians per second.

Figure 3. Angular frequency. Copied from Quora.

Figure 4. Animation showing how the sine function (in red) is graphed from the y-coordinate (red dot) of a point on the unit circle (in green), at an angle of θ. Copied from Wikipedia.

Simple harmonic motion

(Sound) waves can be described as a simple harmonic motion. For a full description and derivation of the formula below I refer to Chapter 15.1 of University Physics Volume 1 (Moebs et al, 2021). A nice explanation is given by the next video.

Here I summarize few relevant points.

When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time. The string vibrates around an equilibrium position, and one oscillation is completed when the string starts from the initial position, travels to one of the extreme positions, then to the other extreme position, and returns to its initial position. We define periodic motion to be any motion that repeats itself at regular time intervals, such as exhibited by the guitar string.

In simple harmonic motion, the net force ( $F$ ) is proportional to the displacement ( $x$ ) and acts in the opposite direction of the displacement: $F_s = -kx$ (Hooke’s law). After doing the math, it follows that the position ( $x$ with a maximum of amplitude $A$ ) as function of time ( $t$ ) for a block on a spring becomes

$x(t) = A cos(\omega t + \phi)$

Here $\omega$ is the angular frequency and $\phi$ represents the phase shift to account for the fact that the oscillation does not always have to start from $x=0$ and $v=0$ (see Figure 5). We will find a similar formula below when mathematically describing mechanical waves.

Figure 5. (a) A cosine function. (b) A cosine function shifted to the left by and angle $\phi$ (phase shift).

Mechanical Waves (sound waves)

A wave is a disturbance that propagates, or moves from the place it was created. There are three basic types of waves: mechanical waves, electromagnetic waves, and matter waves. Here I only consider mechanical waves of which a sound wave is an example.

Basic mechanical waves are governed by Newton’s laws and require a medium. A medium is the substance a mechanical waves propagates through, and the medium produces an elastic restoring force when it is deformed (see video about simple harmonic motion). Mechanical waves transfer energy and momentum (quantify of motion), without transferring mass. Some examples of mechanical waves are water waves, sound waves, and seismic waves. The medium for sound waves is usually air. The disturbance is the oscillation of the molecules of the air. In mechanical waves, energy and momentum transfer with the motion of the wave, whereas the mass oscillates around an equilibrium point.

Mechanical waves exhibit characteristics common to all waves, such as amplitude, wavelength, period, frequency, and energy. All wave characteristics can be described by a small set of underlying principles.

The simplest mechanical waves repeat themselves for several cycles and are associated with simple harmonic motion. These simple harmonic waves can be modeled using some combination of sine and cosine functions. For example, consider the simplified surface water wave that moves across the surface of water as illustrated in Figure 5. In this case water moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. The waves causes a seagull to move up and down in simple harmonic motion as the wave crests and troughs (peaks and valleys) pass under the bird. The crest is the highest point of the wave, and the trough is the lowest part of the wave. Recall:

Time for one complete oscillation of the up-and-down motion is the wave’s period T. The period can be expressed using any convenient unit of time but is usually measured in seconds.
The wave’s frequency (f) is the number of waves that pass through a point per unit time and is equal to $f=\frac{1}{T}$ $f = 1 / T .$ Frequency is usually measured in hertz (Hz), where $1Hz=1s^{-1}$ .
The length of the wave is called the wavelength ( $\lambda$ ), which is measured in any convenient unit of length, such as a centimeter or meter.
The amplitude (A) of the wave is a measure of the maximum displacement of the medium from its equilibrium position.

The wavelength can be measured between any two similar points along the medium that have the same height and the same slope. In Figure 6, the wavelength is shown measured between two crests. As stated above, the period of the wave is equal to the time for one oscillation, but it is also equal to the time for one wavelength to pass through a point along the wave’s path.

