Physics II: Chapter 2

Reference: Beginning Physics II

Chapter 2: SOUND



Sound Velocity, Rms Velocity, Wave-Front, Wave Power (Two Dimensional), Wave Power (Three Dimensional), Intensity, Plane Wave, Reflection, Refraction, Interference, Decibel Scale, Reverberation, Reverberation Time, Absorption Coefficient, Absorbing Power, Quality, Pitch, Beats, Doppler Shift, Shock Waves



For details on the following concepts, please consult Chapter 2.

The velocity of sound in air is,

The root-mean-square velocity of the gas molecules themselves is,

Waves in two and three dimensions have a wave-front. It is circular or spherical as shown below.

The wave-front isan imaginary line or surface drawn through the crest (or trough) of one of the ripples at a given instant of time. We are looking at the same phase of the disturbance at all different locations in the fluid. The circular or spherical shape of the wave-front means that the wave propagation of the disturbance is characteristic of the material through which the wave moves. The direction of propagation of the wave at any location is perpendicular to the wave front at that location.

For water ripples, the power transmitted through a unit length parallel to the wave-front is being diluted as the circular wave-front expands to larger circumference. Since the circumference of a ripple increases in proportion to its growing radius R, the power per unit wave-front length must decrease as 1/R.

Similarly, the energy and power of the wave, per unit area perpendicular to the direction of propagation of the wave fall off as 1/R2.

The power per unit area perpendicular to the direction of propagation is called the intensity, I. The intensity for a three-dimensional wave is given by,

I = P/A

A wave moving through space in which the wave-front is planar is called a plane wave, and is characterized by the fact that every point on the planar wave-front is in phase at the same time. A small window on the spherical wave front is almost planar if the dimensions are small compared to the distance from the source of the wave. The wave equation for such a wave is exactly the same as for the longitudinal waves in a long tube.

When sound wave-fronts hit a barrier, such as, the side of a mountain, part of the wave reflects and part is transmitted into the barrier. The part of the wave that is reflected has diminished amplitude but the same frequency and velocity as the original wave, and hence the same wavelength.

When a wave travels through a medium of varying densities (for example, layers of air at different temperatures) the velocity of different parts of the wave-front are different, and the direction of propagation of the wave changes as a consequence. This is called refraction.

Interference is the effect of having more than one wave passing a given point, and the possibility that the two waves will reinforce or weaken each other as a consequence of the phase difference between the waves.

To describe the range of sound intensities it is useful to create a logarithmic scale called the decibel scale (db), which gives a quantitative measure to “loudness”, which we label n, and define as:

n = 10 log (I/Io)

The persistence of a sound after its source has stopped, caused by multiple reflection of the sound within a closed space.

The reverberation time is defined as the time it takes for the intensity of a given steady sound to drop 60 db (or six orders of magnitude in intensity) from the time the sound source is shut off. Reverberation times depend on the total acoustic energy pervading the room, the surface areas of the absorbing materials and their absorption coefficients. A formula that gives good estimates of the reverberation time is given by:

tr = 0.16V / A

where tr is the reverberation time (s), V is the volume of the room (m3) and A is called the absorbing power of the room.

The absorption coefficient of a surface is defined as the fraction of sound energy that is absorbed at each reflection. Thus, an open window has an absorption coefficient of 1 since all the energy passes out of it and none reflects back in. Heavy curtains have a coefficient of about 0.5, and acoustic ceiling tiles have a coefficient of about 0.6.

The absorbing power A is just the sum of the products of the areas of all the absorbing surfaces (m2) and their respective absorption coefficients.

When a note on a musical instrument is played, the fundamental is typically accompanied by various overtones (harmonics, i.e., integer multiples of the fundamental) with differing intensity relative to that of the fundamental. The sound of harmonics is pleasing to the ear, and while the note is identified by the listener with the fundamental frequency, the same note from different instruments will sound differently as a consequence of the different harmonic content. These different sound recognitions by the human ear are called the quality of the note.

The pitch of a note is the human perception of the note as “high” or “low” and is closely related to the frequency but is not identical to it. The pitch involves human subjective sense of the sound. While a higher frequency will be perceived as a higher pitch, the same frequency will be perceived as having slightly different pitches when the intensity is changed. When the human ear hears a fundamental and harmonics it perceives the pitch as that of the fundamental.

If we have two frequencies that differ only by a few Hz we can indeed detect “interference” effects that oscillate in time slowly enough to be easily detectable. This variable amplitude corresponds to a maximal loudness in the sound, called a beat. The number of beats per second is just the difference of the two frequencies.

The Doppler shift is a change in pitch caused by motion of the source of a sound wave through the air (as in the example of the siren of an ambulance) or by the motion of the listener through the air. If the source is moving toward the listener, the sound waves are bunched up, and the listener would detect shorter wavelengths or higher frequencies. If the source is moving away from the listener, the sound waves are more separated, and the listener would detect longer wavelengths or lower frequencies.

For more general case, when the velocity of the listener is included,

When supersonic (faster than the speed of sound) motion occurs a compressional wave, due to the object cutting through the air, is emitted by the traveling body and forms what is called a shock wave. The shock wave moves at a specific angle relative to the direction of motion of the object through the air, and can sometimes be of sufficient intensity to cause a loud booming sound.

R/x is the ratio of the opposite side to the hypotenuse of a right triangle with angle  as shown. Then:

The direction of propagation of the shock wave is perpendicular to the wave-front and makes an angle (90° – ) to the direction of motion of the object. Shock waves accompany speeding bullets, and an example in a medium other than air is the bow wave of a speed boat in water.

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