Sound as a Longitudinal Wave

A sound begins with a vibration

At its core, a sound wave begins with a vibrating source. The source may be the cone of a loudspeaker, a guitar string, or your own vocal cords. The vibrations of the source nudge the particles of air next to it. Those particles nudge their neighbours in turn. So a chain of pushes and pulls ripples out through the surrounding medium.

Amplitude and frequency

Before we go deeper into sound waves, we need two basic properties of any wave. They are the amplitude and the frequency. Each one describes the wave independently of the other.

The amplitude is the largest displacement of the particles from their equilibrium positions. In a sound wave, a larger amplitude means three things:

• a greater pressure variation, \(\Delta P_{\text{max}}\), in the air;
• more energy carried by the wave;
• a louder sound heard by our ears.

Think of the amplitude as the strength of the vibration. When you turn up the volume of a speaker, you increase the amplitude of its sound wave.

The frequency is the number of complete oscillations in one second. It is measured in hertz (Hz). In a sound wave, the frequency decides the pitch that we hear:

• a higher frequency gives a higher pitch, like a whistle;
• a lower frequency gives a lower pitch, like a bass drum.

The amplitude and the frequency are independent properties. A sound can be loud and high, or quiet and low. The two can change separately.

Seven samples to hear

Now listen to seven samples, and notice what amplitude and frequency each do to the sound. First keep the frequency fixed at \(440\,\text{Hz}\), the musical note A4. Change only the amplitude, from quieter (\(0.5\)) to normal (\(1\)) to louder (\(2\)). The pitch stays the same, and only the loudness changes.

Next keep the amplitude fixed at \(1\) and change the frequency. The loudness stays the same, but the pitch shifts. \(220\,\text{Hz}\) (A3) sounds lower, \(440\,\text{Hz}\) (A4) is in the middle, and \(880\,\text{Hz}\) (A5) is higher. Last, combine both changes. Compare a loud, high note (\(880\,\text{Hz}\) at amplitude \(2\)) with a quiet, low note (\(220\,\text{Hz}\) at amplitude \(0.5\)).

The drawing in Fig 1.1 shows what you hear. A louder chip makes the wave taller but keeps its wavelength. A higher chip packs the crests closer but keeps their height. At \(343\,\text{m/s}\), the speed of sound that this lesson uses, \(440\,\text{Hz}\) has a wavelength of \(0.78\,\text{m}\). \(220\,\text{Hz}\) has \(1.56\,\text{m}\), and \(880\,\text{Hz}\) has \(0.39\,\text{m}\).

These demonstrations show clearly that amplitude and frequency are independent. Changing one does not change the other. The amplitude controls the loudness, and the frequency controls the pitch.

The wave equation for sound

Consider a sinusoidal sound wave travelling through a medium along \(x\). The displacement \(s\) of a particle of the medium depends on its position \(x\) and on the time \(t\). We can write it as

\[ s(x, t) = s_{\text{max}}\cos(kx - \omega t) \]

or, equally well, as \(s(x, t) = s_{\text{max}}\sin(kx - \omega t)\). The two differ only by where the clock starts. In these equations:

• \(s_{\text{max}}\) is the amplitude, the largest displacement;
• \(k = 2\pi/\lambda\) is the wave number, where \(\lambda\) is the wavelength;
• \(\omega = 2\pi f\) is the angular frequency, where \(f\) is the frequency in Hz;
• \(kx - \omega t\) is the phase of the wave.

The speed of the wave

Sound waves obey the same relation between speed, wavelength and frequency as waves on a string. The wave speed \(v\) can be written in several equal ways:

\[ v = \frac{\lambda}{T} = \lambda f = \frac{\omega}{k} \]

Here \(T = 1/f\) is the period, the time for one complete oscillation. These relations hold for every travelling wave. What matters for sound is this: the speed \(v\) depends on the properties of the medium. It does not depend on the frequency or the amplitude of the wave. Lesson 3 explores this.

Compressions and rarefactions

The disturbance moves outward as a longitudinal wave. The particles oscillate parallel to the direction in which the wave advances. As they do so, the medium alternates between two kinds of region. In some regions the particles are squeezed together; these are compressions. In others they are spread apart; these are rarefactions.

In a compression, the particles are crowded together. The local pressure and the local density are higher than average. In a rarefaction, the particles are pulled apart. The local pressure and the local density are lower than average. Fig 1.3 shows both.

Watch the painted column in Fig 1.3. It swings to and fro over its short dashed line and never leaves it. The compressions, however, march steadily down the tube. The particles here move as \(s = s_{\text{max}}\sin(\omega t - kx)\), an equation of the same type as the one above.

It is this rhythmic journey of compressions and rarefactions that travels through the air, or any other medium, and reaches our ears as sound. Whenever you hear a sound, you hear the oscillation of the particles of the medium. That oscillation shows itself as a variation of pressure at your ear. The Listen chip of Fig 1.3 plays a sound wave of \(440\,\text{Hz}\).

Now return to the dhol. The stick sets the skin of the drum vibrating. The skin pushes and pulls the air beside it, and compressions and rarefactions travel out to you at the speed of sound. Each layer of air only swings about its own place, so no wind blows. What reaches your ear, and your chest, is the travelling variation of pressure.