In the figure, the equilibrium position is indicated by the dotted line, which is the height of the water if there were no waves moving through it. In this case, the wave is symmetrical, the crest of the wave is a distance $+A$ above the equilibrium position, and the trough is a distance $-A$ $− A$ below the equilibrium position. The units for the amplitude can be centimeters or meters, or any convenient unit of distance.

The water wave in the figure moves through the medium with a propagation velocity $\overrightarrow{v}$ . The magnitude of the wave velocity is the distance the wave travels in a given time, which is one wavelength in the time of one period, and the wave speed is the magnitude of wave velocity. In equation form, this is

$v = \frac{\lambda}{T}=\lambda f$

Figure 6. An idealized surface water wave passes under a seagull that bobs up and down in simple harmonic motion. The wave has a wavelength λ, which is the distance between adjacent identical parts of the wave. The amplitude $A$ of the wave is the maximum displacement of the wave from the equilibrium position, which is indicated by the dotted line. In this example, the medium moves up and down, whereas the disturbance of the surface propagates parallel to the surface at a speed $v$ .

Transverse and Longitudinal Waves

A simple mechanical wave consists of a periodic disturbance that propagates from one place to another through a medium. In Figure 7(a), the wave propagates in the horizontal direction, whereas the medium is disturbed in the vertical direction. Such a wave is called a transverse wave. In a transverse wave, the wave may propagate in any direction, but the disturbance of the medium is perpendicular to the direction of propagation. In contrast, in a longitudinal wave the disturbance is parallel to the direction of propagation. Figure 7(b) shows an example of a longitudinal wave. The size of the disturbance is its amplitude $A$ and is completely independent of the speed of propagation $v$ .

Figure 7. (a) In a transverse wave, the medium oscillates perpendicular to the wave velocity. Here, the spring moves vertically up and down, while the wave propagates horizontally to the right. (b) In a longitudinal wave, the medium oscillates parallel to the propagation of the wave. In this case, the spring oscillates back and forth, while the wave propagates to the right.

Mathematics of Waves

Here we model a one-dimensional sinusoidal wave using a wave function. Consider the oscillation shown in Figure 8. Here we have a wave with amplitude $A$ that starts at point $a$ and moves with a velocity $v$ to the right. The wavelength is defined by $lambda$ and exactly completes one period ( $2\pi$ radians) of the wave. The wave propagates with one wavelength in one period and, therefore, moves with constant speed $v=\frac{\lambda}{T}$ . Since the amplitude is zero at $x=0$ and $x=\lambda$ we can model this with a sine function.

Figure 8. Snapshots of a transverse wave moving through a string under tension, beginning at time $t=T$ and taken at intervals of $\frac{1}{8}T$ . Colored dots are used to highlight points on the string. Points that are a wavelength apart in the x-direction are highlighted with the same color dots.

Recall that the sine function is a function of the angle $\theta$ (i.e., $y=sin(\theta)$ ), oscillating between +1 and -1, and repeating every $2\pi$ radians (Figure 9). However, the y-position of the medium, or the wave function, oscillates between $+A$ and $-A$ , and repeats every $\lambda$ .

Figure 9. A sine function oscillates between +1 and −1 every $2\pi$ radians.

To construct a model using a periodic function, consider the ratio of the angle and the position:

(1) $\begin{equation*} \begin{align*} \frac{\theta}{x} &= \frac{2\pi}{\lambda} \\ \theta &= 2\pi\frac{x}{\lambda} \end{align} \end{equation*}$

Therefore, we can model our wave as

$y(x) = Asin \left( \frac{2\pi}{\lambda}x \right)$

The wave on the string travels in the positive x-direction with a constant velocity $v$ , and moves a distance $vt$ in a time $t$ . Thus, we can write

(2) $\begin{equation*} \begin{align*} y(x,t) &= Asin \left( \frac{2\pi}{\lambda}(x-vt) \right) \\ &= Asin\left( \frac{2\pi}{\lambda}x-\frac{2\pi}{\lambda}vt \right) \end{align} \end{equation*}$

The value $k=\frac{2\pi}{\lambda}$ is defined as the wave number and has units of inverse meters $[m^-1]$ . Also recall that $\omega=\frac{2\pi}{T}$ and $v=\frac{\lambda}{T}$ and, therefore,

$\frac{2\pi}{\lambda}vt = \frac{2\pi}{\lambda} \left( \frac{\lambda}{T} \right)t = \frac{2\pi}{T}t = \omega t.$

This results in a wave function for a simple harmonic wave on a string:

$y(x,t) = Asin \left( kx \mp \omega t)$

The minus sign is for waves moving in the positive x-direction, and the plus sign for waves moving in the negative x-direction. The velocity $v$ of the wave is

$v=\frac{\lambda}{T}=\frac{\lambda}{T}\frac{2\pi}{2\pi}=\frac{\omega}{k}$

In analogy with the simple harmonic oscillation we can add a phase shift $\phi$ to allow for the fact that the mass may have initial conditions other than $x=+A$ and $v=0$ . Thus we obtain

$y(x,t) = Asin \left( kx \mp \omega t + \phi)$

Also note that

$y(x,t) = Acos \left( kx \mp \omega t + \phi')$

works equally well because it corresponds to a different phase shift $\phi' = \phi - \frac{\pi}{2}$

Velocity of the medium

We have seen that the wave speed is constant and represents the speed of the wave as it propagates through the medium:

The velocity $v$ of the wave:

$v=\frac{\lambda}{T}=\frac{\lambda}{T}\frac{2\pi}{2\pi}=\frac{\omega}{k}$

However, the speed of the particles that make up the medium is not constant since these oscillate around an equilibrium position as the wave propagates though the medium.

In the case of the transverse wave propagating in the x-direction, the particles oscillate up and down in the y-direction, perpendicular to the motion of the wave. The velocity of the medium, can be found by taking the partial derivative of the position equation with respect to time. The partial derivative is found by taking the derivative of the function. In the case of the partial derivative with respect to time $t$ , the position $x$ is treated as a constant since the objective is to find the transverse velocity at a point. We have

(3) $\begin{equation*} \begin{align*} y(x,t) &= Asin (kx-\omega t + \phi) \\ v_y (x,t) &= \frac{\partial y(x,t)}{\partial t} = \frac{\partial}{\partial t} Asin (kx-\omega t + \phi) \\ &= -A\omega cos(kx-\omega t + \phi) \\ &= v_{y max} cos(kx-\omega t + \phi) \end{align} \end{equation*}$

A similar function can be obtained for longitudinal waves.

Speed of Sound

In an ideal gas, like air, the speed of sound is given by $v=\sqrt \frac{\gamma R T_{K}}{M} = 331 m/s$ in air at $0^{\circ}C$ . $R=8.31 J/mol\times K$ (gas constant) and $T_K$ (absolute temperature in Kelvin). $M$ is the molar mass. $\gamma$ is the adiabatic index. For a derivation see [here].

From $v=f\lambda$ we can now calculate the wavelengths for sounds of different frequencies (Figure 10).

For example, at $20^{\circ} C \approx 293K$ we get $v= 331 m/s \times \sqrt \frac{T}{273K} = 331 \ times \sqrt \frac{293}{273} \approx 343 m/s$ .

This gives $\lambda = \frac{343}{f}$ .

$20Hz: \lambda \approx 17m$

$20,000Hz: \lambda \approx 1.7cm$

Figure 10. Because they travel at the same speed in a given medium, low-frequency sounds must have a greater wavelength than high-frequency sounds. Here, the lower-frequency sounds are emitted by the large speaker, called a woofer, whereas the higher-frequency sounds are emitted by the small speaker, called a tweeter. (credit: modification of work by Jane Whitney).

Energy and power in Waves

The amount of energy in a wave is related to its amplitude and its frequency. Large-amplitude earthquakes produce large ground displacements. Loud sounds have high-pressure amplitudes and come from larger-amplitude source vibrations than soft sounds. Consider the example of the seagull and the water wave earlier in the chapter (Figure 6). Work is done on the seagull by the wave as the seagull is moved up, changing its potential energy. The larger the amplitude, the higher the seagull is lifted by the wave and the larger the change in potential energy.

If the energy of each wavelength is considered to be a discrete packet of energy, a high-frequency wave will deliver more of these packets per unit time than a low-frequency wave. We will see that the average rate of energy transfer in mechanical waves is proportional to both the square of the amplitude and the square of the frequency. If two mechanical waves have equal amplitudes, but one wave has a frequency equal to twice the frequency of the other, the higher-frequency wave will have a rate of energy transfer a factor of four times as great as the rate of energy transfer of the lower-frequency wave.

One can derive (see [here]) that the total energy ( $E_{\lambda}$ ) associated with a wavelength is the sum of the potential energy ( $U_{\lambda}$ ) and the kinetic energy ( $K_{\lambda}$ ):

$E_{\lambda} = \frac{1}{4} \mu A^2 \omega^2 \lambda + \frac{1}{4} \mu A^2 \omega^2 \lambda = \frac{1}{2} \mu A^2 \omega^2 \lambda$

The time-averaged power (units of Watt; $W=\frac{J}{s}=\frac{kg \times m^2}{s^3}$ ) of a sinusoidal mechanical wave, which is the average rate of energy transfer associated with a wave as it passes a point, can be found by taking the total energy associated with the wave divided by the time it takes to transfer the energy. If the velocity of the sinusoidal wave is constant, the time for one wavelength to pass by a point is equal to the period of the wave, which is also constant. For a sinusoidal mechanical wave, the time-averaged power is therefore the energy associated with a wavelength divided by the period of the wave. The wavelength of the wave divided by the period is equal to the velocity of the wave,

$P_{avg} = \frac{E_{\lambda}}{T} = \frac{1}{2} \mu A^2 \omega^2 \frac{\lambda}{T} = \frac{1}{2} \mu A^2 \omega^2 v$

Note that this equation for the time-averaged power of a sinusoidal mechanical wave shows that the power is proportional to the square of the amplitude of the wave and to the square of the angular frequency of the wave. Recall that the angular frequency is equal to $\omega = 2 \pi f$ , so the power of a mechanical wave is equal to the square of the amplitude and the square of the frequency of the wave.

Waves intensity

The equations for the energy of the wave and the time-averaged power were derived for a sinusoidal wave on a string. In general, the energy of a mechanical wave and the power are proportional to the amplitude squared and to the angular frequency squared (and therefore the frequency squared).

Another important characteristic of waves is the intensity of the waves. Waves can also be concentrated or spread out. Waves from an earthquake, for example, spread out over a larger area as they move away from a source, so they do less damage the farther they get from the source. Changing the area the waves cover has important effects. These factors are included in the definition of intensity (I) as power per unit area ( $W/m^2$ ):

$I = \frac{P}{A}$

where $P$ is the power carried by the wave through area $A$ . This definition of intensity is valid for any energy in transit, including that carried by waves. Many waves are spherical waves that move out from a source as a sphere. For example, a sound speaker mounted on a post above the ground may produce sound waves that move away from the source as a spherical wave. In general, the farther you are from the speaker, the less intense the sound you hear. As a spherical wave moves out from a source, the surface area of the wave increases as the radius increases ( $A=4 \pi r^2$ ). The intensity for a spherical wave is therefore

$I = \frac{P}{4 \pi r^2}$

If there are no dissipative forces, the energy will remain constant as the spherical wave moves away from the source, but the intensity will decrease as the surface area increases.

References

Moebs W, Ling, SJ, Sanny J (2021) University Physics Volume 1. Web version January 19, 2021. Openstax, Houston. https://openstax.org/details/books/university-physics-volume-1. This book is licensed under Creative Commons Attribution License v4.0

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University Physics Volume 1 (chapter 15-17) (pdf)

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Sound waves

University Physics Volume 1 (chapter 15-17) (pdf)

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